Operator Theory: Advances and Applications Vol. 175
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ZZZELUNKDXVHUFK Contents
Preface ...... vii
T.Ya. Azizov and L.I. Soukhotcheva LinearOperatorsinAlmostKreinSpaces ...... 1 J. Behrndt, A. Luger and C. Trunk Generalized Resolvents of a Class of Symmetric Operators inKreinSpaces ...... 13 J. Behrndt, H. Neidhardt and J. Rehberg Block Operator Matrices, Optical Potentials, Trace Class PerturbationsandScattering ...... 33 V. Derkach, S. Hassi and H. de Snoo Asymptotic Expansions of Generalized Nevanlinna Functions andtheirSpectralProperties ...... 51 A. Fleige A Necessary Aspect of the Generalized Beals Condition for the Riesz Basis Property of Indefinite Sturm-Liouville Problems ...... 89 K.-H. F¨orster and B. Nagy On Reducible Nonmonic Matrix Polynomials with General andNonnegativeCoefficients ...... 95 S. Hassi, H. de Snoo and H. Winkler On Exceptional Extensions Close to the Generalized Friedrichs ExtensionofSymmetricOperators ...... 111 P. Jonas and H. Langer On the Spectrum of the Self-adjoint Extensions of a Nonnegative LinearRelationofDefectOneina KreinSpace ...... 121 M. Kaltenb¨ack and H. Woracek Canonical Differential Equations of Hilbert-Schmidt Type ...... 159 I. Karabash and A. Kostenko Spectral Analysis of Differential Operators with Indefinite Weightsanda LocalPointInteraction ...... 169 vi Contents
C. Mehl and C. Trunk Normal Matrices in Degenerate Indefinite Inner Product Spaces ...... 193 V. Pivovarchik SymmetricHermite-BiehlerPolynomialswithDefect ...... 211 L. Rodman A Note on Indefinite Douglas’ Lemma ...... 225 A. Sandovici Some Basic Properties of Polynomials in a Linear Relation inLinearSpaces ...... 231 Preface
This volume contains papers written by the participants of the 4th Workshop on Operator Theory in Krein Spaces and Applications, which was held at the Technische Universit¨at Berlin, Germany, December 17 to 19, 2004. The workshop covered topics from spectral, perturbation and extension the- ory of linear operators and relations in inner product spaces. They included spec- tral analysis of differential operators, the theory of generalized Nevanlinna func- tions and related classes of functions, spectral theory of matrix polynomials and problems from scattering theory. All these topics are reflected in the present vol- ume. The workshop was attended by 58 participants from 12 countries. It is a pleasure to acknowledge the substantial financial support received from the – Deutsche Forschungsgemeinschaft (DFG), – DFG-Forschungszentrum MATHEON “Mathematik f¨ur Schl¨ussel- technologien”, – Institute of Mathematics of the Technische Universit¨at Berlin. We would also like to thank Petra Grimberger for her great help. Last but not least, special thanks are due to Jussi Behrndt and Christian Mehl for their excellent work in the organisation of the workshop and the preparation of this volume. Without their assistance the workshop might not have taken place.
The Editors Operator Theory: Advances and Applications, Vol. 175, 1–11 c 2007 Birkh¨auser Verlag Basel/Switzerland
Linear Operators in Almost Krein Spaces
Tomas Ya. Azizov and Lioudmila I. Soukhotcheva
Abstract. The aim of this paper is to study the completeness and basicity problems for selfadjoint operators of the class K(H)inalmostKreinspaces and prove criteria for the basicity and completeness of root vectors of linear pencils. Mathematics Subject Classification (2000). Primary 47B50; Secondary 46C50. Keywords. Krein space, operator pencil, completeness and basicity problem.
1. Introduction Let H be a Hilbert space, let A and B be compact operators and let A be addi- tionally a positive operator. Consider the linear operator pencil L(λ)=A−1 − λ(I + B). (1)
Such a pencil appears, for instance, if the following spectral problem in L2(0,π) is considered: ⎧ ⎨ d2f π − + q(t)f(t)=λ(f(t)+ K(t, s)f(s)ds) dt2 ⎩ 0 (2) f(0) = f(π)=0. Assume q is a continuous real function with q(t) > −1, the kernel K(t, s)issym- metric and continuous on [0,π]×[0,π]. It remains to define A and B in the following way: d2f A−1f = − + qf, dt2 −1 dom A = f ∈ L2(0,π) | f,f absolutely continuous on (0,π), d2f − + qf ∈ L2(0,π),f(0) = f(π)=0 dt2
This research is supported partially by the grant RFBR 05-01-00203. 2 T.Ya. Azizov and L.I. Soukhotcheva
π (Bf)(t)= K(t, s)f(s)ds. 0 The completeness and basicity problems for the pencil (1) and for the operator
H = A(I + B)(3) are closely related. The completeness problem for operators (3) under the assump- tion λ = 0 is not an eigenvalue of H (0 ∈/ σp(H)) was considered for the first time by M.V. Keldysh (see, for instance, [6, Theorem 8.1]). The case when B is also selfadjoint and ker H = {0} was studied by I.Ts. Gokhberg and M.G. Krein in [6, p. 322]. There it was shown that there is a Riesz basis in H which consists of root vectors of H. The proof is based on the selfadjointness of the operator H with respect to the indefinite inner product
[·, ·]=((I + B)·, ·), (4) which turns {H, [·, ·]} into a Pontryagin space. General criteria for the completeness and basicity of root vectors of selfadjoint operators in Pontryagin spaces can be found in [2], and, more generally, in [3, Theorem IV.2.12].
If ker(I + B) = 0, the inner product (4) is degenerate and, due to the com- pactness of the operator B,wehave (a) the isotropic part
H0 := {x ∈H|[x, y]=0, for all y ∈H}
of H is finite-dimensional, and (b) the factor-space H = H/H0 is a Pontryagin space. Indefinite inner product spaces H with the properties (a) and (b) are called almost Pontryagin spaces. The operator H in (3) is selfadjoint in the degenerate almost Pontryagin space {H, [·, ·]}. First results about the completeness and ba- sicity problems for compact selfadjoint operators in almost Pontryagin space were obtained in [1]. Recently, spectral properties of operators acting in almost Pontryagin and almost Krein spaces (for a definition of almost Krein spaces we refer to Definition 1 below) and their applications were studied in, for example, [4], [5], [7]–[13].1 The main aim of this paper is to study the completeness and basicity problems for selfadjoint operators of the class K(H) in almost Krein spaces (a definition of the class K(H) see below on p. 3). Moreover, we will prove criteria for the basicity and completeness of root vectors of the pencil (1).
1The authors thank Chr. Mehl (TU Berlin) for his help with the bibliography. Linear Operators in Almost Krein Spaces 3
2. Main definitions We shortly recall some definitions and notions related to Krein spaces. For more details we refer to [3]. A linear space K equipped with an indefinite inner product [·, ·] is called a Krein space if it admits a canonical decomposition
K = K+[+]K−, where K+ is [·, ·]-orthogonal to K−,andK± is a Hilbert space with respect to the scalar product ± [·, ·], respectively.
Definition 1. AspaceK with an indefinite inner product [·, ·] is called an almost Krein space if its isotropic part K0 is finite-dimensional (we do not exclude the case K0 = {0}) and the factor-space
K = K/K0, [ˆx, yˆ]=[x, y], x,ˆ yˆ ∈ Kˆ,x∈ x,ˆ y ∈ y,ˆ (5) is a Krein space with respect to the naturally reduced indefinite inner product [·, ·].
Let us note that each almost Pontryagin space is an almost Krein space. As in the Krein space case we say that a nonnegative/nonpositive subspace ± L± belongs to the class h , if it is represented as a sum of a finite-dimensional neutral and a uniformly positive/negative subspace.
Remark 2. From the definitions it follows immediately that nonnegative/nonpositi- ± ve subspace L± in an almost Pontryagin space belongs to the class h .
All operators below are considered to be linear, everywhere defined and bounded. An operator A in an almost Krein space is called selfadjoint, if [Ax, y]= [x, Ay] for all x, y ∈K. By definition, a selfadjoint operator A belongs to the class H,ifithasa maximal nonpositive and a maximal nonnegative invariant subspaces L± and each such a subspace belongs to h±, respectively. We say that a selfadjoint operator A belongs to K(H), if there is a selfadjoint operator B ∈ H, which commutes with A.
Remark 3. Let K be an almost Pontryagin space. From Remark 2 and the definition of the class H it follows that I ∈ H. Hence each selfadjoint operator in an almost Pontryagin space belongs to the class K(H). Moreover, from Theorem 4 below we have that every selfadjoint operator in an almost Pontryagin space belongs to the class H. 4 T.Ya. Azizov and L.I. Soukhotcheva
3. Invariant dual pairs
{L+, L−} is said to be a dual pair, if L± is a nonnegative/nonpositive subspace and [x+,x−] = 0 for all x± ∈L±. We prove an analog of a known result about invariant subspaces of operators of the class K(H) in a Krein space (see [3, § III.5]) for the case of almost Krein spaces. Theorem 4. Let A ∈ K(H) be a selfadjoint operator in an almost Krein space K and let B be a selfadjoint operator in H which commutes with A. Assume {L+, L−} is an A-invariant dual pair (it is not excluded that L+ = L− = {0}). + − ± Then there exists an A-invariant maximal dual pair {L , L } such that L± ⊂L . If {L+, L−} is also B-invariant, then we can choose a maximal dual pair which is simultaneously A-andB-invariant. In particular, there exists an A-invariant maximal dual pair {L+, L−} such that L± ∈ h±. Proof. Since the isotropic part K0 of K is a part of each maximal semidefinite 0 0 subspace and {L+ + K , L− + K } is A-invariant, without loss of generality we 0 assume that K ⊂L+ ∩L−. Consider the factor-space (5). Let us note that the isotropic part K0 of K is invariant under all selfadjoint operators in K, in particular, it is A-andB- invariant. Therefore the selfadjoint operators A and B in K, generated by A and B, are well defined. Since AB = BA and B ∈ H,wehaveA ∈ K(H). 0 Denote L± = L±/K . The dual pair {L+, L−} is A-invariant. By [3, Corol- lary III.5.13] and [3, Theorem III.1.13] there is an A-invariant maximal dual pair {L+, L−} in K.Then{L+, L−},where L± = {x ∈K|xˆ ∈ L±}, is a desired A-invariant dual pair. If {L+, L−} is an A-andB-invariant dual pair then by [3, Theorem III.5.12], [3, Theorem III.1.13] and the same arguments as above there is an A-andB- invariant maximal dual pair which is an extension of {L+, L−}. In particular, if L+ = L− = {0}, then, by the above statement, there exists an A-andB-invariant maximal dual pair {L+, L−}. B ∈ H implies L± ∈ h±. From Definition 1 it follows that a space K with an indefinite inner product [·, ·] is an almost Krein space if and only if it admits a decomposition in a direct sum 0 K = K + K1, (6) 0 with finite-dimensional isotropic part K and a subspace K1 which is a Krein space with respect to [·, ·]. + − ± 0 ± Let {L , L } be an A-invariant maximal dual pair in K.ThenL = K +L1 , ± ± + − where L1 = L ∩K1. The dual pair {L1 , L1 } is maximal in the Krein space K1. 0 + − 0 0 Let L1 = L1 ∩L1 and let M1 ⊂K1 be a subspace skew-linked with L1. Then: 0 0 + − 0 K = K + L1 + L1 + L1 + M1. (7) Linear Operators in Almost Krein Spaces 5
Introduce in K a Hilbert scalar product (·, ·) such that all subspaces in (7) are ± ± orthogonal to each others with respect to (·, ·)and(·, ·)|L1 = ± [·, ·]|L1 . Then with respect to (7) the operator A has a triangular representation ⎡ ⎤ A00 A01 A02 A03 A04 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 A11 A12 A13 A14⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ A = ⎢ 00A22 0 A24⎥ , (8) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 000A33 A34⎥ ⎣ ⎦
0000A44 ∗ where A0k, k =0, 1, 2, 3, 4, are bounded operators, A11 and A44 are similar, and A22 and A33 are selfadjoint with respect to the introduced scalar product. We omit information about other components of (8) since it is not essential in this paper. Corollary 5. The spectrum of a selfadjoint operator A ∈ K(H) in an almost Krein space is real, except for at most finite number of non-real normal eigenvalues. There are not more than a finite number of real eigenvalues λ of A with degenerate ker(A − λ). In particular, the set of eigenvalues with nontrivial Jordan chains is finite. 4 Proof. The triangular representation (8) implies σ(A)= k=0 σ(Akk). Since A22 and A33 are Hilbert space selfadjoint operators, the non-real spectrum of A coin- cides with the non-real spectra of A00, A11 and A44. Hence the statement about the non-real spectrum of A follows immediately from the finite dimensionality of 0 0 0 K + L1 + M1. Assume that A has an infinite number of real eigenvalues λ such that ker(A− λ) is degenerate. Let L be the closed linear span of the isotropic parts of these kernels. Let B ∈ H be an operator which commutes with A.ThenBL⊂L.This contradicts the assumption B ∈ H. The last statement of the corollary follows from the fact that kernels with non-trivial Jordan chains are degenerate.
4. Criteria for the completeness and basicity In the sequel we will need the following lemma. Lemma 6. Assume K is an almost Krein space and A ∈ K(H) is a selfadjoint operator in K with σ(A)={0}.ThenA is a nilpotent operator of finite rank. Proof. We consider the decomposition (7) of K and the corresponding decompo- sition (8) of A.Notethatσ(A)={0} implies
σ(A00)=σ(A11)=σ(A22)=σ(A33)=σ(A44)={0} and the selfadjointness of A22 and A33 implies A22 =0andA33 =0. 6 T.Ya. Azizov and L.I. Soukhotcheva
Below we use the following notations:
– Lλ(A) is the root subspace of A related to the eigenvalue λ : λ ∈ σp(A); – E(A) is the closed linear span (c.l.s. ) of the root vectors of A: E(A)= c.l.s. {Lλ(A) | λ ∈ σp(A)};
– E0(A)=c.l.s. {ker(A − λ) | λ ∈ σp(A)};
– ER\0(A)=c.l.s. {Lλ(A) | λ ∈ R \{0}}.
A vector system {ek} is called almost orthonormal if it is a union of two { } { }∪{ } { } systems ek = ek ek ,where ek is orthonormal: [ek,ek] =0, [ek,ej]= { } δkj sign [ek,ek], and ek is a finite system with [ek,ej ]=0. Corollary 7. Let A ∈ K(H) be a compact selfadjoint operator in a Krein space K. Then E(A)=K if and only if L0(A) ∩ ER\0(A)={0}. Moreover, if E(A)=K, there exists in K an almost orthonormal Riesz basis composed of root vectors of A.
Proof. Indeed, let L = L0(A)∩ER\0(A). Then L is orthogonal to all root subspaces of A.IfE(A)=K, we have the orthogonality of L to the whole Krein space which is non-degenerate. Hence L = {0}. Assume L = {0}.LetB ∈ H and AB = BA.SinceER\0(A)isA-andB- 0 invariant, the isotropic part ER\0(A) of ER\0(A) is also A-andB-invariant. Hence, 0 taking into account that B ∈ H and ER\0(A) is B-invariant neutral subspace, we 0 obtain from Theorem 4 that dim ER\0(A) < ∞. Since the root subspaces Lλ(A), 0 0 λ = 0 are non-degenerate, we have ER\0(A) ⊂L0(A), and hence ER\0(A) ⊂L, 0 that is, ER\0(A) = {0}. Therefore ER\0(A) is a Krein space. The spectrum of the [⊥] restriction of A to the orthogonal complement ER\0(A) to ER\0(A) consists of [⊥] a unique point λ = 0. Hence, it follows from Lemma 6 that L0(A)=ER\0(A) . The latter implies E(A)=K. The existence of an almost orthonormal basisisprovedin[3,Thm.IV.2.12]. Below we give necessary and sufficient conditions for the basicity and com- pleteness of the set of root vectors of a compact selfadjoint operator in an almost Krein space. We show that in almost Krein spaces, in contrast to the Pontryagin or Krein space case, the completeness of the root vector system is not sufficient for its basicity. Theorem 8. Let A ∈ K(H) be a compact selfadjoint operator in an almost Krein space K and E(A)=K. Then the following assertions are equivalent: 0 (i) K ∩L0(A) ∩ ER\0(A)={0}; (ii) L0(A) ∩ ER\0(A)={0}; (iii) there exists in K an almost orthonormal Riesz basis consisting of root vectors of A. Proof. Corollary 5 implies that the non-real spectrum of A and the real nonzero eigenvalues λ ∈ σp(A) with degenerate kernels ker(A − λ) is a finite subset of the normal eigenvalues. Let P be the Riesz projector related to this part of σ(A). Since Linear Operators in Almost Krein Spaces 7 dim P K < ∞ and P K is both A-andB-invariant, where B ∈ H commutes with A,wehavethatA|(I − P )K satisfies all conditions of the theorem and the basicity property for the root systems of A and A|(I − P )K is equivalent. Moreover, 0 0 K ∩L0(A) ∩ ER\0(A)=(I − P )K ∩L0(A|(I − P )K) ∩ ER\0(A|(I − P )K) and
L0(A) ∩ ER\0(A)={0}⇐⇒L0(A|(I − P )K) ∩ ER\0(A|(I − P )K)={0}. Hence without loss of generality we can assume P = 0, that is, both the set of the non-real eigenvalues and the set of nonzero real eigenvalues with degenerate kernels are empty. 0 (i) ⇒ (ii). Let K ∩L0(A) ∩ ER\0(A)={0}.ThenER\0(A) is a nondegenerate subspace. Indeed, let us suppose the opposite, i.e., that this subspace is degener- 0 0 ate and its isotropic part ER\0(A) is nonzero. The subspace ER\0(A) is finite- dimensional and A-invariant. Therefore, there exists an eigenvalue λ0 and a cor- 0 responding eigenvector x0 ∈ ER\0(A) . Since all kernels ker(A − λ)withλ =0are nondegenerate, we obtain λ0 =0.So,x0 ∈ ker A, it is orthogonal to L0(A)and 0 thus is orthogonal to E(A). Hence, x0 ∈K ∩L0(A) ∩ ER\0(A), and x0 =0.This contradicts the assumption x =0.Wehavethat ER\0(A) is nondegenerate and therefore L0(A) ∩ ER\0(A)={0}. (ii) ⇒ (i) is trivial. (ii) ⇒ (iii). According to the assumption all eigenvalues of A are real and all ker(A − λ), λ = 0, are nondegenerate. Hence the subspace ER\0(A) is nondegen- erate. Let us prove that this subspace is a Krein space. Really, it is B-invariant, where B ∈ H is an operator commuting with A. In each subspace ker(A − λ) {L+ L−} {L+ L−} the operator B has an invariant maximal dual pair λ , λ ,and , with L± {L± | ∈ } E =c.l.s. λ λ σ(A) ,isaB-invariant maximal dual pair in R\0(A). The assumption B ∈ H implies L± ∈ h±. Hence L+ + L− is an almost Krein space. By construction, the defect of this subspace in ER\0(A) is finite. Hence ER\0(A)is also an almost Krein space and, by assumption, it is nondegenerate. So, ER\0(A) is a Krein space. Therefore the orthogonal decomposition [⊥] K = ER\0(A)[+]ER\0(A) holds. Since B ∈ H has in ER\0(A) an invariant maximal dual pair, we have [⊥] B|ER\0(A) ∈ H and B|ER\0(A) ∈ H. Hence, [⊥] A|ER\0(A) ∈ K(H)andA|ER\0(A) ∈ K(H). [⊥] [⊥] By Lemma 6, the subspace ER\0(A) ⊃L0(A) coincides with L0(A|ER\0(A) ) ⊂ [⊥] L0(A). This implies ER\0(A) )=L0(A)and
K = ER\0(A)[+]L0(A). (9)
Since ER\0(A) is a Krein space and A|ER\0(A) ∈ K(H) it follows from [3, Theorem IV.2.12] that there exists in ER\0(A) an orthonormal Riesz basis constructed from 8 T.Ya. Azizov and L.I. Soukhotcheva eigenvectors of A. If we add to this basis an arbitrary almost orthonormal Riesz basis composed of vectors of L0(A), we obtain desired basis in K. (iii) ⇒ (ii). Assume there exists an almost orthonormal Riesz basis in K composed of root vectors of A.Let{ek} be the part of this basis contained in ER\0(A). Using the same arguments as above we can assume, without loss of generality, that all ∈L ∩ E vectors ek are definite and orthogonal to each other. If x0 0(A) R\0(A), then x0 ∈ c.l.s. {ek} : x0 = αkek. On the other hand it is orthogonal to all these vectors. Hence, αk =[x0,ek]sign[ek,ek] = 0, that is, x0 =0. Remark 9. In the proof of Theorem 8 we have used the assumption that the root vector system of A is complete only in the proof of the implication (i) ⇒ (ii). 0 We can rewrite the assumptions (i) and (ii) as 0 ∈/ σp(A|K ∩ ER\0)and 0 ∈/ σp(A|L0(A) ∩ ER\0), respectively. Theorem 10. Let A ∈ K(H) be a compact selfadjoint operator in an almost Krein space K.Then 0 {E(A)=K} ⇐⇒ {L0(A) ∩ ER\0(A) ⊂K }. (10) 0 Proof. {E(A)=K} =⇒{L0(A) ∩ ER\0(A) ⊂K }.
Let x0 ∈L0(A) ∩ ER\0(A). Since L0(A) is orthogonal to c.l.s. {Lλ(A) | λ =0 },the 0 vector x0 is orthogonal to E(A), that is, x0 ∈K . 0 {E(A)=K} ⇐= {L0(A) ∩ ER\0(A) ⊂K }. With respect to decomposition (6) the operator A admits the following matrix representation: ⎡ ⎤ A00 A01 A = ⎣ ⎦ . (11) 0 A11 Consider also the operator A induced by the operator A in the factor space K = 0 0 K/K . It follows from L0(A) ∩ ER\0(A) ⊂K that L0(A) ∩ ER\0(A)={0}.By Theorem 8, there is in K a Riesz basis composed of root vectors of A. Since the operators A and A11 are similar, there is in K1 a Riesz basis composed of root 0 vectors of A11. Hence, E(A11)=K1. It remains to use that E(A)=K [+]E(A11)= K (see (11)).
5. Keldysh type operators Let H be a Hilbert space with the scalar product (·, ·), let A and B be compact selfadjoint operators acting in this space and A>0. Consider an operator H as in (3) and introduce in H an inner product [·, ·]givenby [·, ·]=((I + B)·, ·). Denote K = {H, [·, ·]}.ThenK is an almost Krein space, moreover, K is an almost Pontryagin space. The operator H is selfadjoint in K. Thus, we can apply to H Theorems 8 and 10. Linear Operators in Almost Krein Spaces 9
Theorem 11. Let an operator H as in (3) satisfy the above-mentioned properties. Then (a) E(H)=K; (b) There is an almost orthonormal Riesz basis in the almost Krein space K composed of root vectors of H if and only if one of the following equivalent assumptions holds. (b1) dim ker A(I + B)=dimker(I + B)A; (b2) ker(I + B) ⊂ ran A. Proof. (a) The isotropic part K0 of the almost Krein space K coincides with ker(I+ B). Since A>0, we have L0(H)=kerH =ker(I +B). Hence, L0(H)∩ER\0(H) ⊂ ker(I + B)=K0.Now(a) follows directly from Theorem 10. (b) Let us note that the assumptions (b1) and (b2) are equivalent since the operator A has a trivial kernel. Suppose that there exists an almost orthonormal Riesz basis in K composed of root vectors of H. By Theorem 8(ii) we have
K =kerH[+]ER\0(H). (12)
Since the operator H is diagonal with respect to (12) and the operator H|ER\0(H) is a Pontryagin space selfadjoint, H|ER\0(H) is similar to its Hilbert space adjoint. Hence H = A(I + B) and its Hilbert space adjoint H∗ =(I + B)A are similar too. This implies (b1). Assume (b2). Consider the linear pencil L(λ)=A−1 − λ(I + B). (13) Since the set of root vectors of this pencil and of the operator H are the same, it is sufficient to check that there is in K an almost orthonormal Riesz basis composed of root vectors of the pencil (13). Consider the orthogonal decomposition of H: H =ker(I + B) ⊕ ran (I + B). (14) Let us rewrite the operator A−1 and the pencil (13) in the matrix form with respect to the decomposition (14): ⎡ ⎤ C00 C01 A−1 = ⎣ ⎦ , C10 C11 ⎡ ⎤ C00 C01 L(λ)=⎣ ⎦ . (15) C10 C11 − λ(I + B11)
From (b2) it follows that C00 is a finite-dimensional operator and positive with respect to the Hilbert space scalar product, that C01 and C10 are bounded operators of finite ranks and that the operator C11 is positive with respect to the x0 Hilbert space scalar product and has a compact inverse. Hence, x = is an x1 10 T.Ya. Azizov and L.I. Soukhotcheva
−1 eigenvector of (15), related to λ = 0, if and only if, x0 = −C00 C01x1,wherex1 is an eigenvector of the pencil −1 L1(λ)=C11 − C10C00 C01 − λ(I + B11), (16) corresponding to the same eigenvalue. As in the proof of Theorem 8 we assume without loss of generality that the root subspaces and kernels corresponding to −1 the same eigenvalue coincides. Since C11 − C10C00 C01 is a Hilbert space positive operator in ran (I + B), its inverse is compact, and ker(I + B11)={0},thesetsof eigenvectors of (16) coincide with the sets of eigenvectors of the operator −1 −1 H1 =(C11 − C10C00 C01) (I + B11). (17)
Consider ran (I + B) as a Pontryagin space K1 with indefinite inner product [·, ·]=((I + B11)·, ·). Because the compact operator H1 is positive in K1,the assumption (ii) of Theorem 8 is fulfilled. Hence, there is an almost orthonormal Riesz basis in K1 composed of eigenvectors of H1. Taking into account the relation between eigenvectors of H1 and H and the equality ker(I + B)=kerH,weobtain the existence of an almost orthonormal Riesz basis in K composed of eigenvectors vectors of H. The latter is true if there are no associated vectors. In the general case, there is an almost orthonormal Riesz basis in K composed of root vectors of H.
References [1] Azizov T.Ya. On completely continuous operators that are selfadjoint with respect to a degenerate indefinite metric, Matem. issled., 7 (1972), 4, 237–240. (Russ.) [2] Azizov T.Ya., Iokhvidov I.S. A criterion of the completeness and basicity of root vectors of a completely continuous J-selfadjoint operator in a Pontryagin space Πκ. Matem. issled., 6 (1971), 1, 158–161. (Russ.) [3] T.Ya. Azizov, I.S. Iokhvidov, Foundation of the theory of linear operators in spaces with an indefinite metric, Nauka, Moscow, 1986 (Russ.); English transl.: Linear op- erators in spaces with an indefinite metric, Wiley, New York, 1989. [4] P. Binding and R. Hryniv. Full and partial range completeness. Oper. Theory Adv. Appl. 130:121–133, 2002. [5] Vladimir Bolotnikov, Chi-Kwong Li, Patrick Meade, Christian Mehl, Leiba Rodman. Shells of matrices in indefinite inner product spaces. Electron. J. Linear Algebra, 9: 67–92, 2002. [6] I.Ts. Gokhberg, M.G. Krein. Introduction to the theory of linear non-selfadjoint operators in a Hilbert space. Nauka, Moscow, 1965 (Russian). [7] M. Kaltenb¨ack and H. Woracek. Selfadjoint extensions of symmetric operators in degenerated inner product spaces. Integral Equations Operator Theory, 28 (1997), 289–320. [8] P. Lancaster, A.S. Markus, and P. Zizler. The order of neutrality for linear operators on inner product spaces. LAA 259 (1997), 25–29. Linear Operators in Almost Krein Spaces 11
[9] H. Langer, R. Mennicken, and C. Tretter. A self-adjoint linear pencil Q − λP of ordinary differential operators. Methods Funct. anal. Topology, 2 (1996), 38–54. [10] A. Luger. A factorization of regular generalized Nevanlinna functions.IntegralEqua- tions Operator Theory 43: 326–345, 2002. [11] Christian Mehl, Leiba Rodman. Symmetric matrices with respect to sesquilinear forms. Linear Algebra Appl., 349: 55–75, 2002. [12] Christian Mehl, Andre C.M. Ran, Leiba Rodman. Semidefinite invariant subspaces: degenerate inner products, to appear in Oper. Theory Adv. Appl. [13] H. Woracek. Resolvent matrices in degenerate inner product spaces.Math.Nachr. 213 (2000), 155–175.
Tomas Ya. Azizov and Lioudmila I. Soukhotcheva Department of Mathematics Voronezh State University Universitetskaya pl., 1 394022, Voronezh, Russia e-mail: [email protected] e-mail: [email protected] Operator Theory: Advances and Applications, Vol. 175, 13–32 c 2007 Birkh¨auser Verlag Basel/Switzerland
Generalized Resolvents of a Class of Symmetric Operators in Krein Spaces
Jussi Behrndt, Annemarie Luger and Carsten Trunk
Abstract. Let A be a closed symmetric operator of defect one in a Krein space K and assume that A possesses a self-adjoint extension in K which locally has the same spectral properties as a definitizable operator. We show that the Krein-Naimark formula establishes a bijective correspondence between the compressed resolvents of locally definitizable self-adjoint extensions A of A acting in Krein spaces K×H and a special subclass of meromorphic functions. Mathematics Subject Classification (2000). Primary: 47B50; Secondary: 47B25. Keywords. Generalized resolvents, Krein-Naimark formula, self-adjoint exten- sions, locally definitizable operators, locally definitizable functions, boundary value spaces, Weyl functions, Krein spaces.
1. Introduction Let A be a densely defined closed symmetric operator with defect one in a Hilbert ∗ space K and let {C, Γ0, Γ1} be a boundary value space for the adjoint operator A . ∗ Let A0 be the self-adjoint extension A ker Γ0 of A in K and denote the γ-field and Weyl function corresponding to the boundary value space {C, Γ0, Γ1} by γ and M, respectively. Here M is a scalar Nevanlinna function, that is, it maps the upper half-plane C+ holomorphically into C+ ∪ R and is symmetric with respect to the real axis. It is well known that in this case the Krein-Naimark formula −1 −1 −1 ∗ PK(A − λ) |K =(A0 − λ) − γ(λ) M(λ)+τ(λ) γ(λ) (1.1) establishes a bijective correspondence between the class of Nevanlinna functions τ (including the constant ∞) and the compressed resolvents of self-adjoint extensions A of A in K×H,whereH is a Hilbert space, cf. [22, 32]. The compressed resolvent on the left-hand side of (1.1) is said to be a generalized resolvent of A.Wenote that if A has equal deficiency indices > 1 the generalized resolvents of A can still be described with formula (1.1), where the parameters τ are so-called Nevanlinna families, cf. [11, 23, 30, 31]. 14 J. Behrndt, A. Luger and C. Trunk
Various generalizations of the Krein-Naimark formula in an indefinite setting have been proved in the last decades. The case that A is a symmetric operator in a Pontryagin space K and H is a Hilbert space was investigated by M.G. Krein and H. Langer in [24]. Later V. Derkach considered both K and H to be Pontryagin or even Krein spaces, cf. [10]. In the general situation of Krein spaces K and H one obtains a correspondence between locally holomorphic relation-valued functions τ and self-adjoint extensions A of A with a non-empty resolvent set. Under additional assumptions other variants of (1.1) were proved in [6, 7, 8, 9, 10, 14, 27]. If, e.g., H is a Pontryagin space, then the parameters τ belong to the class of Nκ-families, a class of relation-valued functions which includes the generalized Nevanlinna functions. If H is a Krein space and the hermitian forms [A·, ·]and [A·, ·] both have finitely many negative squares, then τ belongs to a special subclass of the definitizable functions, cf. [7, 19]. It is the aim of this paper to prove a new variant of formula (1.1). Here we allow both K and H to be Krein spaces and we assume that A is of defect one and possesses a self-adjoint extension A0 in K which locally has the same spectral properties as a definitizable operator or relation, cf. [20, 28]. Under the assumption that A is also locally definitizable and that its sign types coincide “in essence” (i.e., with the exception of a discrete set, see Definition 2.6) with the sign types of A0 we prove in Theorem 3.2 that there exists a so-called locally definitizable function τ such that (1.1) holds. The proof is based on a coupling method developed in [11, §5] and a recent perturbation result from [4]. One of the main difficulties here is to show that the symmetric relation A∩H2 possesses a self-adjoint extension in the Krein space H with a non-empty resolvent set and to choose a boundary value space for the adjoint of A ∩H2 in H such that (1.1) holds with the corresponding Weyl function τ. In connection with a class of abstract λ-dependent boundary value problems the converse direction was already proved in [3], i.e., for a given locally definitizable function τ a self-adjoint extension A of A in K×Hsuch that (1.1) holds was constructed. The paper is organized as follows: In Section 2 we recall the definitions and basic properties of locally defini- tizable self-adjoint operators and relations and the class of locally definitizable functions introduced and studied by P. Jonas, see, e.g., [20, 21]. The notion of d- compatibility of sign types of locally definitizable relations and functions is defined in the end of Section 2.3. In the beginning of Section 3 we recall some basics on boundary value spaces and associated Weyl functions. Section 3.2 contains our main result. We prove in Theorem 3.2 that formula (1.1) establishes a bijective correspondence between an appropriate subclass of the locally definitizable functions and the compressed resolvents of locally definitizable K-minimal self-adjoint exit space extensions A of A in a Krein space K×Hwith spectral sign types d-compatible to those of A0. Finally, in the end of Section 3.2, we formulate a variant of the Krein-Naimark formula for self-adjoint extensions A0 and A of A in K and K×H, respectively, which locally have the same spectral Generalized Resolvents of Symmetric Operators in Krein Spaces 15 properties as self-adjoint operators or relations in Pontryagin spaces and functions τ from the local generalized Nevanlinna class.
2. Locally definitizable self-adjoint relations and locally definitizable functions 2.1. Notations and definitions Let (K, [·, ·]) be a separable Krein space with a corresponding fundamental sym- metry J. The linear space of bounded linear operators defined on a Krein space K1 with values in a Krein space K2 is denoted by L(K1, K2). If K := K1 = K2 we simply write L(K). We study linear relations in K, that is, linear subspaces of K2. The set of all closed linear relations in K is denoted by C(K). Linear operators in K are viewed as linear relations via their graphs. For the usual definitions of the linear operations with relations, the inverse etc., we refer to [15] and [16]. . The sum and the direct sum of subspaces in K2 are denoted by and . We define an indefinite inner product on K2 by f g f,ˆ gˆ := i [f,g ] − [f ,g] , fˆ = , gˆ = ∈K2. f g K2 · · J 0 −iJ ∈LK2 Then ( , [[ , ]]) is a Krein space and = iJ 0 ( ) is a correspond- ing fundamental symmetry. If necessary we will indicate the underlying space by subscripts, e.g., [[·, ·]] K2 . Let A be a linear relation in K. The adjoint relation A+ ∈ C(K) is defined as A+ := A[[ ⊥]] = hˆ ∈K2 | h,ˆ fˆ =0forallfˆ ∈ A , where A[[ ⊥]] denotes the orthogonal companion of A with respect to [[·, ·]] . A is said to be symmetric (self-adjoint)ifA ⊂ A+ (resp. A = A+). Let S be a closed linear relation in K. The resolvent set ρ(S)ofS ∈ C(K) is the set of all λ ∈ C such that (S − λ)−1 ∈L(K), the spectrum σ(S)ofS is the complement of ρ(S)inC. The extended spectrum σ(S)ofS is defined by σ(S)=σ(S)ifS ∈L(K)andσ(S)=σ(S) ∪{∞} otherwise. A point λ ∈ C is called a point of regular type of S, λ ∈ r(S), if (S − λ)−1 is a bounded operator. We say that λ ∈ C belongs to the approximate point spectrum of S, denoted by xn σap(S), if there exists a sequence yn ∈ S, n =1, 2,..., such that xn =1and limn→∞ yn − λxn =0.Theextended approximate point spectrum σap(S) of S is defined by ∪{∞} ∈ −1 σap(S) if 0 σap(S ) σap(S):= −1 . σap(S)if0∈ σap(S )
We remark, that the boundary points of σ(S)inC belong to σap(S). Next we recall the definitions of the spectra of positive and negative type of self-adjoint relations, cf. [20] (for bounded self-adjoint operators see [29]). For 16 J. Behrndt, A. Luger and C. Trunk equivalent descriptions of the spectra of positive and negative type we refer to [20, Theorem 3.18].
Definition 2.1. Let A0 be a self-adjoint relation in K.Apointλ ∈ σap(A0)is said to be of positive type (negative type) with respect to A0, if for every sequence xn yn ∈ A0, n =1, 2,..., with xn = 1, limn→∞ yn − λxn =0wehave lim inf [xn,xn] > 0 resp. lim sup [xn,xn] < 0 . n→∞ n→∞
If ∞∈σap(A0), then ∞ is said to be of positive type (negative type) with respect −1 to A0 if 0 is of positive type (resp. negative type) with respect to A0 .Wedenote the set of all points of σap(A0) of positive type (negative type) by σ++(A0)(resp. σ−−(A0)).
We remark that the self-adjointness of the relation A0 yields that the points of positive and negative type introduced in Definition 2.1 belong to R. An open subset ∆ of R is said to be of positive type (negative type)with respect to A0 if each point λ ∈ ∆ ∩ σ(A0) is of positive type (resp. negative type) with respect to A0.Anopensubset∆ofR is called of definite type with respect to A0 if it is either of positive or of negative type with respect to A0. For each λ ∈ σ++(A0)(σ−−(A0)) there exists an open neighborhood Uλ in C such that (Uλ ∩ σ(A0) ∩ R) ⊂ σ++(A0)(resp.(Uλ ∩ σ(A0) ∩ R) ⊂ σ−−(A0)), Uλ\R ⊂ ρ(A0)and −1 −1 (A0 − λ) ≤M|Im λ| holds for some M>0andallλ ∈Uλ\R, cf. [1], [20] (and [29] for bounded operators).
2.2. Locally definitizable self-adjoint relations In this section we briefly recall the notion of locally definitizable self-adjoint rela- tions and intervals of type π+ and type π− from [20]. Let Ω be some domain in C symmetric with respect to the real axis such that Ω ∩ R = ∅ and the intersections of Ω with the upper and lower open half-planes are simply connected.
Definition 2.2. Let A0 be a self-adjoint relation in the Krein space K such that σ(A0) ∩ (Ω\R) consists of isolated points which are poles of the resolvent of A0, and no point of Ω ∩ R is an accumulation point of the non-real spectrum of A0 in Ω. The relation A0 is said to be definitizable over Ω, if the following holds.
(i) Every point µ ∈ Ω ∩ R has an open connected neighborhood Iµ in R such that both components of Iµ\{µ} are of definite type with respect to A0. (ii) For every finite union ∆ of open connected subsets of R, ∆ ⊂ Ω ∩ R,there exists m ≥ 1, M>0 and an open neighborhood U of ∆inC such that −1 2m−2 −m (A0 − λ) ≤M(1 + |λ|) |Im λ| holds for all λ ∈U\R. Generalized Resolvents of Symmetric Operators in Krein Spaces 17
By [20, Theorem 4.7] a self-adjoint relation A0 in K is definitizable over C if and only if A0 is definitizable, that is, the resolvent set of A0 is non-empty and there exists a rational function r =0withpolesonlyin ρ(A0) such that r(A0) ∈L(K) is a nonnegative operator in K,thatis
[r(A0)x, x] ≥ 0 holds for all x ∈K(see [28] and [16, §4and§5]). + Let A0 = A0 be definitizable over Ω and let δ → E(δ) be the local spectral function of A0 on Ω ∩ R. Recall that E(δ) is defined for all finite unions δ of connected subsets of Ω ∩ R, δ ⊂ Ω ∩ R, the endpoints of which belong to Ω ∩ R and are of definite type with respect to A0 (see [20, Section 3.4 and Remark 4.9]). With the help of the local spectral function E(·) the open subsets of definite type in Ω ∩ R can be characterized in the following way. An open subset ∆, ∆ ⊂ Ω ∩ R, is of positive type (negative type) with respect to A0 if and only if for every finite union δ of open connected subsets of ∆, δ ⊂ ∆, such that the boundary points of δ in R are of definite type with respect to A0, the spectral subspace (E(δ)K, [·, ·]) (resp. (E(δ)K, −[·, ·])) is a Hilbert space (cf. [20, Theorem 3.18]). We say that an open subset ∆, ∆ ⊂ Ω ∩ R,isoftype π+ (type π−)with respect to A0 if for every finite union δ of open connected subsets of ∆, δ ⊂ ∆, such that the boundary points of δ in R are of definite type with respect to A0 the spectral subspace (E(δ)K, [·, ·]) is a Pontryagin space with finite rank of negativity (resp. positivity). We shall say that A0 is of type π+ over Ω(type π− over Ω) if Ω ∩ R is of type π+ (resp. type π−) with respect to A0 and σ(A0) ∩ Ω\R consists of eigenvalues with finite algebraic multiplicity. We remark, that spectral points in sets of type π+ and type π− can also be characterized with the help of approximative eigensequences (see [1, 2]).
2.3. Matrix-valued locally definitizable functions In this section we recall the definition of matrix-valued locally definitizable func- tions from [21]. Although in the formulation of the main theorem in Section 3.2 below only scalar locally definitizable functions appear, matrix-valued functions will be used within the proof. Let Ω be a domain as in the beginning of Section 2.2 and let τ be an L(Cn)- valued piecewise meromorphic function in Ω\R which is symmetric with respect to the real axis, that is τ(λ)=τ(λ)∗ for all points λ of holomorphy of τ.If,in addition, no point of Ω∩ R is an accumulation point of nonreal poles of τ we write τ ∈ M n×n(Ω). The set of the points of holomorphy of τ in Ω\R and all points µ ∈ Ω ∩ R such that τ can be analytically continued to µ and the continuations from Ω ∩ C+ and Ω ∩ C− coincide, is denoted by h(τ). The following definition of sets of positive and negative type with respect to matrix functions and Definition 2.4 below of locally definitizable matrix functions can be found in [21]. 18 J. Behrndt, A. Luger and C. Trunk
Definition 2.3. Let τ ∈ M n×n(Ω). An open subset ∆ ⊂ Ω ∩ R is said to be of n positive type with respect to τ if for every x ∈ C and every sequence (µk)of points in Ω ∩ C+ ∩ h(τ) which converges in C to a point of ∆ we have lim inf Im τ(µk)x, x ≥ 0. k→∞ An open subset ∆ ⊂ Ω ∩ R is said to be of negative type with respect to τ if ∆ is of positive type with respect to −τ. ∆ is said to be of definite type with respect to τ if ∆ is of positive or of negative type with respect to τ. Definition 2.4. A function τ ∈ M n×n(Ω) is called definitizable in Ω if the following holds.
(i) Every point µ ∈ Ω ∩ R has an open connected neighborhood Iµ in R such that both components of Iµ\{µ} are of definite type with respect to τ. (ii) For every finite union ∆ of open connected subsets in R, ∆ ⊂ Ω ∩ R,there exist m ≥ 1, M>0 and an open neighborhood U of ∆inC such that τ(λ) ≤M(1 + |λ|)2m |Im λ|−m holds for all λ ∈U\R. The class of L(Cn)-valued definitizable functions in Ω will be denoted by Dn×n(Ω). In the case n =1wewriteD(Ω) instead of D1×1(Ω) and we set D(Ω) := D(Ω) ∪{d∞}, 0 | ∈ C ∈ C C where d∞ denotes the relation c c ( ). A function τ ∈ M n×n(C) which is definitizable in C is called definitizable,see [20]. We note that τ ∈ M n×n(C) is definitizable if and only if there exists a rational function g symmetric with respect to the real axis such that the poles of g belong to h(τ) ∪{∞} and gτ is the sum of a Nevanlinna function and a meromorphic function in C (cf. [20]). For a comprehensive study of definitizable functions we refer to the papers [18, 19] of P. Jonas. We mention only that the generalized Nevanlinna class is a subclass of the definitizable functions. Recall that a function n×n τ ∈ M (C)issaidtobeageneralized Nevanlinna function if the kernel Kτ , τ(λ) − τ(µ) K (λ, µ)= , τ λ − µ has finitely many negative squares (see [25] and [26]). In [21] it is shown that a function τ ∈ M n×n(Ω) is definitizable in Ω if and only if for every finite union ∆ of open connected subsets of R such that ∆ ⊂ Ω∩R, τ can be written as the sum τ = τ0 + τ(0) (2.1) n n of an L(C )-valued definitizable function τ0 and an L(C )-valued function τ(0) which is locally holomorphic on ∆. We say that a locally definitizable function n×n τ ∈D (Ω) is a generalized Nevanlinna function in Ω if the function τ0 in (2.1) can be chosen as a generalized Nevanlinna function. Generalized Resolvents of Symmetric Operators in Krein Spaces 19
The class of L(Cn)-valued generalized Nevanlinna functions in Ω will be de- noted by N n×n(Ω). In the case n =1wewriteN (Ω) instead of N 1×1(Ω) and we set N (Ω) := N (Ω) ∪{d∞}, 0 | ∈ C ∈ C C where d∞ denotes the relation c c ( ). The following theorem is a consequence of [21, Propositions 2.8 and 3.4]. It establishes a connection between self-adjoint relations which are locally definiti- n zable (locally of type π+)andL(C )-valued locally definitizable functions (resp. local generalized Nevanlinna functions).
Theorem 2.5. Let Ω be a domain as above and let A0 be a self-adjoint relation in the Krein space K which is definitizable over Ω.Letγ ∈L(Cn, K) and S = S∗ ∈ n L(C ), fix some point λ0 ∈ ρ(A0) ∩ Ω and define + −1 τ(λ):=S + γ (λ − Re λ0)+(λ − λ0)(λ − λ0)(A0 − λ) γ for all λ ∈ ρ(A0) ∩ Ω. Then the following holds. (i) The function τ is definitizable in Ω, τ ∈Dn×n(Ω). n×n (ii) If A0 is of type π+ over Ω,thenτ belongs to N (Ω). (iii) An open subset ∆ of Ω ∩ R which is of positive type (negative type) with respect to A0 is of positive type (resp. negative type) with respect to τ. In the sequel we shall often assume that the sign types of self-adjoint relations which are definitizable over Ω, and definitizable functions in Ω coincide outside of a discrete set in Ω ∩ R. A notion for this concept is introduced in the next definition, cf. [3, Definition 2.8].
Definition 2.6. Let A0 and A1 be self-adjoint relations which are definitizable over Ω and let τ and τ be L(Cn)-valued definitizable functions in Ω. We shall say that the sign types of A0 and A1 (A0 and τ, τ and τ) are d-compatible in Ω if for every µ ∈ Ω ∩ R there exists an open connected neighborhood Iµ ⊂ Ω ∩ R of µ such that each component of Iµ\{µ} is either of positive type with respect to A0 and A1 (resp. A0 and τ, τ and τ) or of negative type with respect to A0 and A1 (resp. A0 and τ, τ and τ). n×n If A0 is definitizable over Ω and the function τ ∈D (Ω) is defined as in Theorem 2.5, then obviously the sign types of A0 and τ are d-compatible in Ω. A typical nontrivial situation where d-compatibility of sign types of locally definitizable self-adjoint relations appears is shown in Theorem 2.7 below. For a proof we refer to [4, Theorem 3.2].
Theorem 2.7. Let A0 and A1 be self-adjoint relations in K, assume that the set ρ(A0) ∩ ρ(A1) ∩ Ω is non-empty and that A0 is definitizable over Ω.If −1 −1 (A1 − µ) − (A0 − µ) is a finite rank operator for some (and hence for all) µ ∈ ρ(A0) ∩ ρ(A1) ∩ Ω,then A1 is definitizable over Ω and the sign types of A0 and A1 are d-compatible in Ω. 20 J. Behrndt, A. Luger and C. Trunk
3. Generalized resolvents of a class of symmetric operators 3.1. Boundary value spaces and Weyl functions associated with symmetric relations in Krein spaces Let (K, [·, ·]) be a separable Krein space, let J be a corresponding fundamental symmetry and let A be a closed symmetric relation in K.WesaythatA has defect m ∈ N ∪{∞}, if both deficiency indices ∗ ± n±(JA) = dim ker((JA) − λ),λ∈ C , of the symmetric relation JA in the Hilbert space (K, [J·, ·]) are equal to m. Here ∗ denotes the Hilbert space adjoint. We remark, that this is equivalent to the fact that there exists a self-adjoint extension of A in K and that each self- adjoint extension Aˆ of A in K satisfies dim A/Aˆ = m. We shall use the so-called boundary value spaces for the description of the self-adjoint extensions of closed symmetric relations in Krein spaces. The following definition is taken from [10]. Definition 3.1. Let A be a closed symmetric relation in the Krein space K.We + say that {G, Γ0, Γ1} is a boundary value space for A if G is a Hilbert space and + Γ0 + 2 0 1 →G →G Γ , Γ : A are mappings such that Γ := Γ1 : A is surjective, and the relation ˆ ˆ Γf,Γˆg G2 = f,gˆ K2 holds for all f,ˆ gˆ ∈ A+. In the following we recall some basic facts on boundary value spaces which can be found in, e.g., [8] and [10]. For the Hilbert space case we refer to [17], [12] and [13]. Let A be a closed symmetric relation in K.Then + [⊥] Nλ,A+ := ker(A − λ)=ran(A − λ) denotes the defect subspace of A at the point λ ∈ r(A)andweset fλ Nˆ + ∈N + λ,A := λfλ fλ λ,A . ˆ ˆ When no confusion can arise we write Nλ and Nλ instead of Nλ,A+ and Nλ,A+ . If there exists a self-adjoint extension A of A such that ρ(A ) = ∅ then we have + . A = A Nˆλ for all λ ∈ ρ(A ). + In this case there exists a boundary value space {G, Γ0, Γ1} for A such that ker Γ0 = A (cf. [10]). Let in the following A, {G, Γ0, Γ1} and Γ be as in Definition 3.1. It follows that the mappings Γ0 and Γ1 are continuous. The self-adjoint extensions
A0 := ker Γ0 and A1 := ker Γ1 + of A are transversal, i.e., A0 ∩A1 = A and A0 A1 = A . The mapping Γ induces, via −1 + AΘ := Γ Θ= fˆ ∈ A | Γfˆ ∈ Θ , Θ ∈ C(G), (3.1) Generalized Resolvents of Symmetric Operators in Krein Spaces 21 a bijective correspondence Θ → AΘ between the set of all closed linear relations + C(G)inG and the set of closed extensions AΘ ⊂ A of A. In particular (3.1) gives a one-to-one correspondence between the closed symmetric (self-adjoint) extensions of A and the closed symmetric (resp. self-adjoint) relations in G. If Θ is a closed operator in G, then the corresponding extension AΘ of A is determined by AΘ =ker Γ1 − ΘΓ0 . (3.2)
Assume that ρ(A0) = ∅ and denote by π1 the orthogonal projection onto the first component of K×K. For every λ ∈ ρ(A0) we define the operators −1 −1 γ(λ):=π1(Γ0|Nˆλ) ∈L(G, K)andM(λ):=Γ1(Γ0|Nˆλ) ∈L(G). The functions λ → γ(λ)andλ → M(λ) are called the γ-field and Weyl function corresponding to {G, Γ0, Γ1}. They are holomorphic on ρ(A0) and the relations −1 γ(ζ)=(1+(ζ − λ)(A0 − ζ) )γ(λ) (3.3) and M(λ) − M(ζ)∗ =(λ − ζ)γ(ζ)+γ(λ) (3.4) hold for all λ, ζ ∈ ρ(A0) (cf. [10]). It follows that + M(λ)=ReM(λ0)+γ(λ0) (λ − Re λ0) −1 (3.5) +(λ − λ0)(λ − λ0)(A0 − λ) γ(λ0) holds for a fixed λ0 ∈ ρ(A0)andallλ ∈ ρ(A0). If, in addition, the condition K =clsp{Nλ | λ ∈ ρ(A0)} is fulfilled, then it follows from (3.3) and (3.4) that the function M is strict,thatis M(λ) − M(µ)∗ ker = {0} (3.6) λ − µ λ∈h(M) holds for some (and hence for all) µ ∈ h(τ). If Θ ∈ C(G)andAΘ is the corresponding extension of A (see (3.1)), then for every point λ ∈ ρ(A0)wehave
λ ∈ ρ(AΘ) if and only if 0 ∈ ρ(Θ − M(λ)). (3.7)
For λ ∈ ρ(AΘ) ∩ ρ(A0) the well-known resolvent formula −1 −1 −1 + (AΘ − λ) =(A0 − λ) + γ(λ) Θ − M(λ) γ(λ) (3.8) holds (for a proof see, e.g., [10]).
3.2. A variant of the Krein-Naimark formula We choose a domain Ω as in the beginning of Section 2.2. Let A be a (not necessar- ily densely defined) closed symmetric operator in the Krein space K,let{G, Γ0, Γ1} be a boundary value space for A+ and let H be a further Krein space. A self-adjoint extension A of A in K×H is said to be an exit space extension of A and H is 22 J. Behrndt, A. Luger and C. Trunk called the exit space. The exit space extension A of A is said to be K-minimal if ρ(A) ∩ Ω is non-empty and K×H=clsp K, (A − λ)−1K|λ ∈ ρ(A) ∩ Ω holds. Note, that the definition of K-minimality depends on the domain Ω. The elements of K×H will be written in the form {k, h}, k ∈K, h ∈H.LetPK : K×H→H, {k, h} → k, be the projection onto the first component of K×H. Then the compression −1 PK(A − λ) |K,λ∈ ρ(A), of the resolvent of A to K is called a generalized resolvent of A.
Theorem 3.2. Let A be a closed symmetric operator of defect one in the Krein + space K and let {C, Γ0, Γ1}, A0 =kerΓ0, be a boundary value space for A with corresponding γ-field γ and Weyl function M. Assume that A0 is definitizable over Ω and that the condition K =clsp{Nλ,A+ | λ ∈ ρ(A0) ∩ Ω} is fulfilled. Then the following assertions hold. (i) Let A be a K-minimal self-adjoint exit space extension of A in K×Hwhich is definitizable over Ω and assume that the sign types of A and A0 are d- compatible in Ω. Then there exists a locally definitizable function τ ∈ D(Ω) such that the sign types of τ, A and A0 are d-compatible in Ω, ρ(A) ∩ ρ(A0) ∩ h(τ) ∩ Ω is a subset of h((M + τ)−1) and the formula −1 −1 −1 + PK(A − λ) |K =(A0 − λ) − γ(λ) M(λ)+τ(λ) γ(λ) (3.9) holds for all λ ∈ ρ(A) ∩ ρ(A0) ∩ h(τ) ∩ Ω. (ii) Let τ ∈ D(Ω) be a locally definitizable function such that M(µ)+τ(µ) =0 for some µ ∈ Ω, assume that the sign types of τ and A0 are d-compatible in Ω and let Ω be a domain with the same properties as Ω, Ω ⊂ Ω. Then there exists a Krein space H and a K-minimal self-adjoint exit space extension A of A in K×H which is definitizable over Ω , such that the sign types of A, τ and A0 are d-compatible in Ω , −1 ρ(A0) ∩ h(τ) ∩ h (M + τ) ∩ Ω
is a subset of ρ(A) and formula (3.9) holds for all points λ belonging to −1 ρ(A0) ∩ h(τ) ∩ h((M + τ) ) ∩ Ω .
Proof. The proof of assertion (i) consists of four steps. Let A be a K-minimal self- adjoint exit space extension of A in K×Hwhich is definitizable over Ω such that the sign types of A and A0 are d-compatible in Ω. Generalized Resolvents of Symmetric Operators in Krein Spaces 23
1. In this first step we prove assertion (i) for the case H = {0}. Here A is a canonical extension of A and therefore, by (3.1), there exists a self-adjoint constant τ ∈ R ∪{d∞} such that −1 −1 −1 + (A − λ) =(A0 − λ) − γ(λ) M(λ)+τ γ(λ) holds (cf. (3.8)), that is, A coincides with the canonical self-adjoint extension A−τ −1 of A and by (3.7) we have ρ(A−τ ) ∩ ρ(A0) ⊂ h((M + τ) ). Here each point in Ω∩R is of positive as well as of negative type with respect to τ and hence assertion (i) follows. 2. In the following we assume H = {0}. Following the lines of [11, §5] we define in this step a symmetric relation T in H and a special boundary value space for the adjoint T +. Below we will deal with direct products of linear relations. The following notation will be used. If U is a relation in K and V is a relation in H we shall write U × V for the direct product of U and V which is a relation in K×H, {f1,f2} f1 f2 U × V = ∈ U, ∈ V . {f1,f2} f1 f2 {f1,f2} f1 f2 For the pair we shall also write {fˆ1, fˆ2},wherefˆ1 = and fˆ2 = . {f1,f2} f1 f2 The linear relations k {k, 0} S := A ∩K2 = ∈ A k {k , 0} and h {0,h} T := A ∩H2 = ∈ A h {0,h } are closed and symmetric in K and H, respectively, and we have A ⊂ S.LetJK and JH be fundamental symmetries in the Krein spaces K and H, respectively, and choose JK 0 J := ∈L(K×H) 0 JH 2 as a fundamental symmetry in the Krein space K×H.ThenJKS = JA ∩K and 2 JHT = JA ∩H are symmetric relations in the Hilbert spaces (K, [JK·, ·]) and (H, [JH·, ·]), respectively. It follows from [11, §5] that the deficiency indices of JKS and −JHT coincide. As JKS is a symmetric extension of the symmetric operator JKA in the Hilbert space (K, [JK·, ·]) the deficiency indices n±(JKS)ofJKS are (1, 1) or (0, 0). The case n±(JKS) = 0 is impossible here as otherwise also the relation JHT would be self-adjoint in (H, [JH·, ·]) and therefore JA would coincide with JKS × JHT .ButasA = S × T is by assumption a K-minimal exit space extension of A we would obtain H = {0}, a contradiction. 24 J. Behrndt, A. Luger and C. Trunk
Therefore, it remains to consider the case n±(JKS) = 1. Then the operators A and S coincide. Let us show that A+ coincides with PK{k, h} {k, h} k {k, h} R = ∈ A = ∈ A . PK{k ,h} {k ,h} k {k ,h}
In fact, as A is self-adjoint we have g,PK{k ,h} − g ,PK{k, h} = {g,0}, {k ,h} − {g , 0}, {k, h} =0 g + ∈ {ˆ ˆ}∈ ˆ k ˆ h ⊂ for all g A and k, h A, k = k , h = h . Hence A R . Similarly it follows that R+ ⊂ A holds. Therefore A+ coincides with the closure of R and as + A has finite defect and R is an extension of A we conclude A = R. Replacing PK by the projection PH onto the second component of K×Hthesamearguments show + PH{k, h} {k, h} h {k, h} T = ∈ A = ∈ A . PH{k ,h} {k ,h} h {k ,h} We define the mappings PK and PH by + {k, h} k PK : A → A , → {k ,h } k and + {k, h} h PH : A → T , → . {k ,h } h + + In the sequel we denote the elements in A and T by fˆ1 and fˆ2, respectively. −1 −1 As the multivalued part of PH coincides with A it follows that Γ0PKPH and −1 + Γ1PKPH are operators. We define Γ0, Γ1 : T → C by ˆ −1 ˆ ˆ −1 ˆ ˆ + Γ0f2 := −Γ0PKPH f2, Γ1f2 := Γ1PKPH f2, f2 ∈ T , cf. [11]. Taking into account that A is self-adjoint, one verifies that {C, Γ0, Γ1} is a boundary value space for T +.Weset T0 := ker Γ0. (3.10) ˆ ˆ + + ˆ −1 ˆ An element {f1, f2}∈A × T belongs to A if and only if f1 − PKPH f2 is contained in A. Therefore A is the canonical self-adjoint extension of the symmetric relation A × T in K×Hgiven by + + A = {fˆ1, fˆ2}∈A × T | Γ0fˆ1 +Γ0fˆ2 =Γ1fˆ1 − Γ1fˆ2 =0 . (3.11)
3. In order to show that T0 has a non-empty resolvent set we construct in this step an auxiliary self-adjoint extension Tα of T in H such that ρ(Tα) ∩ Ωisnon-empty and with the help of Theorem 2.7 we will show that Tα is definitizable over Ω and that the sign types of Tα are d-compatible with the sign types of A and A0 in Ω. Generalized Resolvents of Symmetric Operators in Krein Spaces 25
2 It is easy to see that {C , Γ0 , Γ1 },where ˆ ˆ ˆ ˆ Γ0f1 ˆ ˆ Γ1f1 ˆ + ˆ + Γ0 {f1, f2} := and Γ1 {f1, f2} := , f1 ∈ A , f2 ∈ T , (3.12) Γ0fˆ2 Γ1fˆ2 is a boundary value space for A+ × T +. Setting ⎛ ⎞ 00−11 ⎜11 0 0⎟ W := ⎜ ⎟ ∈L(C4) ⎝10 0 0⎠ 00 0 1 and & ' Γ0 Γ0 := W (3.13) Γ1 Γ1 2 + + we obtain a boundary value space {C , Γ0, Γ1} for A × T . This follows, e.g., 4 from the fact that W is unitary in the Krein space (C , [[ ·, ·]] 4 ), where C 0 −iIC2 [[ ·, ·]] C4 := J·, · , J = , (3.14) iIC2 0 (see Section 2.1). Here we have A =kerΓ0. (3.15) As A is by assumption definitizable over Ω it follows from Theorem 2.5 and (3.5) ( 2 that the Weyl function M corresponding to {C , Γ0, Γ1} is a definitizable function in Ω, M( ∈D2×2(Ω), and the sign types of M( and A are d-compatible in Ω. It is not difficult to verify that −1 + ran PH(A − λ) |K =ker(T − λ)=Nλ,T + ,λ∈ ρ(A), holds. Since A is an K-minimal exit space extension of A we have −1 H =clsp ran PH(A − λ) |K | λ ∈ ρ(A) ∩ Ω (3.16) =clsp Nλ,T + | λ ∈ ρ(A) ∩ Ω and from the assumption K =clsp{Nλ,A+ | λ ∈ ρ(A0) ∩ Ω} we obtain K×H=clsp Nλ,A+×T + | λ ∈ ρ(A0) ∩ ρ(A) ∩ Ω . This implies that the function M( ∈D2×2(Ω) is strict (see (3.6)). We claim that ∈ R there exists α such that the function 00 λ → M((λ) − 0 α ∈ ∩ ( 2 is invertible for some λ ρ(A) Ω. Indeed, let M(λ)=(mij (λ))i,j=1 and suppose that for all λ ∈ ρ(A) ∩ Ω and every α ∈ R we have ( 00 det M(λ) − = m11(λ)(m22(λ) − α) − m21(λ)m12(λ)=0. 0 α 26 J. Behrndt, A. Luger and C. Trunk
This implies m11(λ)=m12(λ)m21(λ) = 0 and since m12 and m21 are piecewise meromorphic functions in Ω\R and M( is symmetric with respect to the real axis we conclude m12(λ)=m21(λ)=0,λ ∈ ρ(A) ∩ Ω, which contradicts the strictness of M(. It is straightforward to check that the matrix ⎛ ⎞ 0010 ⎜ 0 −αα1⎟ V := ⎜ ⎟ ∈L(C4) (3.17) ⎝−1001⎠ 0 −110
4 2 is unitary in (C , [[ ·, ·]] C4 ), cf. (3.14). Let {C , Γ0, Γ1} be the boundary value space for A+ × T + defined by & ' & ' Γ0 Γ0 Γ0 := V = VW , (3.18) Γ1 Γ1 Γ1
(see (3.13)). From ⎛ ⎞ 1000 ⎜0 −α 01⎟ VW = ⎜ ⎟ ⎝0010⎠ 0 −100 we obtain ˆ ˆ ˆ Γ0f1 ˆ + ˆ + Γ0{f1, f2} = , f1 ∈ A , f2 ∈ T , Γ1fˆ2 − αΓ0fˆ2 and ˆ ˆ ˆ Γ1f1 ˆ + ˆ + Γ1{f1, f2} = , f1 ∈ A , f2 ∈ T . −Γ0fˆ2 We denote the self-adjoint extension ker(Γ1 − αΓ0) ∈ C(H)ofT in H by Tα.Then the self-adjoint extension ker Γ0 of A × T in K×Hcoincides with A0 × Tα. Since (3.18) and (3.17) imply 00 10 A0 × T =kerΓ0 =ker Γ0 + Γ1 α 0 −α α 1 00 =ker Γ1 − Γ0 , 0 α we find from (3.15), (3.2) and (3.7) that a point λ ∈ ρ(A) belongs to the set ρ(A0 × Tα) if and only if 0 belongs to the resolvent set of 00 M((λ) − . 0 α Generalized Resolvents of Symmetric Operators in Krein Spaces 27
But we have chosen α such that this function is invertible for some λ ∈ ρ(A) ∩ Ω, therefore λ belongs to ρ(A0 × Tα). In particular λ ∈ ρ(A0) ∩ ρ(Tα)and ρ(Tα) ∩ ρ(A0) ∩ ρ(A) ∩ Ω = ∅. As A × T is a symmetric relation of defect two and A and A0 × Tα are self-adjoint extensions of A × T in K×Hwe have −1 −1 dim ran (A − λ) − ((A0 × Tα) − λ) ≤ 2 for all λ ∈ ρ(A) ∩ ρ(A0) ∩ ρ(Tα) ∩ Ω. Since A is definitizable over Ω we obtain from Theorem 2.7 that also the self-adjoint relation A0 × Tα is definitizable over Ω and that the sign types of A and the sign types of A0 × Tα are d-compatible in Ω. It is a simple consequence from Definition 2.1 that σ++(A0 × Tα) ∩ σap(Tα) ⊂ σ++(Tα) and σ−−(A0 × Tα) ∩ σap(Tα) ⊂ σ−−(Tα) holds. Hence, real points from σ++(A0 ×Tα)(σ−−(A0 ×Tα)) belong to ρ(Tα)orto σ++(Tα)(resp.σ−−(Tα)). Therefore Tα is definitizable over Ω and the sign types of Tα in Ω are d-compatible with the sign types of A0 × Tα and, hence, with the sign types of A and A0 in Ω.
4. In this step we show that also T0 in (3.10) has a non-empty resolvent set and that formula (3.9) holds with the Weyl function τ corresponding to the boundary value space {C, Γ0, Γ1}. Moreover, we show that τ is locally definitizable and that its sign types are d-compatible with the sign types of A0 and A in Ω. It is straightforward to verify that {C, Γ1 − αΓ0, −Γ0} is a boundary value + space for T and we have Tα =ker(Γ1 − αΓ0)andT0 = ker(−Γ0). The cor- responding Weyl function τα is defined for all λ ∈ ρ(Tα). As Tα is definitizable over Ω the function τα belongs to the class D(Ω) and the sign types of τα are d-compatible with the sign types of Tα, A and A0 in Ω (cf. Theorem 2.5 and (3.5) or [3, Proposition 3.2]). Relation (3.16) implies that τα is strict and in particular τα is not identically equal to zero. ∈ ∩ h −1 Then, by (3.8), for λ ρ(Tα) (τα )wehave 1 − −1 − −1 − + (T0 λ) =(Tα λ) γα(λ) γα(λ) , τα(λ) {C − − } where γα is the γ-field of the boundary value space , Γ1 αΓ0, Γ0 . Therefore the set ρ(Tα)∩ρ(T0)∩Ω is non-empty and by Theorem 2.7 the self-adjoint relation T0 is definitizable over Ω and the sign types of T0 and Tα are d-compatible in Ω. The Weyl function τ corresponding to the boundary value space {C, Γ0, Γ1} satisfies − −1 ∈ h −1 ∩ τ(λ)= τα(λ) + α, λ (τα ) ρ(Tα). and is holomorphic on ρ(T0). It follows from Theorem 2.5 and (3.5) that τ belongs to the class D(Ω) and that its sign types are d-compatible with the sign types of T0 and Tα and hence 28 J. Behrndt, A. Luger and C. Trunk
also with the sign types of A and A0.Theγ-field corresponding to {C, Γ0, Γ1} will be denoted by γ . Since A0 and T0 are both definitizable over Ω the set ρ(A0)∩ρ(T0)∩Ωisnon- empty. The γ-field γ and the Weyl function M corresponding to the boundary value space {C2, Γ , Γ } defined in (3.12) are given by 0 1 γ(λ)0 λ → γ (λ)= ,λ∈ ρ(A0) ∩ ρ(T0) ∩ Ω, (3.19) 0 γ (λ) and M(λ)0 λ → M (λ)= ,λ∈ ρ(A0) ∩ ρ(T0) ∩ Ω, (3.20) 0 τ(λ) respectively. The relation {u, −u} Θ:= u, v ∈ C ∈ C(C2) (3.21) {v, v} is self-adjoint and the corresponding self-adjoint extension of A × T is given by −1 Γ0 + + Θ= {fˆ1, fˆ2}∈A × T | Γ0fˆ1 +Γ0fˆ2 =Γ1fˆ1 − Γ1fˆ2 =0 (3.22) Γ1 and coincides with A (see (3.11)). By (3.7) a point λ ∈ ρ(A0 × T0) belongs to ρ(A) if and only if 0 ∈ ρ(Θ − M (λ)). Hence, for λ ∈ ρ(A0 × T0) ∩ ρ(A) ∩ Ω=ρ(A0) ∩ h(τ) ∩ ρ(A) ∩ Ω −1 {v − M(λ)u, v + τ(λ)u} Θ − M (λ) = u, v ∈ C {u, −u} is an operator. Therefore (M(λ)+τ(λ))u = 0 implies u = 0 and we conclude that −1 the set ρ(A)∩ρ(A0)∩h(τ)∩Ω is a subset of h((M + τ) ). Setting x = v − M(λ)u and y = v + τ(λ)u we obtain −1 −1 u = − M(λ)+τ(λ) x + M(λ)+τ(λ) y for λ ∈ ρ(A0 × T0) ∩ ρ(A) ∩ Ω. This implies −1 −1 −1 −(M(λ)+τ(λ)) (M(λ)+τ(λ)) Θ − M (λ) = . (3.23) (M(λ)+τ(λ))−1 −(M(λ)+τ(λ))−1 For all λ ∈ ρ(A0 × T0) ∩ ρ(A) ∩ Ωtherelation −1 −1 −1 + (A − λ) = (A0 × T0) − λ + γ (λ) Θ − M (λ) γ (λ) (3.24) holds (cf. (3.8)) and it follows from (3.24), (3.19) and (3.23) that the formula −1 −1 −1 + PK(A − λ) |K =(A0 − λ) − γ(λ) M(λ)+τ(λ) γ(λ) holds. This completes the proof of assertion (i). 5. Assertion (ii) was already proved in [3] in a slightly different form. For the convenience of the reader we sketch the proof. Generalized Resolvents of Symmetric Operators in Krein Spaces 29
If τ is identically equal to a real constant, then A−τ := ker(Γ1 + τΓ0)isa canonical self-adjoint extension of A. As the Weyl function M corresponding to + A and {C, Γ0, Γ1} is strict we obtain ρ(A−τ ) ∩ Ω = ∅ and Theorem 2.7 implies that A−τ is definitizable over Ω and that the sign types of A0, A−τ and τ ∈ R are d-compatible. By (3.8) −1 −1 −1 + (A−τ − λ) =(A0 − λ) − γ(λ) M(λ)+τ γ(λ) ∈ ∩ −1 0 | ∈ C holds for all λ ρ(A0) ((M + τ) ). In the case τ = d∞ = c c we have A−τ = A0. Assume now that τ ∈D(Ω) is not equal to a constant and let Ω be a domain with the same properties as Ω, Ω ⊂ Ω. With the help of [21, Theorem 3.8] it was shown in [3, Theorem 3.3] that there exists a Krein space H, a closed symmetric + operator T of defect one in H and a boundary value space {C, Γ0, Γ1} for T such that T0 := ker Γ0 is definitizable over Ω ,thesigntypesofτ and T0 are d-compatible and τ coincides with the Weyl function corresponding to {C, Γ0, Γ1} on Ω ∩ ρ(T0). Moreover the condition H =clsp γ (λ) | λ ∈ ρ(T0) ∩ Ω (3.25)
2 + + is fulfilled. We choose the boundary value space {C , Γ0 , Γ1 } for A × T as in (3.12) with γ-field and Weyl function given by (3.19) and (3.20), respectively. The self-adjoint extension corresponding to Θ in (3.21) via (3.1) is denoted by A.Then A has the form (3.22) and the relation (3.24) holds for all λ ∈ Ω which belong to ρ(A0 × T0)andfulfil0∈ ρ(Θ − M (λ)). From (3.23) we conclude −1 ρ(A0) ∩ h(τ) ∩ h (M + τ) ∩ Ω ⊂ ρ(A) and (3.24) implies that the formula (3.9) holds. Since the minimality condition (3.25) is fulfilled it follows from (3.24) that A is a K-minimal exit space extension of A.AsA0 × T0 is definitizable over Ω the relation (3.24) and Theorem 2.7 imply that A is also definitizable over Ω and the sign types of A, A0 and τ are d-compatible.
The next theorem is a variant of the Krein-Naimark formula for the case that A0 and A are locally of type π+ and τ is a local generalized Nevanlinna function. The proof of Theorem 3.3 below is essentially the same as the proof of Theo- rem 3.2. Instead of the result on finite rank perturbations of locally definitizable self-adjoint relations from [4], cf. Theorem 2.7, one has to use [5, Theorem 2.4] on the stability of self-adjoint operators and relations locally of type π+ under compact perturbations in resolvent sense. We leave the details to the reader.
Theorem 3.3. Let A be a closed symmetric operator of defect one in the Krein + space K and let {C, Γ0, Γ1} be a boundary value space for A with corresponding γ-field γ and Weyl function M. Assume that A0 =kerΓ0 is of type π+ over Ω and that the condition K =clsp{Nλ,A+ | λ ∈ ρ(A0) ∩ Ω} is fulfilled. 30 J. Behrndt, A. Luger and C. Trunk
Then the following assertions hold: (i) For every K-minimal self-adjoint exit space extension A of A in K×H which is of type π+ over Ω there exists a function τ ∈ N (Ω) such that ρ(A) ∩ ρ(A0) ∩ h(τ) ∩ Ω is a subset of h((M + τ)−1) and the formula −1 −1 −1 + PK(A − λ) |K =(A0 − λ) − γ(λ) M(λ)+τ(λ) γ(λ) (3.26) holds for all λ ∈ ρ(A) ∩ ρ(A0) ∩ h(τ) ∩ Ω. (ii) Let τ ∈ N(Ω) be a local generalized Nevanlinna function such that M(µ)+ τ(µ) =0 for some µ ∈ Ω and let Ω be a domain with the same properties as Ω, Ω ⊂ Ω. Then there exists a Krein space H and a K-minimal self-adjoint exit space extension A of A in K×Hwhich is of type π+ over Ω , such that −1 ρ(A0) ∩ h(τ) ∩ h (M + τ) ∩ Ω
is a subset of ρ(A) and formula (3.26) holds for all points λ belonging to −1 ρ(A0) ∩ h(τ) ∩ h((M + τ) ) ∩ Ω.
References
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[2] T.Ya. Azizov, P. Jonas, C. Trunk: Spectral Points of Type π+ and Type π− of Selfadjoint Operators in Krein Spaces, J. Funct. Anal. 226 (2005), 114–137. [3] J. Behrndt: A Class of Abstract Boundary Value Problems with Locally Definitizable Functions in the Boundary Condition, Operator Theory: Advances and Applications 163 (2005), 55–73. [4] J. Behrndt: Finite Rank Perturbations of Locally Definitizable Operators in Krein Spaces, to appear in J. Operator Theory. [5] J. Behrndt, P. Jonas: On Compact Perturbations of Locally Definitizable Selfadjoint Relations in Krein Spaces, Integral Equations Operator Theory 52 (2005), 17–44. [6] J. Behrndt, H.C. Kreusler: Boundary Relations and Generalized Resolvents of Sym- metric Relations in Krein Spaces, submitted. [7] J. Behrndt, C. Trunk: On Generalized Resolvents of Symmetric Operators of Defect One with Finitely many Negative Squares, Proceedings of the Algorithmic Informa- tion Theory Conference, Vaasan Yliop. Julk. Selvityksi¨a Rap., 124,
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[28] H. Langer: Spectral Functions of Definitizable Operators in Krein Spaces, Functional Analysis Proceedings of a Conference held at Dubrovnik, Yugoslavia, November 2–14 (1981), Lecture Notes in Mathematics 948, Springer Verlag Berlin-Heidelberg-New York (1982), 1–46. [29] H. Langer, A. Markus, V. Matsaev: Locally Definite Operators in Indefinite Inner Product Spaces, Math. Ann. 308 (1997), 405–424. [30] H. Langer, B. Textorius: On Generalized Resolvents and Q-functions of Symmetric Linear Relations (Subspaces) in Hilbert Space, Pacific J. Math. 72 (1977), 135–165. [31] M.M. Malamud: On a Formula for the Generalized Resolvents of a Non-densely de- fined Hermitian Operator (Russian), Ukrain. Mat. Zh. 44 (1992), 1658–1688; trans- lation in Ukrainian Math. J. 44 (1993), 1522–1547. [32] M.A. Naimark: On Spectral Functions of a Symmetric Operator, Izv. Akad. Nauk SSSR, Ser. Matem. 7 (1943), 373–375.
Jussi Behrndt Institut f¨ur Mathematik, MA 6-4 Technische Universit¨at Berlin Straße des 17. Juni 136 D-10623 Berlin, Germany e-mail: [email protected] Annemarie Luger Institut f¨ur Analysis und Scientific Computing Technische Universit¨at Wien Wiedner Hauptstraße 8-10 A-1040 Wien, Austria e-mail: [email protected] Carsten Trunk Institut f¨ur Mathematik, MA 6-3 Technische Universit¨at Berlin Straße des 17. Juni 136 D-10623 Berlin, Germany e-mail: [email protected] Operator Theory: Advances and Applications, Vol. 175, 33–49 c 2007 Birkh¨auser Verlag Basel/Switzerland
Block Operator Matrices, Optical Potentials, Trace Class Perturbations and Scattering
Jussi Behrndt, Hagen Neidhardt and Joachim Rehberg
Abstract. For an operator-valued block-matrix model, which is called in quan- tum physics a Feshbach decomposition, a scattering theory is considered. Un- der trace class perturbations the channel scattering matrices are calculated. Using Feshbach’s optical potential it is shown that for a given spectral param- eter the channel scattering matrices can be recovered either from a dissipative or from a Lax-Phillips scattering theory. Mathematics Subject Classification (2000). Primary 47A40; Secondary 47A55, 47B44. Keywords. Feshbach decomposition, optical potential, Lax-Phillips scattering theory, dissipative scattering theory, scattering matrix, characteristic func- tion, dissipative operators.
1. Introduction
Let L and L0 be self-adjoint operators in a separable Hilbert space L and denote ac by P (L0) the orthogonal projection onto the absolutely continuous subspace ac L (L0)ofL0. The pair {L, L0} of self-adjoint operators is said to perform a scattering system if the wave operators W±(L, L0),
itL −itL0 ac W±(L, L0):=s − lim e e P (L0), (1.1) t→±∞ exist, cf. [5]. If the wave operators exist, then they are isometries from the ab- ac solutely continuous subspace L (L0) into the absolutely continuous subspace ac ac L (L), i.e., ran (W±(L, L0)) ⊆ L (L). The scattering system {L, L0} is called ac complete if the ranges of the wave operators W±(L, L0) coincide with L (L), cf. [5]. The operator ∗ S(L, L0):=W+(L, L0) W−(L, L0) is called the scattering operator of the scattering system {L, L0}. The scattering ac ac operator regarded as an operator in L (L0)commuteswithL0 . If the scattering 34 J. Behrndt, H. Neidhardt and J. Rehberg
ac system {L, L0} is complete, then S(L, L0) is a unitary operator in L (L0). In physical applications L0 is usually called the unperturbed or free Hamiltonian while L is called the perturbed or full Hamiltonian. Since S(L, L0)commutes with the free Hamiltonian L0 the scattering operator is unitarily equivalent to a multiplication operator induced by a family {S(λ)}λ∈R of unitary operators in the spectral representation of L0. This family is called the scattering matrix of the complete scattering system {L, L0} and is the most important quantity in the analysis of scattering processes. In this paper we investigate the special case that the Hilbert space L splits into two subspaces H1 and H2,
H1 L = ⊕ , H2 and the unperturbed Hamiltonian L0 is of the form
H1 H1 H1 0 L0 = : ⊕ −→ ⊕ . (1.2) 0 H2 H2 H2
In physics the subspaces Hj and the self-adjoint operators Hj, j =1, 2, are often called scattering channels and channel Hamiltonians, respectively. With respect to the decomposition (1.2) one introduces the channel wave operators
itL −itHj ac W±(L, Hj):=s − lim e Jje P (Hj) t→±∞ where Jj : Hj −→ L is the natural embedding operator. Introducing the channel scattering operators ∗ Sij = W+(L, Hi) W−(L, Hj):Hj −→ Hi,i,j=1, 2, one obtains a channel decomposition of the scattering operator S11 S12 S(L, L0)= . (1.3) S21 S22 In physics the decomposition (1.2) is often motivated either by the exclusive in- terest to scattering data in a certain channel or by the limited measuring process which allows to measure the scattering data only of a certain channel, say H1. ac ac Thus, let us assume that only the channel scattering operator S11 : H1 −→ H1 in the scattering channel H1 is known. This gives rise to the following problem: Is it possible to replace the full Hamiltonian L by an effective one H acting only in H1 such that the scattering operator of the scattering system {H, H1} coincides with S11?SinceS11 is a contraction, in general, this implies that either the scattering system {H, H1} cannot be complete or H is not self-adjoint. The problem has a solution within the scope of dissipative scattering systems developed in [18, 19, 20] for pairs {H, H1} of dissipative and self-adjoint operators Block Operator Matrices and Scattering 35
D in some separable Hilbert space. For such pairs the wave operators W± (H, H1) are defined by ∗ D itH −itH1 ac W+ (H, H1):=s − lim e e P (H1) t→+∞ and D −itH itH1 ac W− (H, H1):=s − lim e e P (H1), t→+∞ and the notion of completeness is generalized, cf. [18, 19, 20]. The scattering op- erator of a dissipative scattering system {H, H1} is defined by D ∗ D SD := W+ (H, H1) W− (H, H1).
It turns out that SD is a contraction acting on the absolutely continuous sub- ac space H1 of H1 which commutes with H1.In[17,18]itwasshownthatforany self-adjoint operator H1 in H1 and any contraction SD acting on the absolutely ac continuous subspace H1 and commuting with H1 there is a maximal dissipa- tive operator H on H1 such that {H, H1} performs a complete scattering system with scattering operator given by SD. In particular, this holds for the self-adjoint operator H1 and the channel scattering operator S11. That means, there is a max- imal dissipative operator H on H1 such that the channel scattering operator S11 is the scattering operator of the complete dissipative scattering system {H, H1}. Hence, roughly speaking, the scattering operator S11 can be always viewed as the scattering operator of a suitable chosen dissipative scattering system on H1.The disadvantage of this fact is that H is not known explicitly. Another approach to this problem was suggested by Feshbach in [10, 11], see also [6, 9]. He proposes a concrete dissipative perturbation V1 of the channel Hamiltonian H1, called “optical potential”, such that the scattering operator S1 of the dissipative scattering system {H1 + V1,H1} approximates S11 with a certain accuracy. To explain this approach in more detail let us assume that the full Hamiltonian L is obtained from L0 by an additive perturbation, L = L0 + V , where V is given by H H 0 G 1 1 V = : ⊕ −→ ⊕ . (1.4) G∗ 0 H2 H2 Introducing the “optical potential” −1 ∗ V1(λ):=−G(H2 − λ − i0) G ,λ∈ R, (1.5) it was shown in [8, Theorem 4.4.4] that under strong assumptions indeed the scat- tering operator S1[λ] of the (in general dissipative) scattering system {H1(λ),H1},
H1(λ):=H1 + V1(λ),λ∈ R, (1.6) coincides with the scattering operator S11 with an error of second order in the coupling constant. We show that Feshbach’s proposal can be made precise in another sense. Note first that the decomposition (1.2) leads not only to the decomposition (1.3) 36 J. Behrndt, H. Neidhardt and J. Rehberg of the scattering operator S but also to a decomposition of the scattering matrix {S(µ)}µ∈R, S11(µ) S12(µ) S(µ):= , S21(µ) S22(µ) where {Sij (µ)}µ∈R are called the channel scattering matrices. Denoting the scat- tering matrix of the dissipative scattering system {H1(λ),H1} by {S1[λ](µ)}µ∈R we prove that
S11(λ)=S1[λ](λ) (1.7) holds for a.e. λ ∈ R. This shows, that Feshbach’s proposal gives in fact a good approximation of the channel scattering matrix {S11(µ)}µ∈R in a neighborhood of the chosen spectral parameter λ of the optical potential V1(λ). Moreover, Feshbach’s proposal implies a second problem. Similarly to the optical potential V1(λ) in the first channel H1 one can introduce an optical potential V2(λ) in the second channel,
∗ −1 V2(λ):=−G (H1 − λ − i0) G, λ ∈ R, (1.8) and define a perturbed operator H2(λ),
H2(λ):=H2 + V2(λ),λ∈ R, (1.9) in H2. We show below that the characteristic function Θ2[λ](ξ), ξ ∈ C−,ofthe dissipative operator H2(λ) and the scattering matrix {S11(λ)}λ∈R are related by
∗ S11(λ)=Θ2[λ](λ) (1.10) for a.e. λ ∈ R. By [1]–[4] the last relation also yields that the scattering matrix {S11(λ)}λ∈R can be regarded as the scattering matrix SLP [λ](µ) of a Lax-Phillips scattering system at the point λ. Below we restrict ourself to a complete scattering system {L,L0}, L=L0 +V , where the perturbation V is a self-adjoint trace class operator. The assumption that V is a trace class operator is made for simplicity. Indeed, it would be sufficient −p −p to assume that the resolvent difference (L − z) − (L0 − z) is nuclear for a certain p ∈ N or, more generally, that the conditions of the so-called “stationary” scattering theory are satisfied, cf. [5, Section 14]. However, we emphasize that in contrast to [8] the smallness of the perturbation V is not assumed. Following the lines of [5] we show in Section 2 how the scattering matrix of the scattering system {L, L0} can be calculated. Under the additional assumptions (1.2) and (1.4) we find in Section 3 the channel scattering matrices {Sij (λ)}λ∈R. In Section 4 we prove relation (1.7). Section 5 is devoted to the proof of (1.10). Moreover, the ∗ Lax-Phillips scattering theory for which {Θ2[λ](µ) }µ∈R is the scattering matrix is indicated. Block Operator Matrices and Scattering 37
2. Scattering matrix
In this section we briefly recall the notion of the scattering matrix {S(λ)}λ∈R of a scattering system {L, L0}, where it is assumed that the unperturbed operator L0 is self-adjoint in the separable Hilbert space L and the perturbed operator L differs from L0 by a self-adjoint trace class operator V ∈B1(L), ∗ L = L0 + V, V = V ∈B1(L). (2.1)
Let E0(·) be the spectral measure of L0 and denote by B(R)thesetofallBorel subsets of the real axis R. Without loss of generality we assume throughout the paper that the condition
L =clospan{E0(∆)ran (|V |):∆∈ B(R)} (2.2) is satisfied, where |V | := (V ∗V )1/2. By Theorem X.4.4 of [13] the scattering system {L, L0} is complete, that is, the ranges of the wave operators W±(L, L0) in (1.1) coincide with the absolutely continuous subspace Lac(L)ofL. The operator V admits the representation V = |V |1/2C|V |1/2, |V | =(V ∗V )1/2,C=sgn(V ), (2.3) 1/2 where |V | belongs to the Hilbert-Schmidt class B2(L) and sgn(·) is the signum function. By Proposition 3.14 of [5] the limits |V |1/2(L − λ ± i0)−1|V |1/2 = lim |V |1/2(L − λ ± i )−1|V |1/2 (2.4) →+0 exist in B2(L) for a.e. λ ∈ R. The same holds for the limits 1/2 −1 1/2 |V | (L0 − λ ± i0) |V | . Moreover by Proposition 3.13 of [5] the derivative 1/2 1/2 |V | E0(dλ)|V | M0(λ):= ≥ 0 (2.5) dλ exists in B1(L) for a.e. λ ∈ R.Weset Qλ := clo ran (M0(λ)) ⊆ L.
By {Q(λ)}λ∈R we denote the family of orthogonal projections from L onto Qλ. One verifies that {Q(λ)}λ∈R is measurable. Let us consider the standard Hilbert space L2(R,dλ,L). On L2(R,dλ,L) we introduce the projection Q (Qf)(λ):=Q(λ)f(λ),λ∈ R,f∈ L2(R,dλ,L), and set Q =ran(Q). Further, in L2(R,dλ,L) we define the multiplication operator ML by (MLf)(λ):=λf(λ),λ∈ R, 2 2 dom (ML):= f ∈ L (R,dλ,L):λf(λ) ∈ L (R,dλ,L) .
Obviously, the multiplication operator ML and the projection Q commute. We set
MQ := ML dom (ML) ∩ Q. 38 J. Behrndt, H. Neidhardt and J. Rehberg
From Section 4.5 of [5] one gets that the absolutely continuous part Lac of the per- turbed operator L and the operator MQ are unitarily equivalent. In the following 2 we denote the subspace Q by L (R,dλ,Qλ) which can be regarded as the direct integral of the family of subspaces {Qλ}λ∈R with respect to the Lebesgue measure dλ on R, cf. [5]. ∗ ac Since the scattering operator S = W+(L, L0) W−(L, L0)actsonL (L0)and ac commutes with L0 there is a measurable family {S(λ)}λ∈R of operators
S(λ):Qλ −→ Qλ such that S is unitarily equivalent to the multiplication operator
(MQ(S)f)(λ):= S(λ)f(λ), 2 dom (MQ(S)) := L (R,dλ,Qλ).
The family {S(λ)}λ∈R is called the scattering matrix of the scattering system {L, L0}. Since the scattering system {L, L0} is complete the operator S(λ)is unitary on Qλ for a.e. λ ∈ R. The following representation theorem of the scattering matrix is a conse- quence of Corollary 18.9 of [5], see also [5, Section 18.2.2].
Theorem 2.1. Let L, L0 and V be self-adjoint operators in L as in (2.1).Then {L, L0} is a complete scattering system and the corresponding scattering matrix matrix {S(λ)}λ∈R admits the representation S λ I − πi M1/2 λ C − C|V |1/2 L − λ − i −1|V |1/2C M 1/2 λ ( )= Qλ 2 0 ( ) ( 0) 0 ( ) for a.e. λ ∈ R.
3. Channel scattering matrices Let us now assume that the Hilbert space L is the orthogonal sum of two subspaces H1 and H2, L = H1 ⊕ H2,thatL0 is a diagonal block operator matrix of the form
H1 H1 H1 0 L0 = : ⊕ −→ ⊕ , (3.1) 0 H2 H2 H2 cf. (1.2), where H1 and H2 are self-adjoint operators in H1 and H2 and that V ∈B1(L) is a self-adjoint trace class operator of the form H H 0 G 1 1 V = : ⊕ −→ ⊕ , (3.2) G∗ 0 H2 H2 see (1.4). The operator G : H2 −→ H1 describes the interaction between the channels. Since V is a trace class operator we have
G ∈B1(H2, H1). Block Operator Matrices and Scattering 39
The perturbed or full Hamiltonian L has the form
H1 H1 H1 G L := L0 + V = ∗ : ⊕ −→ ⊕ . (3.3) G H2 H2 H2 The following lemma is known as the Feshbach decomposition in physics, cf. [10, 11]. We use the notation
H1(z)=H1 + V1(z)andH2(z)=H2 + V2(z),z∈ C\R, (3.4) where −1 ∗ ∗ −1 V1(z)=−G(H2 − z) G and V2(z)=−G (H1 − z) G, (3.5) see (1.6), (1.9), (1.5) and (1.8).
Lemma 3.1. Let L, H1(z) and H2(z), z ∈ C\R, be given by (3.3) and (3.4), respectively. Then we have z ∈ res(Hi(z)), i =1, 2, for all z ∈ C\R and & ' −1 −1 −1 (H1(z) − z) −(H1 − z) G(H2(z) − z) L − z −1 . ( ) = −1 ∗ −1 −1 (3.6) −(H2(z) − z) G (H1 − z) (H2(z) − z) Proof. From 2 −1 ∗ 2 Im (H1(z) − z)h1,h1 =Imz h1 +Imz (H2 − z) G h1 , −1 z ∈ C\R, h1 ∈ H1, we conclude that (H1(z)−z) is a bounded everywhere defined −1 operator for all z ∈ C\R. Analogously one verifies that (H2(z)−z) is a bounded everywhere defined operator for all z ∈ C\R. A straightforward computation shows (L−z)−1 & ' −1 −1 −1 ∗ −1 −1 −1 (H1 −z) +(H1 −z) G(H2(z)−z) G (H1 −z) −(H1 −z) G(H2(z)−z) = −1 ∗ −1 −1 −(H2(z)−z) G (H1 −z) (H2(z)−z) for z ∈ C\R.Fromtheidentity ∗ −1 −1 −1 ∗ ∗ −1 −1 ∗ −1 I − G (H1 − z) G(H2 − z) G = G I − (H1 − z) G(H2 − z) G we obtain −1 −1 −1 ∗ −1 (H1 −z) +(H1 −z) G(H2(z)−z) G (H1 −z) −1 −1 ∗ −1 −1 −1 ∗ −1 =(H1 −z) I +G(H2 −z) I −G (H1 −z) G(H2 −z) G (H1 −z) −1 −1 ∗ −1 −1 =(H1 −z) I +G(H2 −z) G (H1(z)−z) =(H1(z)−z) for all z ∈ C\R, which proves (3.6). In the next lemma we calculate the limit |V |1/2(L − λ − i0)−1|V |1/2, λ ∈ R, cf. (2.4). Here and in the following it is convenient to use the functions ∗ 1/2 −1 ∗ 1/2 N1(z):=|G | (H1 − z) |G | , z ∈ C\R, (3.7) 1/2 −1 1/2 N2(z):=|G| (H2 − z) |G| , 40 J. Behrndt, H. Neidhardt and J. Rehberg and ∗ 1/2 −1 ∗ 1/2 F1(z):=|G | (H1(z) − z) |G | , z ∈ C\R. (3.8) 1/2 −1 1/2 F2(z):=|G| (H2(z) − z) |G| ,
Lemma 3.2. Let V ∈B1(L) be given by (3.2) with G ∈B1(H2, H1) and let U be a partial isometry such that G = U|G|. Then the limits
Ni(λ) := lim Ni(λ + i ) and Fi(λ) := lim Fi(λ + i ),i=1, 2, (3.9) →+0 →+0 exist in B2(H ) for a.e. λ ∈ R and the representation i 1/2 −1 1/2 F1(λ) −N1(λ)UF2(λ) |V | (L − λ − i0) |V | = ∗ (3.10) −F2(λ)U N1(λ) F2(λ) holds for a.e. λ ∈ R.
1/2 ∗ 1/2 Proof. By |G| ∈B2(H2)and|G | ∈B2(H1) the existence of the limits Ni(λ) in (3.9) for a.e. λ ∈ R follows from Proposition 3.13 of [5]. Using the representations ∗ 1/2 −1 ∗ 1/2 1/2 −1 1/2 F1(z)=|G | P1(L − z) H1 |G | and F2(z)=|G| P2(L − z) H2 |G| , z ∈ C\R, which follow from (3.6), and taking into account [5, Proposition 3.13] we again obtain the existence of Fi(λ), i =1, 2, for a.e. λ ∈ R. It is easy to see that |G∗|1/2 0 |V |1/2 =(V ∗V )1/4 = (3.11) 0 |G|1/2 holds. Let U be a partial isometry from H2 into H1 such that G = U|G|.Making use of the factorizations G = |G∗|1/2U|G|1/2 and G∗ = |G|1/2U ∗|G∗|1/2, (3.12) the block matrix representation of (L − z)−1 in Lemma 3.1 and relation (3.11) one verifies (3.10). We note that if U is a partial isometry such that G = U|G| holds and C is defined by 0 U C := , (3.13) U ∗ 0 then the operator V in (3.2) can be written in the form |V |1/2C|V |1/2, cf. (2.3). Let E1(·)andE2(·) be the spectral measures of H1 and H2, respectively. The operator function M0(·) from (2.5) here admits the representation M1(λ)0 M0(λ)= (3.14) 0 M2(λ) for a.e λ ∈ R,wherethederivatives ∗ 1/2 ∗ 1/2 1/2 1/2 |G | E1(dλ)|G | |G| E2(dλ)|G| M1(λ)= and M2(λ)= (3.15) dλ dλ exist in B1(H1)andB1(H2) for a.e. λ ∈ R, respectively. Setting Qj,λ := clo ran (Mj(λ)) ,j=1, 2, Block Operator Matrices and Scattering 41 and Qλ := Q1,λ ⊕ Q2,λ (3.16) for a.e. λ ∈ R we obtain the decomposition 2 2 2 L (R,dλ,Qλ)=L (R,dλ,Q1,λ) ⊕ L (R,dλ,Q2,λ), cf. Section 2. From (2.2) the conditions ∗ H1 =clospan{E1(∆)ran (|G |):∆∈ B(R)}, (3.17) H2 =clospan{E2(∆)ran (|G|):∆∈ B(R)} follow. Moreover, the converse is also true, that is, condition (3.17) implies (2.2). Hence, without loss of generality we assume that condition (3.17) is satisfied.
Therefore the reduced multiplication operators MQj , ∩ 2 R Q MQj := MHj dom (MHj ) L ( ,dλ, j,λ), where (MH f)(λ):=λf(λ),λ∈ R, j ∈ 2 R H ∈ 2 R H dom (MHj ):= f L ( ,dλ, j ):λf(λ) L ( ,dλ, j) . ac are unitary equivalent to the absolutely continuous parts Hj of the operators Hj, j =1, 2. With respect to the decomposition (3.16) the scattering matrix {S(λ)}λ∈R admits the decomposition Q1,λ Q1,λ S11(λ) S12(λ) S(λ)= : ⊕ −→ ⊕ (3.18) S21(λ) S22(λ) Q2,λ Q2,λ for a.e. λ ∈ R.Theentries{Sij (λ)}λ∈R, i, j =1, 2, are called channel scattering ma- trices. We note that the multiplication operators induced by the channel scattering matrices are unitary equivalent to the channel scattering operators Sij = PiSPj, i, j =1, 2, where Pi is the orthogonal projection in L onto the subspace Hj and S is the scattering operator of the complete scattering system {L, L0}. In the next proposition we give a more explicit description of the channel scattering matrices Sij (λ). The proof is an immediate consequence of Theorem 2.1, Lemma 3.2 and relations (3.14), (3.15) and (3.13).
Proposition 3.3. Let L0, V and L be given in accordance with (3.1), (3.2) and (3.3), respectively. Then the scattering matrix {S(λ)}λ∈R of the complete scattering system {L, L0} admits the representation (3.18) with entries Sij (λ) given by 1/2 ∗ 1/2 S11(λ)=IQ1,λ +2πiM1(λ) UF2(λ)U M1(λ) , 1/2 ∗ 1/2 S12(λ)=−2πiM1(λ) {U + UF2(λ)U N1(λ)U}M2(λ) , (3.19) 1/2 ∗ ∗ ∗ 1/2 S21(λ)=−2πiM2(λ) {U + U N1(λ)UF1(λ)U }M1(λ) , 1/2 ∗ 1/2 S22(λ)=IQ2,λ +2πiM2(λ) U F1(λ)UM2(λ) . for a.e. λ ∈ R. 42 J. Behrndt, H. Neidhardt and J. Rehberg
4. Dissipative channel scattering
In this section we consider the (dissipative) scattering system {H1(λ),H1} for a.e. λ ∈ R,where H1(λ)=H1 + V1(λ), (4.1) is defined for a.e. λ ∈ R,andH1 is the self-adjoint operator in H1 from (3.1). The limit V1(λ) = lim→+0 V1(λ + i ) (see Lemma 4.1) is called the optical potential of the channel H1. We recall that a linear operator T in a Hilbert space is said to be dissipative if Im (Tf,f) ≤ 0, f ∈ dom (T ), and T is called maximal dissipative if T is dissipative and does not admit a proper dissipative extension. In Theorem 4.4 below we establish a connection between the scattering matrices corresponding to the scattering systems {H1(λ),H1} and the channel scattering matrix S11(λ)from (3.18) and (3.19). −1 ∗ Lemma 4.1. Let V1(z)=−G(H2 − z) G , z ∈ C\R, be defined by (3.5) with G ∈B1(H2, H1). Then the limit V1(λ) = lim→+0 V1(λ + i ) exists in B1(H1) and V1(λ) is dissipative for a.e. λ ∈ R. Proof. Using the factorizations (3.12) of G and G∗ we find ∗ 1/2 ∗ ∗ 1/2 V1(z)=−|G | UN2(z)U |G | ,z∈ C\R, (4.2) where N2(z) is given by (3.7). According to Lemma 3.2 the limit lim→+0 N2(λ+i ) ∗ 1/2 exists in B2(H1) and since |G | ∈B2(H1) we conclude that the limit ∗ 1/2 ∗ ∗ 1/2 V1(λ) = lim V1(λ + i ) = lim −|G | UN2(λ + i )U |G | →+0 →+0 exists in B1(H1) for a.e. λ ∈ R. It is not difficult to see that Im V1(z) ≤ 0for + z ∈ C and therefore also the limit V1(λ) is dissipative for a.e. λ ∈ R.
It follows from Lemma 4.1 that for a.e. λ ∈ R the operator H1(λ)=H1+V1(λ) is maximal dissipative and therefore {H1(λ),H1} is a dissipative scattering system in the sense of [19, 20]. By Theorem 4.3 of [20] the corresponding wave operators ∗ D itH1(λ) −itH1 ac W+ (H1(λ),H1)=s − lim e e P (H1) t→+∞ and D −itH1(λ) itH1 ac W− (H1(λ),H1)=s − lim e e P (H1) t→+∞ exist and are complete which yields that {H1(λ),H1} performs a complete dissipa- tive scattering system for a.e. λ ∈ R, see [19] for details. The associated scattering operators are defined by D ∗ D SD[λ]:=W+ (H1(λ),H1) W− (H1(λ),H1) ac and act on the absolutely continuous subspaces H1 (H1). Since SD[λ]commutes with H1 the scattering operator is unitary equivalent to a multiplication operator 2 in the spectral representation L (R,dλ,Q1,µ)ofH1 induced by a family of con- tractions {SD[λ](µ)}µ∈R. The family {SD[λ](µ)}µ∈R is called the scattering matrix of the complete dissipative scattering system {H1(λ),H1}. Block Operator Matrices and Scattering 43
Using the fact that every maximal dissipative operator admits a self-adjoint dilation, i.e., there exists a self-adjoint operator in a (in general) larger Hilbert space such that its compressed resolvent coincides with the resolvents of the maxi- mal dissipative operator for all z ∈ C+, cf. [7, Section 7], see also [12], one concludes from Proposition 3.14 of [5] that the limit
F1[λ](µ) = lim F1[λ](µ + i ) →+0 exist in B1(H1) for a.e. µ ∈ R,where ∗ 1/2 −1 ∗ 1/2 F1[λ](z):=|G | (H1(λ) − z) |G | ,z∈ C+, is defined for a.e. λ ∈ R. The next proposition is a direct consequence of Theorem 2.2 of [16], see also [15].
Proposition 4.2. Let G ∈B1(H2, H1) and H1(λ) be given by (4.1). Then for a.e. λ ∈ R the scattering matrix {SD[λ](µ)}µ∈R of the complete dissipative scattering system {H1(λ),H1} admits the representation 1/2 ∗ ∗ 1/2 SD[λ](µ)=IQ1,µ +2πiM1(µ) U N2(λ)+N2(λ)U F1[λ](µ)UN2(λ) U M1(µ) for a.e. (µ, λ) ∈ R2 with respect to the Lebesgue measure in R2.
In the next lemma we show that the limit F1[λ](λ), ∗ 1/2 −1 ∗ 1/2 F1[λ](λ) = lim F1[λ](λ + i ) = lim |G | (H1(λ) − λ − i ) |G | , (4.3) →+0 →+0 exist in B2(H1) for a.e. λ ∈ R.
Lemma 4.3. Let L0, V and L be given by (3.1), (3.2) and (3.3), respectively, with G ∈B1(H2, H1).Further,letF1(λ) be as in Lemma 3.2 and let H1(λ) be defined by (4.1). Then the limit F1[λ](λ) in (4.3) exists in B2(H1) for a.e. λ ∈ R and the relation F1[λ](λ)=F1(λ) (4.4) holds for a.e. λ ∈ R. Proof. We have ∗ 1/2 −1 −1 ∗ 1/2 F1(z) − F1[λ](z)=|G | (H1(z) − z) − (H1(λ) − z) |G | (4.5) ∗ 1/2 −1 −1 ∗ 1/2 = |G | (H1(z) − z) (V1(λ) − V1(z))(H1(λ) − z) |G | . From (4.2) we obtain ∗ 1/2 ∗ ∗ 1/2 V1(λ) − V1(z)=|G | U N2(z) − N2(λ) U |G | and inserting this expression into (4.5) and using the definitions of F1(z) in (3.8) and F1[λ](z) yields ∗ F1(z) − F1[λ](z)=F1(z)U(N2(z) − N2(λ))U F1[λ](z). Hence ∗ F1(z)={IH1 + F1(z)U(N2(z) − N2(λ))U } F1[λ](z) 44 J. Behrndt, H. Neidhardt and J. Rehberg and for z = λ + i , >0 sufficiently small, the operator ∗ {IH1 + F1(z)U(N2(z) − N2(λ))U } is invertible. Therefore we conclude ∗ −1 {IH1 + F1(λ + i )U(N2(λ + i ) − N2(λ))U } F1(λ + i )=F1[λ](λ + i ).
From this representation we get the existence of F1[λ](λ)inB2(H1) and the equality (4.4) for a.e. λ ∈ R.
The next theorem is the main result of this section. We show how the channel scattering matrix S11(λ) of the scattering system {L, L0} is connected with the scattering matrices SD[λ](µ) of the dissipative scattering systems {H1(λ),H1}.
Theorem 4.4. Let {L, L0} be the scattering system from Section 3,whereL0, V and L are given by (3.1), (3.2) and (3.3), respectively, and G ∈B1(H2, H1).Fur- ther, let {Sij (λ)}, i, j =1, 2, be the corresponding scattering matrix from (3.18) and let SD[λ](µ) be the scattering matrices of the dissipative scattering systems {H1(λ),H1}. Then the scattering matrix SD[λ](λ) exists for a.e λ ∈ R and satis- fies the relation
SD[λ](λ)=S11(λ) for a.e. λ ∈ R.
Proof. From Proposition 4.2 and Lemma 4.3 we obtain that SD[λ](λ)existsfor a.e λ ∈ R and has the form 1/2 ∗ ∗ 1/2 SD[λ](µ)=IQ1,µ +2πiM1(µ) U N2(λ)+N2(λ)U F1(λ)UN2(λ) U M1(µ) (4.6) A similar calculation as in the proof of Lemma 3.1 shows ∗ F2(z)=N2(z)+N2(z)U F1(z)UN2(z),z∈ C+. If z tends to λ ∈ R,thenweget ∗ F2(λ)=N2(λ)+N2(λ)U F1(λ)UN2(λ) (4.7) for a.e. λ ∈ R. Inserting (4.7) into (4.6) we obtain
1/2 ∗ 1/2 SD[λ](λ)=IQ1,λ +2πiM1(λ) UF2(λ)U M1(λ) and by Proposition 3.3 this coincides with S11(λ) for a.e. λ ∈ R.
5. Lax-Phillips channel scattering
Similarly to Lemma 4.1 one verifies that V2(λ) = lim→+0 V2(λ + i )existsin B1(H2) for a.e. λ ∈ R. The limit V2(λ), which is called the optical potential of the channel H2, is dissipative for a.e. λ ∈ R. The optical potential defines the maximal dissipative operator H2(λ):=H2 + V2(λ) Block Operator Matrices and Scattering 45 for a.e. λ ∈ R. The operator H2(λ) decomposes for a.e. λ ∈ R into a self-adjoint part and a completely non-self-adjoint part. Let Θ2[λ](ξ), ξ ∈ C−, be the charac- teristic function, cf. [12], of the completely non-self-adjoint part of H2(λ). We are ∗ going to verify S11(λ)=Θ2[λ](λ) for a.e. λ ∈ R which shows that for a.e. λ ∈ R the scattering matrix S11(λ) can be regarded as the result of a certain Lax-Phillips scatteringtheory,cf.[1,2,3,4,14]. There is an orthogonal decomposition H Hcns ⊕ Hself 2 = 2,λ 2,λ ∈ R Hcns Hself for a.e. λ such that 2,λ and 2,λ reduce H2(λ) into a completely non-self- cns self adjoint operator H2 (λ) and a self-adjoint operator H2 (λ), cns self H2(λ)=H2 (λ) ⊕ H2 (λ). Taking into account Proposition 3.14 of [5] we get that 1/2 ∗ 1/2 m(V2(λ)) = −π|G| U M1(λ)U|G| for a.e. λ ∈ R. Let us introduce the operator ) 1/2 α(λ):= 2πM1(λ) U|G| . (5.1) Notice that clo{ran (α(λ))} = Q1,λ for a.e. λ ∈ R. With the completely non-self-adjoint part Hcns(λ) one associates the characteristic function Θ2[λ](·):Q1,λ −→ Q1,λ defined by − ∗ − −1 ∗ Θ2[λ](ξ):=IQ1,λ iα(λ)(H2(λ) ξ) α(λ) ,
ξ ∈ C−. The characteristic function is a contraction-valued holomorphic function in C−. From [12, Section V.2] we get that the boundary values
Θ2[λ](µ):=s − lim Θ2[λ](µ − i ) →+0 exist for a.e. µ ∈ R.
Theorem 5.1. Let L0, V and L be given by (3.1), (3.2) and (3.3). If the condition G ∈B1(H2, H1) is satisfied, then the limit Θ2[λ](λ),
Θ2[λ](λ):=s − lim Θ2[λ](λ − i ) →+0 exists for a.e. λ ∈ R and the relation ∗ S11(λ)=Θ2[λ](λ) holds for a.e. λ ∈ R. Proof. We set ∗ ∗ − −1 ∗ Θ2[λ](ξ):=Θ2[λ](ξ) = IQ1,λ + iα(λ)(H2(λ) ξ) α(λ)
ξ ∈ C+.Using(5.1)weget ) ) ∗ F ∗ Θ2[λ](ξ)=IQ1,λ +2πi M1(λ) U 2[λ](ξ)U M1(λ) 46 J. Behrndt, H. Neidhardt and J. Rehberg for a.e. λ ∈ R,where 1/2 −1 1/2 F2[λ](ξ):=|G| (H2(λ) − ξ) |G| ,ξ∈ C+.
Similar to the proof of Lemma 4.3 one verifies that the limit F2[λ](λ)
F2[λ](λ) = lim F2[λ](λ + i ) →+0 exist in B2(H2) for a.e. λ ∈ R and satisfies the relation F2[λ](λ)=F2(λ). Hence the ∗ ∗ limit Θ2[λ](λ)=s − lim→+0 Θ2[λ](λ − i ) exists for a.e. λ ∈ R and the relation ) ) ∗ F ∗ Θ2[λ](λ)=IQ1,λ +2πi M1(λ) U 2[λ](λ)U M1(λ) ∗ holds for a.e λ ∈ R. From (3.19) we obtain that S11(λ)=Θ2[λ](λ) for a.e. λ ∈ R. ∗ Since the limit Θ2[λ](λ) exists for a.e. λ ∈ R one concludes that
Θ2[λ](λ):=s − lim Θ2[λ](λ − i ) →+0 ∗ ∗ exists for a.e. λ ∈ R and Θ2[λ](λ) =Θ2[λ](λ) is valid. This completes the proof Theorem 5.1.
The last theorem admits an interpretation of the scattering matrix S11(λ) as the result of a Lax-Phillips scattering. Indeed, let us introduce the minimal self-adjoint dilation K2(λ) of the maximal dissipative operator H2(λ). We set
K2,λ = D−,λ ⊕ H2 ⊕ D+,λ, where 2 D±,λ := L (R±,dx,Q1,λ). Further, we define ⎛ ⎞ ⎛ ⎞ − d f− i dx f− ⎝ ⎠ ⎝ 1 ∗ ⎠ K2(λ) f := e(H2(λ))f − 2 α(λ) [f+(0) + f−(0)] d + − f i dx f+ for elements of the domain⎧⎛ ⎞ ⎫ ⎨ f− f ∈ dom (H2(λ)) ⎬ ⎝ ⎠ ∈ 1,2 R Q dom (K2(λ)) := ⎩ f : f± W ( ±,dx, 1,λ) ⎭ . f+ f+(0) − f−(0) = −iα(λ)f
The operator K2(λ) is self-adjoint and is a minimal self-adjoint dilation of the maximal dissipative operator H2(λ), that is,
−1 K2,λ −1 2 − 2 − H2 (H (λ) z) = PH2 (K (λ) z) for z ∈ C+ and K H ∈B R 2,λ =clospan EK2(λ)(∆) 2 :∆ ( ) , · D where EK2(λ)( ) is the spectral measure of K2(λ). It turns out that ±,λ are incoming and outgoing subspaces with respect to K2(λ), i.e.,
−itK2(λ) e D+,λ ⊆ D+,λ,t≥ 0, Block Operator Matrices and Scattering 47 and −itK2(λ) e D−,λ ⊆ D−,λ,t≤ 0. However, we remark that the completeness condition −itK2(λ) K2,λ =clospan e D±,λ : t ∈ R (5.2) is in general not satisfied. Condition (5.2) holds if and only if the maximal dissipa- tive operator H2(λ) is completely non-selfadjoint and H2 is singular, that means, ac the absolutely continuous part H2 of H2 is trivial. On the subspace Dλ, 2 Dλ = D−,λ ⊕ D+,λ = L (R,dx,Q1,λ) ⊆ K2,λ, let us define the operator K0(λ),
d 1,2 (K0(λ)g)(x):=−i g(x), dom (K0(λ)) := W (R,dx,Q1 ). dx ,λ The self-adjoint operator K0(λ) generates the shift group, i.e,
−itK0(λ) (e g)(x)=g(x − t),g∈ Dλ. 2 2 Using the Fourier transform F : L (R,dx,Q1,λ) −→ L (R,dµ,Q1,λ), 1 (Ff)(µ)=√ dx e−iµxf(x), 2π R the operator K0(λ) transforms into the multiplication operator on the Hilbert 2 space L (R,dµ,Q1,λ). Furthermore, one has
−itK2(λ) −itK0(λ) e D+,λ = e D+,λ,t≥ 0, and −itK2(λ) −itK0(λ) e D−,λ = e D−,λ,t≤ 0. The last properties yield the existence of the Lax-Phillips wave operators
LP itK2(λ) −itK0(λ) W± [λ]:=s − lim e J±(λ)e , t→±∞ cf. [5, 14] where J±(λ):D±,λ −→ K2,λ is the natural embedding operator. The Lax-Phillips scattering operator SLP (λ) is defined by LP ∗ LP SLP [λ]:=W+ [λ] W− [λ], 2 cf. [5, 14]. With respect to the spectral representation L (R,dµ,Q1,λ)theLax- ∗ Phillips scattering matrix {SLP [λ](µ)}µ∈R coincides with {Θ2[λ](µ) }µ∈R,see[1, 2, 3, 4]. Hence the scattering matrix {S11(λ)}λ∈R can be regarded as the result of a Lax-Phillips scattering for a.e. λ ∈ R.
Acknowledgement The support of the work by DFG, Grant 1480/2, is gratefully acknowledged. The authors thank Prof. P. Exner for discussion and literature hints. 48 J. Behrndt, H. Neidhardt and J. Rehberg
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Jussi Behrndt Technische Universit¨at Berlin Institut f¨ur Mathematik Straße des 17. Juni 136 D–10623 Berlin, Germany e-mail: [email protected] Hagen Neidhardt Weierstraß-Institut f¨ur Angewandte Analysis und Stochastik Mohrenstr. 39 D–10117 Berlin, Germany e-mail: [email protected] Joachim Rehberg Weierstraß-Institut f¨ur Angewandte Analysis und Stochastik Mohrenstr. 39 D–10117 Berlin, Germany e-mail: [email protected] Operator Theory: Advances and Applications, Vol. 175, 51–88 c 2007 Birkh¨auser Verlag Basel/Switzerland
Asymptotic Expansions of Generalized Nevanlinna Functions and their Spectral Properties
Vladimir Derkach, Seppo Hassi and Henk de Snoo
Abstract. Asymptotic expansions of generalized Nevanlinna functions Q are investigated by means of a factorization model involving a part of the gener- alized zeros and poles of nonpositive type of the function Q. The main results in this paper arise from the explicit construction of maximal Jordan chains in the root subspace R∞(SF ) of the so-called generalized Friedrichs extension. A classification of maximal Jordan chains is introduced and studied in ana- lytical terms by establishing the connections to the appropriate asymptotic expansions. This approach results in various new analytic characterizations of the spectral properties of selfadjoint relations in Pontryagin spaces and, con- versely, translates analytic and asymptotic properties of generalized Nevan- linna functions into the spectral theoretical properties of self-adjoint relations in Pontryagin spaces. Mathematics Subject Classification (2000). Primary 46C20, 47A06, 47B50; Sec- ondary 47A10, 47A11, 47B25. Keywords. Generalized Nevanlinna function, asymptotic expansion, Pontrya- gin space, symmetric operator, selfadjoint extension, operator model, factor- ization, generalized Friedrichs extension.
1. Introduction
Let Nκ be the class of generalized Nevanlinna functions, i.e., meromorphic func- tions on C \ R with Q(¯z)=Q(z) and such that the kernel Q(z) − Q(λ) N (z,λ)= ,z,λ∈ ρ(Q),z= λ,¯ Q z − λ¯
The research was supported by the Academy of Finland (project 212150) and the Research Institute for Technology at the University of Vaasa. 52 V.A. Derkach, S. Hassi and H.S.V. de Snoo has κ negative squares on the domain of holomorphy ρ(Q)ofQ,see[20].Ifthe function Q ∈ Nκ belongs to the subclass Nκ,−2n, n ∈ N, (see [6]) then it admits the following asymptotic expansion 2-n+1 s −1 1 Q(z)=γ − j + o ,z.→∞, (1.1) zj z2n+1 j=1 where γ,sj ∈ R and z.→∞ means that z tends to ∞ nontangentially (0 <ε< arg z<π− ε). Asymptotic expansions for Q ∈ Nκ of the form (1.1) (with γ =0) were introduced in [21]. They naturally appear, for instance, in the indefinite mo- ment problem considered in [22]. The expansion (1.1) is equivalent to the following operator representation of the function Q ∈ Nκ,−2n: Q(z)=γ +[(A − z)−1ω,ω], (1.2) where ω ∈ dom An and A is a selfadjoint operator in a Pontryagin space H;see [21, Satz 1.10] and Corollary 3.4 below. The representation (1.2) can be taken to be minimal in the sense that ω is a cyclic vector for A, i.e., H = span { (A − z)−1ω : z ∈ ρ(A) }, in which case the negative index sq−(H)ofH is equal to κ. The representation (1.2) shows that ∞ is a generalized zero of the function Q(z) − γ,orequivalently, that ∞ is a generalized pole of the function Q∞(z)=−1/(Q(z) − γ). This means that the underlying symmetric operator S is nondensely defined in H with dom S = { f ∈ dom A :[f,ω]=0} (1.3) and that SF = S+({0}×span {ω}) (1.4) is a selfadjoint extensions of S in H with ∞∈σp(SF ). Here + stands for the componentwise sum in the Cartesian product H×H. In other words, the extension SF is multivalued and, in fact, can be interpreted as the generalized Friedrichs extension of S, see [5] and the references therein. It follows from (1.1) and (1.2) that s0 =[ω,ω] ∈ R.
If κ>0 then it is possible that s0 ≤ 0, in which case ∞ is a generalized pole of nonpositive type (GPNT) of the function Q∞, cf. [23]. More precisely, if ∞ is a GPNT of Q∞ with multiplicity κ∞ := κ∞(Q∞) (see (2.2) below for the definition), then in (1.1) one automatically has
s0 = ···= sj =0, for every j<2κ∞ − 2.
Furthermore, if m is the first nonnegative index in (1.1) such that sm =0(ifit exists), then, equivalently, the function Q∞ admits an asymptotic expansion of the form m+1 2+1 2+1 Q∞(z)=pm+1z + ···+ p2+1z + o z ,z.→∞, (1.5) Asymptotic expansions 53
where pm+1 =1/sm, pi ∈ R, i = m+1,...,2+1, and the integers m, n,and are connected by = m − n with m ≥ 2; see Theorem 5.4 below for further details. It turns out that (1.5) holds for some ≤ 0 if and only if ∞ is a regular critical point of SF , or equivalently, if and only if the corresponding root subspace R { ∈ H { }∈ k ∈ N } ∞(SF )= h : 0,h SF for some k of the generalized Friedrichs extensions SF in (1.4) is nondegenerate. In this case the GPNT ∞ of Q∞ as well as the corresponding root subspace R∞(SF )are shortly called regular. On the other hand, if ∞ is a singular critical point of SF , then in (1.5) >0 and, moreover, the minimal integer such that the expansion 0 (1.5) exists coincides with the dimension κ∞ of the isotropic subspace of the root subspace R∞(SF ), see Theorem 5.6. In this case the GPNT ∞ of Q∞ and the corresponding root subspace R∞(SF ) are shortly called singular with the index of 0 singularity κ∞. The above-mentioned results reflect the close connections between the asymptotic expansions (1.1), (1.5), and the root subspace R∞(SF )ofSF .The given assertions are examples of the results in the present paper which have been derived by means of the factorization model of the function Q∞ recently con- structed by the authors in [9]. This model is based on the following “proper” factorization of the function Q∞ ∈ Nκ: Q∞(z)=q(z)q (z)Q0(z), (1.6)
where q is a (monic) polynomial, q (z)=q(¯z), and Q0 ∈ Nκ such that κ∞(Q0)=0 and κ = κ − deg q, see Lemma 4.3 below. Such a factorization for Q∞ is in general not unique, but the factorization model based on such a factorization carries the complete information about the root subspace R∞(SF )ofSF . A major part of the results presented in this paper is associated with the structure of the root subspace R∞(SF )ofSF in a model space and the various connections to the asymptotic expansions (1.1) and (1.5). By using the factoriza- tion model based on a proper factorization (1.6) of Q∞ maximal Jordan chains in R∞(SF ) are constructed in explicit terms. Their construction leads to three dif- ferent types of maximal Jordan chains in R∞(SF ). Each of these three types of maximal Jordan chains admits its own characteristic features, reflecting various properties of the root subspace R∞(SF ). The construction shows explicitly, for instance, when the root subspace R∞(SF ) is regular and when it is singular. The length of the maximal Jordan chain as well as the signature of the root subspace R∞(SF ) can be easily read off from their construction. In the case that the root subspace R∞(SF ) is regular, the three types of maximal Jordan chain can be char- acterized by their length. The first type of maximal Jordan chain is of length 2k+1, where k =degq = κ∞(Q∞), and the second and third type of maximal Jordan chains are of length 2k and 2k−1, respectively. The classification of these maximal Jordan chains remains the same in the case when the root subspace R∞(SF )is 54 V.A. Derkach, S. Hassi and H.S.V. de Snoo
0 singular. In that case the index of singularity κ∞ as introduced above enters to the formulas, while the difference κ−(R∞(SF )) − κ+(R∞(SF )) of the negative and the positive index of R∞(SF ) remains unaltered, see Theorem 4.12. All of these facts can be translated into the analytical properties of the functions Q∞ and Q = γ − 1/Q∞ via the asymptotic expansions (1.1) and (1.5), and conversely. The classification of maximal Jordan chains in R∞(SF ) motivates an analo- gous classification of generalized zeros and poles of nonpositive type of the function Q ∈ Nκ, which turns out to be connected with the characterization of the mul- tiplicities of GZNT and GPNT of the function Q due to H. Langer in [24]; see Subsection 3.2 for the definitions of generalized zeros and poles of types (T1)– (T3). This induces a classification for the asymptotic expansions for the functions Q and Q∞; see Theorems 5.3 and 5.4. Some further characterizations of the three different types of generalized zeros and poles are obtained by means of the fac- torized integral representations of the functions Q and Q∞, which are based on their canonical factorizations, see [11]; for definitions, see Subsection 2.1, cf. also [5]. In particular, Theorem 6.1 and Theorem 6.3 extend some earlier results by the authors in [6] (where κ = 1) and in [8], from the regular case to the singular case 0 in an explicit manner involving the index of singularity κ∞, which is characterized in Theorem 5.6 below. The construction of the maximal Jordan chains in R∞(SF )usingthefac- torization model for Q∞ in (1.6) is carried out in Section 4. The most careful treatment of the model is required in the construction of maximal Jordan chains which are of the third type (T3). The reason is that the factorization of Q∞ does not produce a minimal model for the function Q∞ directly. In the minimal fac- torization model the maximal Jordan chains of type (T3) are roughly speaking the shortest ones, cf. (4.21), (4.24), (4.28); see also Theorem 5.3 and Theorem 6.3. The results in Lemma 4.11 and part (iii) of Theorem 4.12 characterize maximal Jordan chains of type (T3). In this case the underlying symmetric relation S(Q)is multivalued (before the auxiliary part of the space is factored out). This statement is true more generally: for an arbitrary Nκ-function Q the occurrence of general- ized zeros and poles of type (T3) in R ∪{∞}is an indication that point spectrum σp(S(Q)) of S(Q) is nonempty, see Lemma 6.4 below, which by part (i) of Theo- rem 4.6 is equivalent to S(Q) being not simple. In fact, the existence of maximal Jordan chains of type (T3) or, equivalently, the existence of GZNT and GPNT of type (T3) can be used to give criteria for minimality of various factorization models for Nκ-functions, see Propositions 6.6 and 6.7 below. The topics considered in this paper have connections to some other recent studies involving asymptotic expansions of Nκ-functions, see in particular [6], [8], [9], [12], [13], [15], and their canonical factorization, see, e.g., [3], [5], [7], [11], [14]. For instance, in [13] the authors investigate the subclass of Nκ-functions with κ = κ∞(Q) and extend some results, e.g., from [6], [8]. General operator models based on the canonical factorization of Nκ-functions have been introduced in [3]; for another model not using the canonical factorization of Q, see [18]. The construction of a minimal canonical factorization model by using reproducing kernel Pontryagin Asymptotic expansions 55 space methods has been recently worked out in [12], cf. also [3, Theorem 4.1]. Some of the results in the present paper can be naturally augmented by the results which can be found from [15], where characteristic properties of the generalized zeros and poles of Nκ-functions have been studied with the aid of their operator representations. The present paper forms a continuation of the paper [9], where the details concerning the construction of the announced factorization model can be found. Some basic definitions and concepts which will be used throughout the paper are given in Section 2. In Section 3 some additions concerning the subclasses Nκ,− as introduced in [6] are given, including a proof for [6, Proposition 6.2] as announced in that paper, cf. Theorem 3.3 below; see also Theorem 5.4 for an extension of these results. Asymptotic expansions are introduced in Section 3 and a classification of generalized zeros and poles is given. In Section 4 the main ingredients concerning the factorization model are given and the construction of maximal Jordan chains in R∞(SF ) is carried out. The connection between the properties of the root subspace R∞(SF ) and the asymptotic expansions of the form (1.1) and (1.5) is investigated in Section 5. Finally, in Section 6 the classification of GZNT and GPNT is connected with factorized integral representations of the functions Q and Q∞(z). In this section also the generalized zeros and poles of nonpositive type of Nκ-functions which belong to R are briefly treated and some consequences as announced above are established.
2. Preliminaries
2.1. Canonical factorization of Q ∈ Nκ The notions of generalized poles and generalized zeros of nonpositive type were introduced in [23]. The following definitions are based on [24]. A point α ∈ R is called a generalized pole of nonpositive type (GPNT) of the function Q ∈ Nκ with multiplicity κα (= κα(Q)) if −∞ < lim (z − α)2κα+1Q(z) ≤ 0, 0 < lim (z − α)2κα−1Q(z) ≤∞. (2.1) z.→α z.→α Similarly, the point ∞ is called a generalized pole of nonpositive type (GPNT) of Q with multiplicity κ∞ (= κ∞(Q)) if
≤ Q(z) ∞ −∞ ≤ Q(z) 0 lim 2 +1 < , lim 2 −1 < 0. (2.2) z.→∞ z κ∞ z.→∞ z κ∞ Apointβ ∈ R is called a generalized zero of nonpositive type (GZNT) of the function Q ∈ Nκ if β is a generalized pole of nonpositive type of the function −1/Q. The multiplicity πβ (= πβ(Q)) of the GZNT β of Q can be characterized by the inequalities:
Q(z) ≤∞ −∞ Q(z) ≤ 0 < lim 2 +1 , < lim 2 −1 0. (2.3) z.→β (z − β) πβ z.→β (z − β) πβ 56 V.A. Derkach, S. Hassi and H.S.V. de Snoo
Similarly, the point ∞ is called a generalized zero of nonpositive type (GZNT) of Q with multiplicity π∞ (= π∞(Q)) if −∞ ≤ lim z2π∞+1Q(z) < 0, 0 ≤ lim z2π∞−1Q(z) < ∞. (2.4) z.→∞ z.→∞
It was shown in [23] that for Q ∈ Nκ the total number (counting multiplic- ities) of poles (zeros) in C+ and generalized poles (zeros) of nonpositive type in R ∪{∞} is equal to κ.Letα1,...,αl (β1,...,βm) be all the generalized poles (zeros) of nonpositive type in R and the poles (zeros) in C+ with multiplicities κ1,...,κl (π1,...,πm). Then the function Q admits a canonical factorization of the form p Q(z)=r(z)r (z)Q00(z),Q00 ∈ N0,r= , (2.5) q / / m − πj l − κj where p(z)= j=1(z βj) and q(z)= j=1(z αj ) are relatively prime polynomials of degree κ−π∞(Q)andκ−κ∞(Q), respectively; see [11], [5]. It follows from (2.5) that the function Q admits the (factorized) integral representation 1 t p − Q(z)=r(z)r (z) a + bz + 2 dρ(t) ,r= , (2.6) R t − z 1+t q where a ∈ R, b ≥ 0, and ρ(t) is a nondecreasing function satisfying the integrability condition dρ(t) ∞ 2 < . (2.7) R t +1
2.2. The subclasses Nκ,1 and Nκ,0
A function Q ∈ Nκ is said to belong to the subclass Nκ,1,if Q (z) ∞ |Im Q (iy) | lim =0and dy < ∞, .→∞ z z η y with η>0 large enough. Similarly Q ∈ Nκ is said to belong to the subclass Nκ,0,if Q(z) lim = 0 and lim sup |z Im Q(z)| < ∞, z.→∞ z z.→∞ see [5]. In the following theorems the subclasses Nκ,1 and Nκ,0 are characterized both in terms of the integral representation (2.6) and in terms of operator repre- sentations of the form (1.2). Let Et be a spectral function of a selfadjoint operator A in a Pontryagin space H0, see [1]. Denote by H := H(A), ∈ N,thesetofall ∈ H | | ∞ ±∞ elements h such that ∆ t d[Eth, h] < for some neighborhood ∆ of . Moreover, let H−(A), ∈ N, be the corresponding dual spaces. Here, for instance, H−1(A) can be identified as the set of all generalized0 elements obtained by com- −1 pleting H with respect to the inner product ∆(1 + |t|) d[Eth, h] < ∞ with some neighborhood ∆ of ±∞. The operator A admits a natural continuation A from H into H−1, see [5] for further details. The classes Nκ,1 and Nκ,0 are characterized in the following two theorems, see [5]. Asymptotic expansions 57
Theorem 2.1. ([5]) For Q ∈ Nκ the following statements are equivalent:
(i) Q belongs to Nκ,1; (ii) Q(z)=γ +[(A − z)−1ω,ω], z ∈ ρ(A), for some selfadjoint operator A in a Pontryagin space H, a cyclic vector ω ∈ H−1,andγ ∈ R; (iii) Q has the integral representation (2.6) with deg q − deg p = π∞(Q) > 0,or with deg p =degq (π∞(Q)=0), b =0,and (1 + |t|)−1dρ(t) < ∞. (2.8) R
Theorem 2.2. ([5]) For Q ∈ Nκ the following statements are equivalent:
(i) Q belongs to Nκ,0; (ii) Q (z)=γ + O (1/z), z.→∞; (iii) Q(z)=γ +[(A − z)−1ω,ω], z ∈ ρ(A), for some selfadjoint operator A in a Pontryagin space H, a cyclic vector ω ∈ H,andγ ∈ R; (iv) Q has the integral representation (2.6) with deg q − deg p = π∞(Q) > 0,or with deg p =degq (π∞(Q)=0), b =0,and dρ(t) < ∞. (2.9) R
Remark 2.3. If Q ∈ Nκ,0, then the operator representation of Q in part (iii) of Theorem 2.2 implies that lim −z(Q(z) − γ)=[ω,ω]. z.→∞ Hence, the statement (ii) in Theorem 2.2 can be strengthened in the sense that for every function Q ∈ N 0 there are real numbers γ and s0, such that κ, s0 1 Q (z)=γ − + o ,z.→∞. (2.10) z z
3. Asymptotic expansions of generalized Nevanlinna functions Asymptotic expansions of generalized Nevanlinna functions (as in (2.10)) can be used for studying operator and spectral theoretical properties of selfadjoint exten- sions of symmetric operators in Pontryagin and Hilbert spaces, see [20], [16]. In this section a subdivision of the class Nκ of generalized Nevanlinna functions is given along the lines of [16], [6]. Moreover, a classification for generalized zeros of nonpositive type is introduced and interpreted via asymptotic expansions.
3.1. The subclasses Nκ,− of generalized Nevanlinna functions
Definition 3.1. A function Q ∈ Nκ is said to belong to the subclass Nκ,−2n, n ∈ N, if there are real numbers γ and s0,...,s2 −1 such that the function ⎛ n ⎞ -2n s −1 Q(z)=z2n ⎝Q(z) − γ + j ⎠ (3.1) zj j=1 58 V.A. Derkach, S. Hassi and H.S.V. de Snoo
is O(1/z)asz.→∞.Moreover,Q ∈ Nκ is said to belong to the subclass Nκ,−2n+1 if the function Q in (3.1) belongs to Nκ ,1 for some κ ∈ N.
The next lemma clarifies the above definition of the subclasses Nκ,−, ∈ N.
Lemma 3.2. If the function Q belongs to the subclass Nκ,−2n (Nκ,−2n+1) for some n ∈ N, then the function Q in (3.1) belongs to the subclass Nκ ,0 (resp. Nκ ,1) with κ ≤ κ. Moreover, the following inclusions are satisfied
···⊂Nκ,−2n−1 ⊂ Nκ,−2n ⊂ Nκ,−2n+1 ⊂···⊂Nκ,0 ⊂ Nκ,1. (3.2)
Proof. Rewrite the expression for the function Q in (3.1) in the form Q(z)= z2nQ(z). Then Q(z) as a sum of two generalized Nevanlinna functions is also a generalized Nevanlinna function, and therefore, in view of (2.5), Q is a generalized Nevanlinna function, too. Next it is shown that the inequality κ(Q) ≤ κ(Q)is satisfied. First observe that the condition Q(z)=O(1) and hence, in particular, the condition Q(z)=O(1/z)asz.→∞ implies that κ∞(Q)=0, (3.3) cf. (2.2), (2.4). Clearly, κα(Q)=κα(Q) for every α =0 , ∞, while for α =0one derives from (2.1) the estimate κ0(Q) ≤ κ0(Q). (3.4)
Therefore, one can conclude from (3.3) and (3.4) that κ(Q) ≤ κ(Q). Now by Theorem 2.2 the condition Q(z)=O(1/z), z.→∞, is equivalent to Q ∈ Nκ ,0 with κ ≤ κ, which proves the first statement for the subclasses Nκ,−2n.IfQ ∈ Nκ,−2n+1,thenQ(z)=O(1) and since κ(Q) ≤ κ(Q), one actually has Q ∈ Nκ ,1 for κ ≤ κ. Since Nκ,0 ⊂ Nκ,1 the inclusions Nκ,−2n ⊂ Nκ,−2n+1, n ∈ N, follow from thefirstpartofthelemma.NowletQ ∈ Nκ,−2n−1. Then by definition 2 z Q(z)+zs2n + s2n+1 ∈ Nκ ,1, (3.5) where Q is as in (3.1) and κ ≤ κ. It is clear from (3.5) (see Theorem 2.1) that Q(z)=O(1/z)asz.→∞. Hence, Q ∈ Nκ,−2n and this proves the remaining inclusions in (3.2).
The subclasses Nκ,−, ∈ N, are now characterized by means of the operator and the integral representation of Q in (1.2) and (2.6), respectively.
Theorem 3.3. For Q ∈ Nκ the following statements are equivalent:
(i) Q ∈ Nκ,−, ∈ N; (ii) Q(z)=γ +[(A − z)−1ω,ω], z ∈ ρ(A), for some selfadjoint operator A in a Pontryagin space H, a cyclic vector ω ∈ H,andγ ∈ R; Asymptotic expansions 59
(iii) Q has an integral representation (2.6) with π∞(Q)=degq− deg p ≥ 0 (and b =0if π∞(Q)=0), such that (1 + |t|)−2π∞ dρ(t) < ∞. (3.6) R
Proof. (i) ⇒ (iii) Let Q ∈ Nκ,−,where is either 2n or 2n − 1, n ∈ N.Inviewof (3.2) and Theorems 2.1, 2.2 one has π∞(Q)=degq− deg p ≥ 0, and if π∞(Q)=0 then b = 0 and (2.8) or (2.9) is satisfied. By Lemma 3.2 the function Q in (3.1) belongs to Nκ ,2n− with κ ≤ κ. Hence, Q admits the factorization 1 t p2 − Q = r(z)r (z) a + bz + 2 dρ(t), r = , (3.7) R t − z 1+t q2 where p2 and q2 are the polynomials associated to Q, cf. (2.5). Moreover, the inequality π∞(Q)=degq2 − deg p2 ≥ 0 holds by Theorems 2.1 and 2.2. On the other hand, it follows from (2.6) and (3.1) that Q admits also the representation 1 t 2n − Q(z)=z r(z)r (z) a + bz + 2 dρ(t)+p1(z), (3.8) R t − z 1+t where p1 is a polynomial with deg p1 ≤ 2n. An application of the generalized Stieltjes inversion formula (see [19]) shows that the measures dρ(t) in (3.7) and dρ(t) in (3.8) are connected by |r(t)|2dρ(t)=t2n|r(t)|2 dρ(t). (3.9) Therefore, if Q ∈ Nκ ,1 \ Nκ ,0 so that =2n − 1, then deg p2 =degq2 and dρ(t) satisfies the condition (2.8) in Theorem 2.1. The condition (3.6) follows now from (3.9). If Q ∈ Nκ ,0 so that =2n, then either deg p2 =degq2 in which case dρ(t) satisfies the condition (2.9) in Theorem 2.2, or π∞(Q)=degq2 − deg p2 > 0in which case dρ(t) satisfies the condition (2.7). In both cases |r(t)|2dρ(t) < ∞ for M>0 large enough. |t|>M Hence, again the condition (3.6) follows from (3.9).
(ii) ⇔ (iii) Let Et be the spectral function of a selfadjoint operator A in the minimal representation (1.2) of Q. It follows from (1.2), (2.6), and the generalized Stieltjes inversion formula that 2 d[Etω,ω]=|r(t)| dρ(t),t∈ ∆, in some neighborhood ∆ of ±∞. This implies that
−2π∞ (1 + |t|) dρ(t) < ∞ if and only if (1 + |t|) d[Etω,ω] < ∞, R ∆ i.e., ω ∈ H, which proves the equivalence of (ii) and (iii). 60 V.A. Derkach, S. Hassi and H.S.V. de Snoo
n (ii) ⇒ (i) First consider the case =2n.Thenω ∈ H means that ω ∈ dom A . Define the function Q in (3.1) by setting j j n sj =[A ω,ω],sn+j =[A ω,A ω],j=0,...,n. (3.10) Then a straightforward calculation shows that Q admits the operator representa- tion Q(z)=[(A − z)−1ω ,ω ],ω = Anω ∈ H. (3.11) Therefore, Q(z)=O(1/z)andQ ∈ Nκ,−. n Now let =2n − 1. Then ω ∈ H means that ω := A ω ∈ H−1. Hence s2n−1 := [Aω ,ω ] is well defined. Moreover, by defining s0,...,s2n−2 as in (3.10) it follows that the function Q in (3.1) admits the operator representation −1 n Q(z)=[(A − z) ω ,ω ],ω= A ω ∈ H−1. (3.12) Hence, by Theorem 2.1 Q ∈ Nκ ,1 for some κ ∈ N and thus Q ∈ Nκ,−.This completes the proof. Inthecaseofevenindices =2n the equivalence of (i) and (ii) in Theorem 3.3 coincides with the following result of M.G. Kre˘ın and H. Langer, see [21, Satz 1.10].
Corollary 3.4. ([21]) The function Q ∈ Nκ admits an operator representation −1 n Q(z)=γ +[(A − z) ω,ω] with γ ∈ R and ω ∈ H2n (= dom A ) if and only if there are real numbers γ and s0,...,s2n, such that 2-n+1 s −1 1 Q(z)=γ − j + o ,z.→∞. (3.13) zj z2n+1 j=1
In this case the numbers s0,...,s2n are given by (3.10). Proof. The proof of Theorem 3.3 shows that the condition ω ∈ dom An is equiv- alent to the operator representation (3.11) of the function Q(z) in (3.1). Now by applying (2.10) in Remark 2.3 to the function Q(z) in (3.11) and taking into account (3.1) the equivalence to the expansion (3.13) follows. The criterion of M.G. Kre˘ın and H. Langer formulated in Corollary 3.4 does not hold in the case of an odd index =2n − 1. However, it is clear that if ω ∈ H2n−1 then the analog of the expansion (3.13) exists.
Corollary 3.5. If the function Q ∈ Nκ admits an operator representation Q(z)= −1 n−1/2 γ +[(A − z) ω,ω] with γ ∈ R and ω ∈ H2n−1 (= dom |A| ) then there are real numbers γ and s0,...,s2n, such that -2n s −1 1 Q(z)=γ − j + o ,z.→∞. (3.14) zj z2n j=1
Proof. Since ω ∈ H2n−1 the operator representation (3.12) in the proof of Theo- rem 3.3 shows that Q(z)=o(1). The expansion (3.14) for the function Q follows now from the definition of Q in (3.1). Asymptotic expansions 61
It is emphasized that the existence of the expansion (3.14) does not imply n that ω ∈ H2n−1 or, equivalently, that ω := A ω belongs to H−1.Inthiscase [Aω ,ω ] need not be defined and hence it cannot coincide with the coefficient s2n−1 in (3.14). 3.2. A classification of generalized zeros of nonpositive type For what follows it will be useful to give a classification for generalized zeros and poles of nonpositive type of a function Q ∈ Nκ. Let ∞ be a GZNT of Q with multiplicity π∞ > 0. It follows from (2.4) that precisely one of the following three cases can occur: − 2π∞+1 2π∞−1 (T1) s2π∞ := limz.→∞ z Q(z) < 0, limz.→∞ z Q(z)=0; 2π∞+1 2π∞−1 (T2) limz.→∞ z Q(z)=∞, limz.→∞ z Q(z)=0; 2π∞+1 ∞ − 2π∞−1 (T3) limz.→∞ z Q(z)= , s2π∞−2 := limz.→∞ z Q(z) > 0. In these cases ∞ is said to be a generalized zero of type (T1), (T2), or (T3), respec- tively; the shorter notations GZNT1, GZNT2, and GZNT3 are used accordingly. The corresponding classification for a finite generalized zero β ∈ R of Q is defined analogously: Q(z) Q(z) (T1) limz.→β > 0, limz.→β =0; (z − β)2πβ +1 (z − β)2πβ −1 Q(z) Q(z) (T2) limz.→β = ∞, limz.→β =0; (z − β)2πβ +1 (z − β)2πβ −1 Q(z) Q(z) (T3) limz.→β = ∞, limz.→β < 0. (z − β)2πβ +1 (z − β)2πβ −1 A generalized pole of nonpositive type β ∈ R ∪{∞}of Q is said to be of type (T1), (T2), or (T3), if β is a generalized zero of nonpositive type of the function −1/Q which is of type (T1), (T2), or (T3), respectively. To give some immediate implications of the above classification consider the generalized zero ∞ of Q. If it is of the first type, then it follows from (T1) that ∈ N Q κ,−2π∞ .Moreover,Q has the following asymptotic expansion: s2 1 − π∞ .→∞ Q(z)= + o ,z ,s2π∞ > 0. (3.15) z2π∞+1 z2π∞+1 ∞ ∈ N If the generalized zero of Q is of type (T3), then Q κ,−2π∞−2 and Q has the following asymptotic expansion s2 −2 1 − π∞ .→∞ Q(z)= + o ,z ,s2π∞−2 < 0. (3.16) z2π∞−1 z2π∞−1 In the case that the generalized zero ∞ is of type (T2) there are two possibilities: N either Q belongs to κ,−2π∞ , in which case both of the moments s2π∞−1 and s2π∞ are finite and Q has the asymptotic expansion s2 −1 s2 1 − π∞ − π∞ .→∞ Q(z)= + o ,z ,s2π∞−1 =0, (3.17) z2π∞ z2π∞+1 z2π∞+1 62 V.A. Derkach, S. Hassi and H.S.V. de Snoo
N \ N or Q belongs to κ,−2(π∞−1) κ,−2π∞ and it has the asymptotic expansion s2 −1 1 Q(z)=− π∞ + o ,z.→∞, (3.18) z2π∞ z2π∞ or 1 Q(z)=o ,z.→∞. (3.19) z2π∞−1 Observe, that the expansions (3.17) and (3.18) are also special cases of the ex- pansion (3.19). Hence, if ∞ is a generalized zero of type (T2), then Q has an expansion of the form (3.16), but now with s2π∞−2 = 0; however, Q does not have an expansion of the form (3.15). Similar observations remain true for generalized zeros β ∈ R and poles α ∈ R ∪{∞}. For instance, to get the analogous expansions for a generalized zero β ∈ R apply the transform −Q(1/(z − β)) to the expansions in (3.15)–(3.19); cf. also [15]. The role of the above classification for generalized zeros and poles of nonpositive type will be described in detail in Sections 4–6.
4. An operator model for the generalized Friedrichs extension 4.1. Boundary triplets and Weyl functions The construction of the model uses the notion of a boundary triplet in a Pontryagin space setting. Let H be a Pontryagin space with negative index κ,letS be a closed symmetric relation in H with defect numbers (n, n), and let S∗ be the adjoint of S. n ∗ AtripletΠ={C , Γ0, Γ1 } is said to be a boundary triplet for S , if the following two conditions are satisfied: ∗ n n (i) the mapping Γ : f →{Γ0f,Γ1f} from S to C ⊕ C is surjective; (ii) the abstract Green’s identity [f ,g] − [f,g ]=(Γ1f,Γ0g) − (Γ0f,Γ1g) (4.1) holds for all f = {f,f }, g = {g,g }∈S∗, see, e.g., [2], [10]. It is easily seen that A0 =kerΓ0 and A1 =kerΓ1 are selfadjoint extensions of S. Associated to every boundary triplet there is the Weyl function Q defined by Q(z)Γ0fz =Γ1fz,z∈ ρ(A0), ∗ where fz := {fz,zfz}∈Nz,andNz =ker(S − z) denotes the defect subspace of S at z ∈ C. It follows from (4.1) that the Weyl function Q is also a Q-function of the pair {S, A0} in the sense of Kre˘ın and Langer, see [20]. If S is simple,sothat
H = span {Nz : z ∈ ρ(A0)}, then the Weyl function Q belongs to the class Nκ,otherwiseQ ∈ Nκ with κ ≤ κ. Moreover, if S is simple and H is a selfadjoint extension of S in H, then the point spectrum of H is also simple, that is, every eigenspace of H is one-dimensional, and if α ∈ R ∪{∞}, then the root subspace at α is at most 2κ + 1-dimensional. Asymptotic expansions 63
In the case where S is given by (1.3) one can define a boundary triplet for S∗ as follows. Proposition 4.1. (cf. [5]) Let A be a selfadjoint operator in a Pontryagin space H and let the restriction S of A be defined by (1.3) with ω ∈ H. Then the adjoint S∗ of S in H is of the form S∗ = {{f,Af + cω} : f ∈ dom A, c ∈ C } ∞ ∞ ∞ ∗ and a boundary triplet Π = {C, Γ0 , Γ1 } for S is determined by ∞ ∞ ∗ Γ0 f =[f,ω], Γ1 f = c, f = {f,Af + cω}∈S .
The corresponding Weyl function Q∞ is given by 1 Q∞(z)=− ,z∈ ρ(A). (4.2) [(A − z)−1ω,ω]
4.2. The model operator S(Q∞) corresponding to a proper factorization Operator models for generalized Nevanlinna functions whose only generalized pole of nonpositive type is at ∞ have been constructed in [6] and [14]. Such functions admit a canonical factorization of the form Q∞(z)=q(z)q (z)Q0(z), (4.3) k k−1 where Q0 ∈ N0, q(z)=z + qk−1z + ···+ q0 is a polynomial, and q (z)= q(¯z). In general, models which are based on the canonical factorization of Q ∈ Nκ are not necessarily minimal, i.e., the underlying model operator S(Q∞) need not be simple and it can even be a symmetric relation (multivalued operator). However, with the canonical factorization the nonsimple part of S(Q∞)canbe easily identified and factored out to produce a simple symmetric operator from S(Q∞), cf. [3]. The model constructed for S(Q∞) in [6] uses an orthogonal coupling of two symmetric operators. In [9] this model was adapted to the case where the function Q0 is allowed to be a generalized Nevanlinna function, too. In this case the situation becomes more involved and, in general, one cannot represent S(Q∞) as an orthogonal sum of a simple symmetric operator and a selfadjoint relation. However, such a simple orthogonal decomposition for S(Q∞) can still be obtained if the factorization (4.3) of Q∞ is proper. This concept is defined as follows.
Definition 4.2. ([9]) The factorization Q∞(z)=q(z)q (z)Q0(z)issaidtobeproper if q is a divisor of degree κ∞(Q∞) > 0 of the polynomial q in the canonical factorization of the function Q∞,cf.(2.5).
Clearly, proper factorizations of Q∞ ∈ Nκ always exist, but they are not unique if q has more than one zero and κ∞(Q∞) <κ. Proper factorizations Q∞ = qq Q0 can be characterized also without using the canonical factorization of Q∞.
Lemma 4.3. Let Q∞ ∈ Nκ have a factorization of the form Q∞(z)=q(z)q (z)Q0(z), deg q = k ≥ 1, (4.4) 64 V.A. Derkach, S. Hassi and H.S.V. de Snoo where q(z) is a monic polynomial, and let α ∈ σ(q) beazeroofq with multiplicity kα. Then the following statements are equivalent:
(i) the factorization (4.4) of Q∞ is proper; (ii) the multiplicities κ∞(Q∞) and πα(Q∞) satisfy the following relations:
κ∞(Q∞)=degq and πα(Q∞) ≥ kα for all α ∈ σ(q); (4.5)
(iii) κ∞(Q0) and κ(Q∞)=κ satisfy the following identities
κ∞(Q0)=0and κ(Q∞)=degq + κ(Q0). (4.6)
Proof. (i) ⇔ (ii) In a proper factorization (4.4) κ∞(Q∞)=degq and clearly the inequalities in (4.5) just mean that q divides the polynomial q in the canonical factorization of Q∞. (i) ⇒ (iii) If the factorization (4.4) is proper, then in the canonical factor- ization of the function Q0 the numerator q0 and denominator p0 (= p)ofthe corresponding rational factor r0 are of the same degree κ(Q0), and this implies (4.6). (iii) ⇒ (i) It follows from the second equality in (4.6) that q and the poly- nomial p0 in the canonical factorization of the function Q0 are relatively prime and, therefore, q is a factor of the polynomial q in the canonical factorization of Q∞.Moreover,π∞(Q0) = 0. Now the assumption κ∞(Q0) = 0 implies that κ∞(Q∞)=degq.
The construction of factorization models is now briefly described. Let q be a polynomial as in (4.4) of degree k =degq. Define the k × k matrices Bq and Cq by ⎛ ⎞ ⎛ ⎞ q1 ... qk−1 1 01... 0 ⎜ ⎟ ⎜ . . ⎟ ⎜ . . . ⎟ ⎜ . .. 10⎟ ⎜ . .. .. 0 ⎟ Bq = ⎜ ⎟ , Cq = ⎜ ⎟ , ⎝ . . .⎠ ⎝ 00... 1 ⎠ q −1 .. .. . k − − − 10... 0 q0 q1 ... qk−1 so that σ(Cq)=σ(q). Moreover, let Hq be a 2k-dimensional Pontryagin space defined by 0 B (Ck ⊕ Ck, B·, ·), B = q . Bq 0
A general factorization model for functions Q∞ of the form (4.4) was constructed in [9] and can be applied, in particular, for proper factorizations of Q∞.
Theorem 4.4. (cf. [9]) Let Q∞ ∈ Nκ be a generalized Nevanlinna function and let Q∞(λ)=q(λ)q (λ)Q0(λ), (4.7) be a proper factorization of Q∞,whereq is a monic polynomial of degree k = deg q ≥ 1.LetS0 be a closed symmetric relation in a Pontryagin space H0 with 0 0 0 the boundary triplet Π = {C, Γ0, Γ1} whose Weyl function is Q0. Asymptotic expansions 65
Then:
(i) the function Q0 in (4.7) belongs to the class Nκ−k; (ii) the linear relation ⎧ ⎧⎛ ⎞ ⎛ ⎞⎫ ⎫ ⎪ ∗ ⎪ ⎨ ⎨ f0 f0 ⎬ f0 = {f0,f0}∈S0 , ⎬ 0 ⎝ f ⎠ ⎝ C f ⎠ S(Q∞)=⎪ ⎩ , q ⎭ : f1 =Γ1f0, ⎪ (4.8) ⎩ 0 ⎭ f Cqf +Γ0f0ek f1 =0
is closed and symmetric in H := H0 ⊕ Hq and has defect numbers (1, 1); ∗ (iii) the adjoint S(Q∞) of S(Q∞) is given by ⎧ ⎧⎛ ⎞ ⎛ ⎞⎫ ⎫ ∗ ⎨ ⎨ f0 f0 ⎬ f0 = {f0,f0}∈S0 , ⎬ ∗ ⎝ ⎠ ⎝ ⎠ 0 S(Q∞) = f , Cq f + ϕek : f1 =Γ1f0, ; ⎩ ⎩ 0 ⎭ ⎭ f Cqf +Γ0f0ek ϕ ∈ C ∗ (iv) a boundary triplet Π={C, Γ0, Γ1} for S(Q∞) is determined by ∗ Γ0(f0 ⊕ F )=f1, Γ1(f0 ⊕ F )=ϕ, f0 ⊕ F ∈ S(Q∞) ; (4.9)
(v) the corresponding Weyl function coincides with Q∞. Proof. Since the factorization (4.7) is proper the statement (i) is immediate from the equality (4.6) in Lemma 4.3. All the other statements are contained in [9].
In fact, the statement (iv) in Theorem 4.4 can be obtained directly also from Proposition 4.1, since S(Q∞) in (4.8) is a restriction of the selfadjoint relation ⎧ ⎧⎛ ⎞ ⎛ ⎞⎫ ⎫ ⎨ ⎨ f0 f0 ⎬ ∗ ⎬ f0 = {f0,f0}∈S0 , ⎝ ⎠ ⎝ ⎠ A(Q∞)= f , Cq f : 0 (4.10) ⎩ ⎩ 0 ⎭ f1 =Γ1f0 ⎭ f Cqf +Γ0f0ek to the subspace H ω0,whereω0 =col(0,ek, 0); compare (1.3). The generalized Friedrichs extension of S(Q∞)isgivenby ⎧ ⎧⎛ ⎞ ⎛ ⎞⎫ ⎫ ⎪ ∗ ⎪ ⎨ ⎨ f0 f0 ⎬ f0 = {f0,f0}∈S0 , ⎬ 0 ⎝ f ⎠ ⎝ C f + ϕe ⎠ SF (Q∞)=⎪ ⎩ , q k ⎭ : f1 =Γ1f0, ⎪ . (4.11) ⎩ 0 ⎭ f Cqf +Γ0f0ek f1 =0, ϕ ∈ C
According to (4.2) in Proposition 4.1 the Weyl function Q∞(z) corresponding to the boundary triplet (4.9) is of the form
− 1 Q∞(z)= −1 . [(A(Q∞) − z) ω0,ω0]
Thus the function Q = −1/Q∞ has the representation −1 Q(z)=[(A(Q∞) − z) ω,ω], (4.12) which is, however, not necessarily minimal, since mul S and ker (S − α), α ∈ σ(q), can be nontrivial. The following lemma describes these subspaces. 66 V.A. Derkach, S. Hassi and H.S.V. de Snoo
Lemma 4.5. ([9]) Under the assumptions of Theorem 4.4 let S0 be a simple closed H 0 0 ⊃ symmetric operator in the Pontryagin space 0 and let Ai =kerΓi ( S0), i =0, 1. Then: 0 (i) mul S(Q∞) is nontrivial if and only if mul A1 is nontrivial and in this case
0 0 mul S(Q∞)={ (g,0, Γ0ge k) : g = {0,g}∈A1 }; (4.13)
(ii) if mul S(Q∞) is nontrivial, then it is spanned by a positive vector; 0 (iii) if mul A0 is nontrivial, then it is spanned by a positive vector; 0 0 (iv) σp(S(Q∞)) = σp(A0) ∩ σ(q ) and for α ∈ σp(A0) ∩ σ(q ) one has
0 0 ker (S(Q∞) − α)={ (g0, Γ1g0Λ|λ=α, 0) : g0 ∈ ker (A0 − α) }, (4.14)
where Λ=(1,λ,...,λk−1), λ ∈ C; 0 (v) if ker (S(Q∞) − α) or, equivalently, ker (A0 − α) is nontrivial, then it is spanned by a positive vector.
It follows from (ii) and (iv) that the linear relation S(Q∞) can be decomposed into a direct sum of an operator S with an empty point spectrum and a selfadjoint part in a Hilbert space which is the sum of mul S(Q∞)andker(S(Q∞) − α), 0 α ∈ σp(A0) ∩ σ(q ). The next theorem shows that the reduced operator S is simple.
Theorem 4.6. Let the assumptions of Theorem 4.4 and Lemma 4.5 be satisfied and let S(Q∞), A(Q∞),andSF (Q∞) be given by (4.8), (4.10),and(4.11), respectively. Then:
(i) S(Q∞) is simple if and only if σp(S(Q∞)) = ∅. In this case the linear rela- tions S = S(Q∞), A = A(Q∞),andSF = SF (Q∞) satisfy the equalities (1.3) and (1.4) with ω = ω0 and the operator representation (4.12) of Q = −1/Q∞ is minimal. (ii) If S(Q∞) is not simple, then the subspace
0 H =span{ mul S(Q∞), ker (S(Q∞) − α): α ∈ σp(A0) ∩ σ(q) } (4.15)
is positive and reducing for S(Q∞). The simple part of S(Q∞) coincides with the restriction S of S(Q∞) to H := H H . The compressions S , A ,and H SF of S(Q∞), A(Q∞),andSF (Q∞) to the subspace satisfy the equalities (1.3) and (1.4),withω ∈ H given by ω0, if k>1, ω = (g,−1/Γ0g, Γ0g) , if k =1,
and the function Q = −1/Q∞ admits the minimal representation
−1 Q(λ)=−1/Q∞(λ)=[(A − λ) ω,ω]. Asymptotic expansions 67
4.3. The root subspace of the generalized Friedrichs extension at ∞ Let the assumptions of Theorem 4.6 be satisfied and let the linear relations S(Q∞), A(Q∞), SF (Q∞)andS , A , SF be as defined in that theorem. Since S is a simple H symmetric operator in the Pontryagin space its selfadjoint extension SF has a simple point spectrum. In particular, the multivalued part mul SF of SF is at most one-dimensional. The corresponding root subspace R { ∈ H { }∈ k ∈ N } ∞(SF )=span g : 0,g SF , for some k R is at most 2κ+1-dimensional. The root subspace ∞(SF ) is spanned by the vectors R ωj which form a maximal Jordan chain in ∞(SF ): R { ∈ H { }∈ − } ∞(SF )=span ωj : ωj−1,ωj SF ,j=0,...,ν 1 , R where ν =dim ∞(SF )andω−1 =0. R The main properties of the root subspace ∞(SF ) of the selfadjoint extension SF of S are given in Lemmas 4.7–4.11 below. The proofs of these lemmas are constructive, since they are based on the factorization model for S(Q∞) given in Theorem 4.4, cf. also [9]. As a consequence a detailed description for the structure of the corresponding Jordan chains in the factorization model is obtained. It turns out that three different types of maximal Jordan chains can appear in this model. In each of these cases maximal Jordan chains in R∞(SF ) can be regular or singular and the signature of R∞(SF ) has its own specific nature in each case. The first lemma concerns the dimension and nondegeneracy of the root sub- R space ∞(SF ); further equivalent conditions and descriptions (without a specific model space) for arbitrary generalized poles and zeros of Q ∈ Nκ with direct (nonconstructive) proofs can be found from [15]. ∈ N Lemma 4.7. Let Q∞ κ,letκ∞(Q∞)=degq(= k) > 0,andletS , SF ,and ω ∈ H, [ω,ω] ≤ 0, be as in Theorem 4.6.Then: R ≥ (i) dim ∞(SF ) deg q; R ∈ ν−1 \ ν (ii) dim ∞(SF ) is equal to ν if and only if ω dom SF dom SF ; R (iii) ∞(SF ) is a regular subspace of dimension ν if and only if ∈ ν−1 \ ν { }∈ ν−1 ω dom SF dom SF and [ω,ων−1] =0for ω,ων−1 SF ; R (iv) ∞(SF ) is a singular subspace of dimension ν if and only if ∈ ν−1 \ ν { }∈ ν−1 ω dom SF dom SF and [ω,ων−1]=0for ω,ων−1 SF . (4.16)
Proof. (i) Consider a proper factorization (4.7) for Q∞ and the model operator S(Q∞) constructed in Theorem 4.4 and denote by A0 = ker Γ0 the selfadjoint extension S (Q∞) in (4.11) of S(Q∞) in (4.8). Clearly, the vectors F ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 0 0 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ w0 = ek ,...,wk−2 = e2 ,wk−1 = e1 (4.17) 0 0 0 form a Jordan chain in R∞(A0), that is, w0 ∈ mul A0 and {wj−1,wj}∈A0 for − j =1,...,k 1. If S(Q∞) is simple then A0 = SF and this proves (i). In the case 68 V.A. Derkach, S. Hassi and H.S.V. de Snoo
when S(Q∞) is not simple, but mul S(Q∞) is trivial the vectors w0,...,wk−1 still R ∈ 0 ∩ belong to ∞(SF ) since for all α σp(A0) σ(q ) these vectors are orthogonal to the eigenspaces ker (S(Q∞) − α) which were described in (4.14). Assume that mul S(Q∞) is not trivial. Again it follows from Lemma 4.5 that the vectors w0,...,wk−1 are orthogonal to ker (S(Q∞) − α) for all α ∈ 0 σp(A0) ∩ σ(q ). Moreover, by using (4.13) in Lemma 4.5 it is seen that the vectors w0,...,wk−2 are orthogonal to mul S(Q∞). Since mul S(Q∞)[⊥]ker (S(Q∞) − α) ∈ 0 ∩ for all α σp(A0) σ(q ), it is easy to check that the projections wj = P wj of wj, j =1,...,k− 1, to the subspace H = H H ,whereH is given by (4.15), take the form ⎛ ⎞ 0 −(Γ0g)g ⎝ ⎠ w0 = w0,...,wk−2 = wk−2,wk−1 = e1 , (4.18) 0 2 −|Γ0g| ek 0 0 0 where g ∈ mul A1,[g,g]=1,g = {0,g}∈A1,andΓ0g = 0. Therefore, the sequence { } R R ≥ w0,...,wk−1 forms a Jordan chain in ∞(SF ) and hence again dim ∞(SF ) k. (ii) This statement holds true since SF has a simple spectrum. R { }∈ (iii)&(iv) Let ω0,...,ων−1 be a maximal chain in ∞(SF ). Since 0,ω0 SF one obtains immediately [ω0,ωj] = 0 for all j<ν− 1. Therefore, the subspace R ∞(SF ) is regular if and only if [ω0,ων−1] = 0. This proves (iii) and (iv).
The explicit construction of a maximal Jordan chain {ω0,...,ων−1} span- R ning ∞(SF ) in the factorization model will be carried out by a continuation of the chains (4.17) and (4.18), which correspond to the cases mul S(Q∞)={0} and mul S(Q∞) = {0}, respectively. Observe, that by Lemma 4.5 mul S(Q∞)isnon- 0 trivial if and only if mul A1 is nontrivial. Moreover, since S0 is simple at most one 0 0 of the selfadjoint extensions A0 or A1 of S0 can be multivalued.
Case I: mul S(Q∞)={0}. The proof of Lemma 4.7 shows that {ω0,...,ων−1} can be constructed as a continuation of the chain (4.17) if ν>kwith ωj = wj , j =0,...,k− 1. The formula (4.11) implies that the vector ωk should be of the form ωk =(g0,f,f) ,where ∗ 0 0 g0 = {0,g0}∈S0 , Γ1g0 =1,f= Cq e1 + ϕe k, f =Γ0g0ek.
By choosing ϕe k = −Cq e1 one obtains ⎛ ⎞ g0 ⎝ ⎠ ∗ 0 ωk = 0 , g0 = {0,g0}∈S0 , Γ1g0 =1. (4.19) 0 Γ0g0ek
Here g0 =0ifandonlyif ωk = 0. Therefore, one can continue the chain (4.17) if ∗ and only if mul S0 is nontrivial. This proves the following ∗ { } R Lemma 4.8. If mul S0 = 0 ,then ∞(SF ) is isotropic and R R dim ∞(SF )=κ0( ∞(SF )) = k. Asymptotic expansions 69
∗ Now assume that mul S0 is nontrivial, so that one can continue the chain (4.17) by a vector ωk = 0 of the form (4.19). Here two further different cases can 0 0 ∗ 0 occur: either mul A0 is nontrivial, in which case mul A0 =mulS0 ,ormulA0 = {0}. These two cases give rise to two different types of maximal Jordan chains in R ∞(SF ). 0 R Lemma 4.9. Let mul A0 be nontrivial. Then ∞(SF ) is regular if and only if ν = R ≤ ≤ 2k +1.If ∞(SF ) is singular then k +1 ν 2k and R − R − − R − κ0( ∞(SF )) = 2k +1 ν, κ−( ∞(SF )) = ν k 1,κ+( ∞(SF )) = ν k. (4.20)
0 0 Proof. Since mul A0 = {0} one has Γ0g0 = 0. It follows from (4.11) and (4.19) that the chain {ωk,...,ων−1} can be taken to be of the form ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ g0 g1 gν−k−1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ωk = 0 ,ωk+1 = 0 ,...,ων−1 = 0 , (4.21) 0 f(1) f(ν−k−1) 0 with {gj−1,gj}∈A1 and -j−2 f(j) = aek−j+1 + cj,iek−i,cj,i ∈ C,j=1,...,ν− k − 1, i=0 where 0 a := Γ0{g0,g1} =[ωk−1,ωk+1]=[ωk,ωk]=[g0,g0] > 0, 0 since the vector g0 ∈ mul A0 is positive in view of Lemma 4.5. There are two 0 reasons for the chain to break: either gν−k−1 ∈ dom A1, or one has (f(ν−k−1))1 =0 in which case ν =2k +1.Since[ων−1,ω0]=(fν−k−1)1, the subspace R∞(A0)is regular if and only if ν =2k +1. Observe, that the maximal chain {ω0,...,ων−1} in R∞(SF )isalsoamax- R ∈ R − imal chain in ∞(SF ), because the vectors ωj ∞(SF ), j =0,...,ν 1, are 0 orthogonal to the eigenspaces ker (S(Q∞) − α)forα ∈ σp(A0) ∩ σ(q). R ≤ ≤ R If ∞(SF ) is singular then k+1 ν 2k. The isotropic subspace of ∞(SF ) is spanned by the vectors ωj, j =0,...,2k − ν,and
[ω2k−ν+1,ων−1]=[ωk,ωk]=[g0,g0]=a>0.
Therefore, the Gram matrix of the chain ω0,...,ων−1 is a Hankel matrix of the form ⎛ ⎞ 0 a 00 ⎝ . ⎠ G = with G0 = .. , (4.22) 0 G0 a ∗ where the left upper corner of G is the matrix 02k+1−ν . The other two equalities in (4.20) are implied by the structure of G0 in (4.22), since a>0. 70 V.A. Derkach, S. Hassi and H.S.V. de Snoo
0 0 { } ∗ { } R Lemma 4.10. Let mul A0 =mulA1 = 0 and let mul S0 = 0 .Then ∞(SF ) is R ≤ ≤ − regular if and only if ν =2k.If ∞(SF ) is singular then k +1 ν 2k 1 and R − R R − κ0( ∞(SF )) = 2k ν, κ−( ∞(SF )) = κ+( ∞(SF )) = ν k. (4.23) 0 Proof. By assumptions Γ0g0 = 0. In this case the chain {ωk,...,ων−1} takes the form ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ g0 g1 gν−k−1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ωk = 0 ,ωk+1 = 0 ,...,ων−1 = 0 , (4.24) aek f(1) f(ν−k−1) 0 0 where a := Γ0g0 =0, {gj−1,gj}∈A1,andf(j) are given by -j−1 f(j) = aek−j + cj,iek−i,cj,i ∈ C,j=1,...,ν− k − 1. (4.25) i=0 R As in Lemma 4.9 it is seen from (4.24) that the subspace ∞(SF ) is regular if and only if ν =2k. R ≤ ≤ − If ∞(SF ) is singular then k +1 ν 2k 1. The isotropic subspace of R − − ∞(SF ) is spanned by the vectors ωj, j =0,...,2k 1 ν,and
[ω2k−ν ,ων−1]=[ωk,ωk−1]=a =0 .
Therefore, the Gram matrix of the chain ω0,...,ων−1 takes the form (4.22), where the left upper corner of G is the matrix 02k−ν . The equalities in (4.23) are implied by the structure of G.
Case II: mul S(Q∞) = {0}. The proof of Lemma 4.7 shows that a maximal Jordan { } R chain ω0,...,ων−1 in ∞(SF ) can now be constructed as a continuation of the − chain (4.18) with ωj = wj , j =0,...,k 1. { } R Lemma 4.11. Let mul S(Q∞) = 0 .Then ∞(SF ) is regular if and only if ν = − R ≤ ≤ − 2k 1.If ∞(SF ) is singular then 1 0 0 2 Proof. If k =1thenω0 =(−(Γ0g)g,1, −|Γ0g| ) and it follows from (4.11) that the chain (4.18) cannot be continued, in which case ν = k =1and[ω0,ω0]= 0 2 −|Γ0g| < 0. Now let k>1 and consider the continuation of the chain (4.18). According to (4.11) the condition {ωk−1,ωk}∈A0 for some ωk means that for some vector g1 one has 0 ∗ 0 h := {−(Γ0g)g,g1}∈S0 , Γ1h =1, (4.27) 0 0 where g = {0,g}∈A1,[g,g]=1,Γ0g = 0. Observe, that here the conditions 0 Γ1h =1and[g,g] = 1 are equivalent, since by Green’s identity (4.1) 0 0 0 0 0 0 0 −Γ0g[g,g]=Γ1gΓ0h − Γ0gΓ1h = −Γ0gΓ1h. Asymptotic expansions 71 0 0 Observe also, that g ∈ dom A1,sinceg ∈ mul A1 and the vector g is positive. Moreover, it follows from (4.11) that the continuation of the chain (4.18) can be taken to be of the form ⎛ ⎞ ⎛ ⎞ g1 gν−k ⎝ ⎠ ⎝ ⎠ ωk = 0 ,...,ων−1 = 0 , (4.28) f(1) f(ν−k) 0 where {gj−1,gj}∈A1 for j =2,...,ν− k and f(j), j =1,...,ν− k,aregivenby (4.25), where 0 2 a := −|Γ0g| =[ωk−1,ωk−1] < 0. By (4.13) the vector ωj is orthogonal to mul S(Q∞) for all j = k,...,ν − 2. In addition, one can select gν−k in (4.28) such that [gν−k,g]=0andthenalso ων−1 is orthogonal to mul S(Q∞). Moreover, the vectors ωj, j = k,...,ν− 1, are 0 orthogonal to the eigenspaces ker (S(Q∞) − α), α ∈ σp(A0) ∩ σ(q). Hence (4.18) R together with (4.28) forms a maximal Jordan chain in ∞(SF ). Moreover, the R − subspace ∞(SF ) is regular if and only if ν =2k 1. R ≤ ≤ − If ∞(SF ) is singular then k>1andk ν 2k 2, see (4.27). The isotropic R − − subspace of ∞(SF ) is spanned by the vectors ωj, j =0,...,2k ν 2, and [ω2k−ν−1,ων−1]=[ωk−1,ωk−1]=a<0. Therefore, the Gram matrix of the chain ω0,...,ων−1 is a Hankel matrix of the form (4.22), where the left upper corner of G is the matrix 02k−1−ν and a<0. This proves the equalities (4.26). The above considerations show that three different types of maximal Jordan R chains in ∞(SF ) can occur, cf. (4.21), (4.24), and (4.28). The longest Jordan 0 chains appear in the first case, where mul A0 = {0}, see (4.21), while the shortest 0 Jordan chains appear in the third case, where mul A1 = {0},orequivalently, mul S(Q∞) = {0}, see (4.28). The main properties associated with each of these maximal Jordan chains in Lemmas 4.8–4.11 are collected in the next theorem. ∈ N R Theorem 4.12. Let Q∞ κ, κ∞(Q∞)=k>0, ν =dim( ∞(SF )),andletS , SF be as in Theorem 4.6. Then one of the following three cases occurs: 0 { } ≤ ≤ R R − (i) If mul A0 = 0 ,thenk +1 ν 2k +1, κ−( ∞(SF )) = κ+( ∞(SF )) 1, R − R and κ0( ∞(SF )) = 2k +1 ν. Moreover, ∞(SF ) is singular if and only if ν ≤ 2k. 0 0 { } ≤ ≤ R R (ii) If mul A0 =mulA1 = 0 ,thenk ν 2k, κ−( ∞(SF )) = κ+( ∞(SF )), R − R and κ0( ∞(SF )) = 2k ν. Moreover, ∞(SF ) is singular if and only if ν ≤ 2k − 1. 0 { } ≤ ≤ − R R (iii) If mul A1 = 0 ,thenk ν 2k 1, κ−( ∞(SF )) = κ+( ∞(SF )) + 1, R − − R and κ0( ∞(SF )) = 2k 1 ν. Moreover, ∞(SF ) is singular if and only if ν ≤ 2k − 2 and k>1. Proof. It is enough to prove the statement (ii). For this one combines the results ∗ in Lemma 4.8 and Lemma 4.10. Indeed, if mul S0 = {0} then by Lemma 4.8 one R ≤ − R − has ν =dim ∞(SF )=k 2k 1andκ0( ∞(SF )) = k =2k ν. 72 V.A. Derkach, S. Hassi and H.S.V. de Snoo 5. Analytic characterizations of the root subspace at ∞ Let Q∞ ∈ Nκ have a minimal operator representation (1.2) with γ =0,whereA is a selfadjoint operator in H, and let S and SF be defined by (1.3) and (1.4), re- spectively. In Section 4 one such minimal operator representation was constructed in Theorem 4.6 by using the factorization (4.2) of Q∞ in Theorem 4.4. By unitary equivalence general statements concerning S, A,andSF in a minimal representa- tion of the function Q∞ can be obtained by considering the corresponding objects S , A ,andSF in the model of Theorem 4.6. In this section the results concerning the root subspace of the generalized Friedrichs extension SF in Section 4 are con- nected with the asymptotic expansions in Section 3. In particular, the connection between the classification of generalized poles and zeros introduced in Subsec- tion 3.2 and the three different types of maximal Jordan chains in Section 4 is explained. 5.1. The root subspace of the generalized Friedrichs extension at ∞ and operator representations In order to establish the connection between the Jordan chains in the root subspace R∞(SF ) of the generalized Friedrichs extension SF and the asymptotic expansions in Section 3 the following lemma will be useful. Lemma 5.1. Let A be a selfadjoint operator in a Pontryagin H,letω ∈ H,andlet S and SF be defined by (1.3) and (1.4).Then: (i) dom Sn = { f ∈ dom An :[Ajf,ω]=0for all j [ωi,ωj]=0 for all i + j Proof. (i) If f ∈ dom Sn, then the definition of S in (1.3) shows that f ∈ dom An and [Aj f,ω] = 0 for all j ωj = Sωj−1 + cjω for some cj ∈ C,j=1,...,n. (5.3) ∈ n ∈ ∈ 2 Hence, if ω dom SF ,thenωj dom S and ωj−1 dom S for all j (iii) It follows from (5.1) and (5.3) with l ≥ 1that [ω0,ω1]=[ω0,Sω0]. i j Now, if [S ω0,S ω0]=[ωi,ωj] = 0 for all i+j The statements (iii) and (iv) in Lemma 4.7 can now be reformulated in terms j of the moments sj =[A ω,ω] of the operator A, cf. also [15]. Proposition 5.2. Let S, SF , ω ∈ H,and[ω,ω] ≤ 0,beasin(1.3) and (1.4),ina minimal representation of Q∞ ∈ Nκ with κ∞(Q∞)=k>0.Then: (i) R∞(SF ) is a regular subspace of dimension ν if and only if ν−1 ω ∈ dom A , and sν−1 =0 ,sj =0for all j<ν− 1; (5.4) (ii) R∞(SF ) is a singular subspace of dimension ν if and only if ν−1 ν ω ∈ dom A \ dom A , and sj =0for all j ≤ ν − 1. (5.5) Proof. (i) Assume that R∞(SF ) is a regular subspace of dimension ν.Thenby ∈ ν−1 \ ν ∈ ν−1 ⊂ ν−1 Lemma 4.7 ω dom SF dom SF and by Lemma 5.1 ω dom S dom A . j j This implies that sj =[A ω,ω]=[S ω,ω]=0forj<ν− 1. Moreover, if { }∈ ν−1 ω,ων−1 SF then ν−1 ν−1 sν−1 =[A ω,ω]=[S ω,ω]=[ων−1,ω] =0 , so that (5.4) follows. ∈ ν−1 Conversely, if (5.4) holds, then ω dom SF by Lemma 5.1. Moreover, for { }∈ ν−1 ω,ων−1 SF one has ν−1 [ων−1,ω]=[A ω,ω]=sν−1 =0 , ∈ ν R so that ω dom SF . Hence, ∞(SF ) is a regular subspace of dimension ν. (ii) Assume that R∞(SF ) is a singular subspace of dimension ν.Thenit follows from Lemma 4.7 and Lemma 5.1 that ω ∈ dom Sν−1 ⊂ dom Aν−1 and [Ajω,ω]=[Sjω,ω] = 0 for all j ≤ ν − 1. If ω ∈ dom Aν , then by Lemma 5.1 ∈ ν ∈ ν ω dom SF , a contradiction to (4.16). Thus, ω dom A and (5.5) follows. ∈ ν−1 \ ν Conversely, assume that (5.5) holds. Then one has ω dom SF dom SF by Lemma 5.1. Moreover, by (5.2) one has ν−1 [ων−1,ω]=[A ω,ω]=sν−1 =0. Hence, (4.16) holds and R∞(SF ) is a singular subspace of dimension ν. 74 V.A. Derkach, S. Hassi and H.S.V. de Snoo 5.2. Asymptotic expansions and the classification for the generalized zero (pole) ∞ in the model space The classification of generalized poles of Q∞ or, equivalently, of the generalized zeros of Q = −1/Q∞ in Subsection 3.2 is now connected with the maximal Jordan chains constructed in Section 4. For this purpose observe that the assumptions of Theorem 4.12 can be expressed in terms of the Weyl function Q0(λ)ofS0 in the following equivalent form: 0 Q0(z) mul A0 = {0}⇔ lim =0, (5.6) z.→∞ z 0 mul A1 = {0}⇔ lim zQ0(z)=∞. (5.7) z.→∞ Now one can reformulate Theorem 4.12 in the form which makes clear the con- nection with the classification of generalized zeros and poles of nonpositive type introduced in Section 3. Theorem 5.3. Let Q ∈ Nκ have a minimal representation (1.2) and let S and SF be defined by (1.3) and (1.4).Let∞ be a generalized zero of negative type of Q with multiplicity π∞(Q)=k>0 and let the root subspace R∞(SF ) be of dimension ν. Then R∞(SF ) is regular if and only if Q has an asymptotic expansion of the form sν−1 s2ν−2 1 Q(z)=− −···− + o ,z.→∞,s−1 =0 . (5.8) zν z2ν−1 z2ν−1 ν Moreover, precisely one of the following three cases occurs: (i) If ∞ is a GZNT1, then k+1 ≤ ν ≤ 2k+1 and Q has the asymptotic expansion s2 s2 −2 1 Q(z)=− k −···− ν + o ,z.→∞, (5.9) z2k+1 z2ν−1 z2ν−1 where s2k > 0. In this case κ−(R∞(SF )) = κ+(R∞(SF )) − 1, κ0(R∞(SF )) = 2k +1− ν,andR∞(SF ) is singular if and only if ν ≤ 2k. (ii) If ∞ is a GZNT2, then k ≤ ν ≤ 2k and for ν ≥ k +1 Q has the asymptotic expansion s2 −1 s2 −2 1 Q(z)=− k −···− ν + o ,z.→∞, (5.10) z2k z2ν−1 z2ν−1 where s2k−1 =0 ,andforν = kQhas the asymptotic expansion 1 Q(z)=o ,z.→∞. (5.11) z2ν−1 In this case κ−(R∞(SF )) = κ+(R∞(SF )), κ0(R∞(SF )) = 2k−ν,andR∞(SF ) is singular if and only if ν ≤ 2k − 1. (iii) If ∞ is a GZNT3, then k ≤ ν ≤ 2k − 1 and Q has the asymptotic expansion s2 −2 s2 −2 1 Q(z)=− k −···− ν + o ,z.→∞, (5.12) z2k−1 z2ν−1 z2ν−1 Asymptotic expansions 75 where s2k−2 < 0. In this case κ−(R∞(SF )) = κ+(R∞(SF ))+1, κ0(R∞(SF )) = 2k − 1 − ν,andR∞(SF ) is singular if and only if ν ≤ 2k − 2 and k>1. Proof. Since the root subspace R∞(SF )isofdimensionν it follows from Proposi- ν−1 tion 5.2 that ω ∈ dom A . By Theorem 3.3 this means that Q ∈ Nκ,−2(ν−1).Now Corollary 3.4 and (5.4) show that in the regular case the asymptotic expansion is of the form (5.8). If the root subspace R∞(SF ) is singular then Proposition 5.2 and Theorem 3.3 ν−1 ν yield ω ∈ dom A \ dom A and Q ∈ Nκ,−2(ν−1) \ Nκ,−2ν . Now consider the classification given in Subsection 3.1. (i) Assume that ∞ is a GZNT1 of Q. Then by (T1) the following limit exists: lim z2k+1Q(z) < 0. z.→∞ It follows from the factorization (4.7) that Q0(z) Q∞(z) − 1 lim = lim 2 +1 = lim 2 +1 > 0. z.→∞ z z.→∞ z k z.→∞ z k Q(z) 0 According to (5.6) this means that mul A0 = {0}. The asymptotic expansion (5.9) is implied by (3.15) and Corollary 3.4. The remaining statements are obtained from part (i) of Theorem 4.12. (ii) Assume that ∞ is a GZNT2 of Q. Then by (T2) one has lim z2k+1Q(z)=∞, lim z2k−1Q(z)=0. z.→∞ z.→∞ It follows from the factorization (4.7) that Q0(z) 1 1 − − ∞ lim = lim 2 +1 =0, lim zQ0(z)= lim 2 −1 = . z.→∞ z z.→∞ z k Q(z) z.→∞ z.→∞ z k Q(z) 0 0 According to (5.6) and (5.7) this means that mul A0 =mulA1 = {0}.Theas- ymptotic expansion (5.10) is implied by (3.17), (3.18), and Corollary 3.4, while the expansion (5.11) is obtained from (3.18), (3.19). The remaining statements are obtained from part (ii) of Theorem 4.12. (iii) Finally, assume that ∞ is a GZNT3 of Q. Then by (T3) the following limit exists: lim z2k−1Q(z) > 0. z.→∞ It follows from (4.7) that − 1 lim zQ0(z)= lim 2 −1 < 0, z.→∞ z.→∞ z k Q(z) 0 and in view of (5.7) this means that mul A1 = {0}. The asymptotic expan- sion (5.12) is implied by (3.16) and Corollary 3.4, and the remaining statements are obtained from part (iii) of Theorem 4.12. 76 V.A. Derkach, S. Hassi and H.S.V. de Snoo The characterizations in Theorem 5.3 can be translated also for the gener- alized poles of nonpositive type of the function Q∞ by means of the following theorem. In fact, this result is an extension of [6, Theorem 5.2] and can be seen to augment also the result stated in Theorem 3.3. Theorem 5.4. Let Q ∈ Nκ, Q =0 ,withlimz.→∞ Q(z)=0and let Q∞ = −1/Q. Then the following statements are equivalent: (i) Q ∈ Nκ,−2n \ Nκ,−2n−2 and m ≥ 0 is the maximal integer such that sj =0 for all j ≤ m − 1(≤ 2n); (ii) Q(z)=[(A − z)−1ω,ω], z ∈ ρ(A), for some selfadjoint operator A in a Pontryagin space H and a cyclic vector ω ∈ dom An \ dom An+1 satisfying [Ajω,Aiω]=0for all i + j ≤ m − 1(≤ 2n), i, j ≤ n; (iii) Q has an asymptotic expansion of the form s s2 1 Q(z)=− m −···− n + o ,z.→∞, (5.13) zm+1 z2n+1 z2n+1 where m(≤ 2n +1) and n are maximal nonnegative integers, such that (5.13) holds; (iv) Q∞ = −1/Q has an asymptotic expansion of the form m+1 2+1 2+1 Q∞(z)=pm+1z + ···+ p2+1z + o z ,z.→∞, (5.14) where pm+1 =0 if m ≥ 2 and ∈ Z (with 2 ≤ m +1) is minimal such that (5.14) holds. In this case the integers m, n ≥ 0 and are connected by = m − n. Moreover, in (5.13) sm =0 if and only if pm+1 =0 in (5.14),inwhichcasepm+1 =1/sm and m ≤ 2n or, equivalently, 2 ≤ m. Proof. (i) ⇔ (ii) This equivalence follows from Theorem 3.3 and the formulas (3.10) for the moments sj, j ≤ 2n. (i) ⇔ (iii) If (i) holds then by Corollary 3.4 Q has an asymptotic expansion of the form (5.13) and maximality of n in this expansion follows from the assumption Q ∈ Nκ,−2n+2. The converse statement is also clear. (iii) ⇔ (iv) Assume that Q satisfies (5.13). If m ≤ 2n then sm =0.Otherwise m =2n + 1 and the expansion (5.13) reduces to 1 Q(z)=o ,z.→∞. (5.15) z2n+1 In the case that m ≤ 2n, sm = 0 and the expansion (5.13) can be rewritten in the form s2 1 zm+1Q(z)=−s −···− n + o ,z.→∞. (5.16) m z2n−m z2n−m Since Q∞ = −1/Q, by inverting the expansion (5.16) one concludes that the expansion (5.13) for Q with sm = 0 is equivalent for Q∞ to admit an expansion Asymptotic expansions 77 of the form m+1 2(m−n)+1 2(m−n)+1 Q∞(z)=pm+1z + ···+ p2(m−n)+1z + o z ,z.→∞, where pm+1 =1/sm = 0. This means that Q∞ has an asymptotic expansion of the form (5.14), where the integer = m − n,2 2n+3 Q∞(z)=o(z ),z.→∞, so that Q∞ has an expansion of the form (5.14) with := n+1 > 0andm+1 = 2. Moreover, here = n+1 is the minimal integer, such that Q∞ admits an expansion of the form (5.14). Observe, that since = n + 1 is minimal, the maximal m in (5.13) is equal to m =2n + 1, so that the equality = m − n holds also in this case. In particular, (5.13) and (5.14) are still equivalent if m =2n +1,inwhich case one can take m +1=2. To establish the classification of the asymptotic expansions for the function Q∞ along the lines of Theorem 5.3 it is enough to consider expansions of the form (5.14) with = m − n ≥ 0 (so that in (5.13) n ≤ m) and then apply the last statement of Theorem 5.3. In this case (5.14) takes the form 2+1 Q∞(z)=P (z)+o z ,z.→∞, where m+1 2+1 P (z):=pm+1z + ···+ p2+1z ,≥ 0, is a real polynomial of degree deg P = m + 1, whose leading coefficient is given by pm+1 =1/sm if deg P>2.Ifm +1=2, then one can take P =0. 5.3. Asymptotic expansions and the index of singularity As another consequence of Theorem 5.4 some characterizations for the index of sin- gularity of the generalized pole (zero) of nonpositive type of Q∞ (of Q = −1/Q∞) at ∞ are given. The motivation for this notion is given in the end of this section. Definition 5.5. Let ∞ be a generalized pole of Q∞ (zero of Q)ofnonpositive R ν−1 × type of order ν (= dim ∞(SF )) and let Hν =(si+j )i,j=0 be the ν ν Hankel matrix which is determined by the finite moments sj,0≤ j ≤ 2ν − 2, of Q.Then 0 κ∞ =dim(kerHν ) is called the index of singularity of Q∞ at ∞. Theorem 5.6. Let n and m be maximal nonnegative integers with n ≤ m, such that Q ∈ Nκ,−2n and sj =0for all j ≤ m − 1(≤ 2n),andletS, A, SF , ω be from the minimal operator representation (1.2) of Q. 78 V.A. Derkach, S. Hassi and H.S.V. de Snoo Then the following assertions are equivalent: (i) ∞ is a generalized zero of nonpositive type of Q with the index of singularity 0 equal to κ∞; 0 (ii) κ∞ = m − n(≥ 0); (iii) Q∞ = −1/Q admits an asymptotic expansion of the form 0 2κ∞+1 Q∞(z)=Pn+1(z)+o(z ),z.→∞, (5.17) 0 0 where Pn+1 is a polynomial of degree n+1 ≥ 2κ∞ and κ∞ ≥ 0 is the minimal integer such that (5.17) holds; 0 (iv) κ∞ is equal to the dimension of the isotropic subspace of R∞(SF ). Proof. (i) ⇔ (ii) It follows from Theorem 5.3 and the maximality of n and m, n ≤ m,thatn = ν −1, where ν =dimR∞(SF ), cf. [15, Theorem 5.2]. Since clearly dim (ker Hν )=m − n(≥ 0) the equivalence of (i) and (ii) is shown. (ii) ⇔ (iii) This follows immediately from Theorem 5.4 with = m − n = 0 0 κ∞ ≥ 0. Here minimality of κ∞ ≥ 0 in (5.17) is equivalent to the maximality of 0 n (= m − κ∞) in (5.13). (ii) ⇔ (iv) To prove this equivalence the indices m and n are calculated in each of the cases (i)–(iii) in Theorem 5.3. In the case (i) one obtains from (5.9) m =2k, n = ν − 1. Hence 0 κ∞ = m − n =2k − ν +1=κ0(R∞(SF )). Similarly in the case (ii) for both expansions (5.10) and (5.11) one has m = 2k − 1, n = ν − 1. Thus 0 κ∞ = m − n =2k − ν = κ0(R∞(SF )). Finally in the case (iii) m =2k − 2, n = ν − 1, and 0 κ∞ = m − n =2k − 1 − ν = κ0(R∞(SF )). This completes the proof. The equivalence of (i) and (iv) in Theorem 5.6 is also a direct consequence of [15, Corollary 4.4]. From Theorem 5.6 one obtains immediately the following characterization for the regularity of a critical point. Corollary 5.7. ([6, Theorem 4.1], [17, Proposition 1.6]) The root subspace R∞(SF ) is nondegenerate (equivalently ∞ is a regular critical point of SF ) if and only if Q∞ admits the representation Q∞(z)=P (z)+Q0(z), where Q0(z)=o(z),z.→∞. Proof. This is immediate from the equivalence of (iii) and (iv) in Theorem 5.6. 0 The index κ∞ measures the degree of singularity of the singular critical point 0 ∞ of SF . The characterization of κ∞ via the asymptotic expansion (5.17) in The- orem 5.6 is particularly appealing: it extends the result stated in Corollary 5.7 to the case of singular critical points in an explicit manner. Asymptotic expansions 79 6. Spectral characterizations via the underlying Weyl functions In this section the structure of the underlying root subspace corresponding to the three different types of maximal Jordan chains constructed in Section 4 is studied by means of the factorized integral representations of the underlying Weyl func- tions. First detailed results are presented for the point ∞. Then the case of finite generalized zeros and poles of nonpositive type is treated briefly. Furthermore, it is shown how the classification of all generalized zeros and poles of Q ∈ Nκ belonging to R ∪{∞}can be applied in establishing analytic criteria for the minimality of (not necessarily canonical) factorization models of Nκ-functions. 6.1. The classification of generalized zeros and poles at ∞ The canonical factorization of generalized Nevanlinna functions in (1.6) implies that Q(z)=−1/Q∞(z) has the following integral representation: p(z)p(z) 1 t − ∞ − Q(z)= 1/Q (z)= a + bz + 2 dρ(t) , (6.1) q(z)q (z) R t − z t +1 where p and q are as in (2.5), a ∈ R, b ≥ 0, and ρ(t)satisfies dρ(t) ∞ 2 < . (6.2) R t +1 0 0 ∞ − 2 ∈ R In the case that R dρ(t) < ,denotea0 = a R t/(t +1)dρ(t) . In the next theorem the classification of the Jordan chains is characterized via the spectral properties of the function Q using the factorized integral repre- sentation (6.1). Theorem 6.1. Let Q∞ ∈ Nκ,letk = κ∞(Q∞) > 0,andletQ(z)=−1/Q∞(z) have the factorized integral representation (6.1).Then: (i) the GZNT ∞ of Q is regular and of type (T1) if and only if 2k b = a0 =0, dρ(t) > 0, and (1 + |t|) dρ(t) < ∞; (6.3) R R (ii) the GZNT ∞ of Q is regular and of type (T2) if and only if 2(k−1) b =0,a0 =0 , and (1 + |t|) dρ(t) < ∞; (6.4) R (iii) the GZNT ∞ of Q is regular and of type (T3) if and only if b>0 and (1 + |t|)2(k−2)dρ(t) < ∞; (6.5) R (iv) the GZNT ∞ of Q is singular and of type (T1) with the index of singularity 0 κ∞(> 0) if and only if b = a0 =0and 2( − 0 ) 2( − 0 +1) (1 + |t|) k κ∞ dρ(t) < ∞, (1 + |t|) k κ∞ dρ(t)=∞; (6.6) R R 80 V.A. Derkach, S. Hassi and H.S.V. de Snoo (v) the GZNT ∞ of Q is singular of type (T2) with the index of singularity 0 (0 <)κ Proof. By Proposition 5.2 the root subspace R∞(SF ) is regular (singular) of dimen- sion ν if and only if (5.4) holds (respectively (5.5) holds). According to Theorem 3.3 the condition ω ∈ dom Aν−1 is equivalent to the condition (1 + |t|)2(ν−k−1)dρ(t) < ∞. (6.10) R This leads to the integrability conditions in (6.3)–(6.5) in the regular case and to the integrability conditions (6.7)–(6.9) in the singular case, see Theorem 4.12. It follows from the expansions (5.9)–(5.12) of Q in Theorem 5.3 and the factorized integral representation of Q in (6.1) that sj =0forj<2k − 2,s2k−2(Q)=−b, (6.11) and moreover, that if dρ(t) < ∞ and b =0, then s2k−1 = −a0, (6.12) R and if dρ(t) < ∞ and b = a0 =0, then s2k = dρ(t). (6.13) R R All the statements of the theorem can now be obtained from Theorem 5.3 by combining Proposition 5.2 with (6.10)–(6.13). Observe that the regular cases (i)–(iii) are obtained from the singular cases 0 (iv)–(vi) by taking κ∞ = 0 and excluding the second condition in (6.6), (6.7), and (6.9), respectively. All of the conditions in Theorem 6.1 are based on the canonical factorization of Q = rr Q00 in (6.1) involving the ordinary Nevanlinna function Q00 ∈ N0 in (2.5). In the factorization model of Theorem 4.4 the factor Q0 belongs to the class Nκ−k,wherek = κ∞(Q∞). The classification of Jordan chains was 0 0 described in Theorem 4.12 by means of the selfadjoint extensions A0 and A1 of S0 in the Pontryagin space H0 whose negative index is equal to κ(Q0)=κ(Q∞) − κ∞(Q∞). Analogous descriptions remain true also for the canonical factorization of Q∞. Asymptotic expansions 81 Proposition 6.2. Let Q = rr Q00 be the canonical factorization of Q ∈ Nκ in (6.1) with k = π∞(Q) > 0,letS00 be a simple symmetric operator in a Hilbert space 00 00 H00 with a boundary triplet Π00 = {C, Γ0 , Γ1 } whose Weyl function is equal to 00 00 00 00 Q00,andletA0 = ker Γ0 and A1 = ker Γ1 be the corresponding selfadjoint extensions of S00 in H00.Then: 00 (i) the GZNT ∞ of Q is of type (T1) if and only if mul A1 = {0}; 00 00 (ii) the GZNT ∞ of Q is of type (T2) if and only if mul A0 =mulA1 = {0}; 00 (iii) the GZNT ∞ of Q is of type (T3) if and only if mul A0 = {0}. Proof. The identities (6.11) and (6.12) concern the function0Q00. The conditions which describe GZNT ∞ of type (T1) are b = a0 =0and R dρ(t) < ∞.These conditions are equivalent to − limz.→∞ zQ00(z) < ∞, which holds if and only if 1 00 − lim > 0 ⇔ mul A1 = {0}, z.→∞ zQ00(z) where the last equivalence follows from the simplicity of the operator0 S00 in H00. ∞ Similarly, for type (T2) one0 has the conditions b =0anda0 =0if R dρ(t) < , ∞ or the conditions b =0and R dρ(t)= . These conditions are equivalent to Q00(z) 00 00 lim =0, lim zQ00 = ∞⇔ mul A0 =mulA1 = {0}. z.→∞ z z.→∞ Finally, for type (T3) one has the condition b>0 which is equivalent to Q00(z) 00 lim > 0 ⇔ mul A0 = {0}, z.→∞ z which completes the proof. The spectral theoretic characterization in Theorem 6.1 was based on the canonical factorization of the function Q = −1/Q∞.Since∞ is a GPNT of the function Q∞ it is natural to translate this result for the canonical factorization of Q∞: q(z)q(z) 1 t ∞ ∞ ∞ − ∞ Q (z)= a + b z + 2 dσ (t) , (6.14) p(z)p (z) R t − z t +1 where p, q are as0 in (6.1), a∞ ∈ R, b∞ ≥ 0, and σ∞(t)0 satisfies the analog of (6.2). 2 In the case that R dσ∞(t) < ∞,denoteγ∞ = a∞ − R t/(t +1)dσ∞(t) ∈ R. Theorem 6.3. Let Q∞ ∈ Nκ with k = κ∞(Q∞) > 0 have the factorized integral representation (6.14).Then: (i) the GPNT ∞ of Q∞ is regular and of type (T1) if and only if b∞ > 0 and 2(k−1) (1 + |t|) dσ∞(t) < ∞; (6.15) R (ii) the GPNT ∞ of Q∞ is regular and of type (T2) if and only if b∞ =0, γ∞ =0 , and the integrability condition (6.15) is satisfied; (iii) the0 GPNT ∞ of Q∞ is regular and of type (T3) if and only if b∞ = γ∞ =0, R dρ(t) > 0, and the integrability condition (6.15) is satisfied; 82 V.A. Derkach, S. Hassi and H.S.V. de Snoo (iv) the GPNT ∞ of Q∞ is singular and of type (T1) with the index of singularity 0 κ∞(> 0) if and only if b∞ > 0 and 2(k−1−κ0 ) 2(k−κ0 ) (1 + |t|) ∞ dσ∞(t) < ∞, (1 + |t|) ∞ dσ∞(t)=∞; (6.16) R R (v) the GPNT ∞ of Q∞ is singular and of type (T2) with the index of singu- 0 larity (0 <)κ∞ Proof. In each case the conditions concerning the parameters b∞, a∞,andγ∞ are immediate from Proposition 6.2. It remains to establish the integrability conditions for σ∞(t). First observe that the integrability conditions for ρ(t) in Theorem 6.1 concern the function Q00 in (2.5), while the integrability conditions for σ∞(t) concern the function −1/Q00. Now if the GPNT ∞ is of type (T1) then due to the conditions b = a0 =0themeasuredρ(t) has two more finite moments than the measure dσ∞(t). Therefore, the integrability conditions in (6.3) and (6.6) are equivalent to those in (6.15) and (6.16), respectively; cf. [16, Theorem 4.2]. Similarly, due to b∞ = γ∞ = 0, the integrability conditions in (6.5) and (6.9) are equivalent to those in (6.15) and (6.16), respectively. Moreover, by [16, Theorem 4.2] the integrability conditions in (6.4) and (6.7) are equivalent to those in (6.15) and (6.16), respectively, while the0 conditions b = 0 and (6.8) are clearly equivalent to the conditions b∞ =0and R dσ∞(t)=∞. In the case that κ(Q∞)=κ∞(Q∞) = 1 the result in Theorem 6.3 simplifies to [8, Theorem 4.1]. Observe also that if the function σ∞(t) has a compact support in0R, then (6.16) shows that ∞ cannot be a singular critical point of SF .Moreover, if R dσ∞(t)=0andκ∞(Q∞) > 0, then only the cases (i) and (ii) in Theorem 6.3 can occur. 6.2. The classification of generalized zeros and poles in R The classification of generalized zeros and poles of nonpositive type has been stud- ied in detail at the point z = ∞. Similar results hold true also for finite generalized zeros and poles of nonpositive type of Q ∈ Nκ which belong to R.Heresomemain characterizations for the classification of finite generalized zeros β ∈ R and finite generalized poles α ∈ R of Q are presented. First the classification of zeros and poles is characterized by means of the canonical factorization of Q ∈ Nκ. Lemma 6.4. Let Q ∈ Nκ, κ>0,befactorizedasin(6.1). Then the types (T1)– (T3) of a generalized zero β ∈ R of Q are characterized as follows: Asymptotic expansions 83 00 (i) the GZNT β ∈ R of Q is of type (T1) if and only if β ∈ σp(A1 ); 00 00 (ii) the GZNT β ∈ R of Q is of type (T2) if and only if β ∈ σp(A0 ) ∪ σp(A1 ); 00 (iii) the GZNT β ∈ R of Q is of type (T3) if and only if β ∈ σp(A0 ). Moreover, the types (T1)–(T3) of a generalized pole α ∈ R of Q are characterized as follows: 00 (iv) the GPNT α ∈ R of Q is of type (T1) if and only if α ∈ σp(A0 ); 00 00 (v) the GPNT α ∈ R of Q is of type (T2) if and only if α ∈ σp(A0 ) ∪ σp(A1 ); 00 (vi) the GPNT α ∈ R of Q is of type (T3) if and only if α ∈ σp(A1 ). Proof. (i)–(iii) In view of the canonical factorization of Q in (6.1) one obtains the following representation for the limits in (2.3): Q(z) − − − − ≤ lim 2 −1 = lim (z β)Q00(z)= (ρ(β+) ρ(β )) ( 0) (6.17) z.→β (z − β) πβ z.→β 0 dρ(t) ∞ and if R (t−β)2 < and limz.→β Q00(z)=0,then Q(z) Q00(z) dρ(t) lim = lim = b + (> 0), (6.18) 2πβ +1 2 z.→β (z − β) z.→β (z − β) R (t − β) and otherwise the limit in (6.18) is not finite. Observe that the limit in (6.17) 00 is negative if and only if β ∈ σp(A0 ). The limit in (6.18) is finite if and only if for the function −1/Q00 the limit in (6.17) is negative, which is equivalent to 00 β ∈ σp(A1 ). The statements (i)–(iii) are now obvious from the defining properties of the classifications given in Subsection 3.2. (iv)–(vi) Apply the characterizations of generalized zeros in the first part of the lemma to the function −1/Q. Next some characteristic properties of the underlying root subspaces associ- ated with the classification of generalized zeros β ∈ R (generalized poles α ∈ R) are established in a minimal representation of Q (of −1/Q, respectively). Let S(Q) be a simple symmetric operator in a Pontryagin space H,let ∗ {Γ0, Γ1, H} be a boundary triplet for S such that the corresponding Weyl func- tion is the given Nκ-function Q, and let A(Q)=kerΓ0 and A(−1/Q)=kerΓ1. Moreover, let Rα(A(Q)) and Rβ(A(−1/Q)) be the root subspaces of the selfadjoint extension A(Q)andA(−1/Q)ofS(Q) associated with the generalized pole α and the generalized zero β ∈ R of Q, respectively. The following result is analogous to Theorem 4.12. For simplicity, the classification of a GPNT α ∈ R and a GZNT β ∈ R of Q is characterized here by using the signature of the corresponding root subspace. Here the following notations will be used: να := dim Rα(A(Q)),νβ := dim Rβ(A(−1/Q)), α β κ± := κ±(Rα(A(Q))),κ± := κ±(Rβ(A(−1/Q))), α β κ0 := κ0(Rα(A(Q))),κ0 := κ±(Rβ(A(−1/Q))). Proposition 6.5. With the notations given above the following assertions hold for aGPNTα ∈ R andaGZNTβ ∈ R of Q ∈ Nκ: 84 V.A. Derkach, S. Hassi and H.S.V. de Snoo α α (i) α is of type (T1) if and only if κ− = κ+ −1, in this case κα +1 ≤ να ≤ 2κα +1 α and κ0 =2κα +1− να; α α (ii) α is of type (T2) if and only if κ− = κ+, in this case κα ≤ να ≤ 2κα and α κ0 =2κα − να; α α (iii) α is of type (T3) if and only if κ− = κ+ +1, in this case κα ≤ να ≤ 2κα − 1 α and κ0 =2κα − 1 − να; β β (iv) β is of type (T1) if and only if κ− = κ+ −1, in this case πβ +1 ≤ νβ ≤ 2πβ +1 β and ν0 =2πβ +1− νβ; β β (v) β is of type (T2) if and only if κ− = κ+, in this case πβ ≤ νβ ≤ 2πβ and β ν0 =2πβ − νβ; β β (vi) β is of type (T3) if and only if κ− = κ+ +1, in this case πβ ≤ νβ ≤ 2πβ − 1 β and ν0 =2πβ − 1 − νβ. Proof. The statements (i)–(iii) are obtained from Theorem 4.12 by considering the transform Q∞(z):=−Q(α +1/z). Namely, the root subspace Rα(A(Q)) associ- R ated with the generalized pole α of Q coincides with the root subspace ∞(SF ) associated with the generalized pole ∞ of the function Q∞. The statements (iv)–(vi) follow by applying the results in (i)–(iii) to the function −1/Q. The classification for a GZNT β ∈ R and a GPNT α of Q can be characterized also via asymptotic expansions of the function Q in a neighborhood of these points. The defining properties of the types (T1)–(T3) are reflected in these expansions in a similar manner as in Theorems 5.3 and 5.4 above. Such expansions have been studied in [15] and, for instance, the analog of Theorem 5.3 for a GZNT β ∈ R of Q can be easily derived from [15, Theorem 5.2] and for a GPNT α of Q form [15, Theorem 5.4]. Moreover, these expansions can be characterized also via the canonical factorization of Q along the lines of Theorems 6.1 and 6.3 by using Lemma 6.4; see also [15, Section 6]. A detailed formulation of these results is left for the reader. 6.3. Analytic criteria for the minimality of (localized) factorization models The classification of generalized zeros and poles of nonpositive type of Q ∈ Nκ can be used to give analytic criteria for general (localized) factorizations models based on (proper) factorizations of Q of the form p Q(z)=r(z)r (z)Q0, r = , (6.19) q where p and q are divisors of the polynomials p and q in the canonical factorization of Q in (6.1), with multiplicity of a zero equal to its original multiplicity. Such factorization models can be used, for instance, in studying localspectralproperties of the function Q ∈ Nκ, along the lines carried out in the previous sections of the present paper at ∞ with the aid of the model in Theorem 4.6 for proper factorizations of Q at ∞. Asymptotic expansions 85 The basic observation here is that according to Lemma 4.11 the occurrence of a maximal Jordan chain of type (T3) means that mul S(Q∞) = {0}. Therefore, the existence of such a Jordan chain is connected with the non-minimality of the factorization model of Q. The next result is formulated for the canonical factor- ization model as constructed in [3, Theorem 3.3]; by unitary equivalence the result holds for all other canonical factorization models, too; cf. [12]. Proposition 6.6. The canonical factorization model constructed for Q ∈ Nκ in [3, Theorem 3.3] is simple if and only if all generalized zeros β ∈ R ∪{∞} and generalized poles α ∈ R ∪{∞}of Q of nonpositive type are either of type (T1) or of type (T2). Proof. By Lemma 6.4 a real GZNT β (a real GPNT α)ofQ is not of type (T3) 00 00 if and only if β ∈ σp(A0 ) (respectively α ∈ σp(A1 )). Moreover, by Proposi- tion 6.2 the GZNT β = ∞ (the GPNT α = ∞) is not of type (T3) if and 00 00 only if mul A0 = {0} (respectively, mul A1 = {0}). Now, according to [3, Theo- rem 4.1], these conditions characterize the simplicity of the corresponding factor- ization model for Q. These observations lead to the following analytic characterization for the simplicity of the factorization model in Theorem 4.4 ([9, Theorem 4.2]) which is based on a proper factorization of Q∞ at ∞ (see Definition 4.2). Proposition 6.7. Let Q∞ ∈ Nκ with k = κ∞(Q∞) > 0 and let Q∞ = qq Q0 be a proper factorization of Q∞ with some monic polynomial q, deg q = k. Moreover, let the symmetric operator S0 be simple in the Pontryagin space H0 (see Lemma 4.5). Then the factorization model for Q∞ in Theorem 4.4 is minimal if and only if the following two conditions are satisfied: (1) α = ∞ is not a GPNT of Q∞ of type (T3); (2) all the real zeros of q as GZNT of Q∞ are either of the type (T1) or (T2). Proof. By Theorem 4.6 the symmetric relation S(Q∞) is simple if and only if it has an empty point spectrum, or equivalently, the subspace H in (4.15) is trivial. In view of Lemma 4.5 the last condition is equivalent to 0 0 mul A1 = {0} and σp(A0) ∩ σ(q)=∅. (6.20) By Theorem 4.12 the first condition in (6.20) is equivalent to the property formu- 0 lated in part (1). The condition α ∈ σp(A0) ∩ σ(q)meansthatq(α)=0and lim (z − α)Q0(z) < 0, (6.21) z.→α 0 since by Lemma 4.5 ker (A0 −α) is spanned by a positive vector, cf. [9, Lemma 2.3]. Hence, if the multiplicity of α as a root of q is κα,then Q∞(z) lim 2 −1 < 0, (6.22) z.→α (z − α) κα and thus πα(Q∞)=κα and α is of type (T3). Conversely, (6.22) implies (6.21) with κα = πα(Q∞). This completes the proof. 86 V.A. Derkach, S. Hassi and H.S.V. de Snoo Since the polynomial q can be selected to be an arbitrary divisor of degree k = κ∞(Q∞) of the polynomial q in the canonical factorization of Q∞, the con- dition (2) in Proposition 6.7 can be satisfied if, for instance, the total number (counting multiplicities) of all generalized zeros in C+ and all generalized zeros of nonpositive type in R of Q∞ which are of type (T1) or (T2) is at least equal to κ∞(Q∞). Therefore, the factorization model in Theorem 4.4 can be minimal, while the canonical factorization model of Q∞ need not be minimal. Remark 6.8. Similar facts can be derived for other (localized) factorization mod- els which are build on some (proper) factorization of Q ∈ Nκ,cf.(6.19).The construction of such (localized) factorization models can be based on the (orthog- onal) coupling of two minimal models (a method studied in another context in [4]), one of which is a finite-dimensional Pontryagin space model for the rational 2 × 2-matrix function 0 r(z) R(z)= , (6.23) r(z)0 whose reproducing kernel space model has been identified in matrix terms with Bezoutians and companion operators in [3, Proposition 3.3]. The other is a minimal Pontryagin space model for the factor Q0. The model for the product Q = rr Q0 in (6.19) is obtained from the orthogonal sum of the models for R and Q0 as a straightforward extension of the model constructed in [3, Theorem 3.3], simply by allowing the factor Q0 to be a generalized Nevanlinna function, too. This procedure can also be described in pure function theoretic terms: perform suitable “block transforms” to the orthogonal sum R ⊕ Q0 to get the product Q = rr Q0,cf.[3, Section 3]. Such factorization models are not minimal in general. Non-minimality of such models reflects the fact that some poles and zeros of rr and Q0 may cancel each others when these functions are multiplied. The simplest example here is the function Q(z)=−z, which belongs to N1 and whose canonical factorization is given by 1 Q(z)=−z = z2 − , z where Q0(z)=−1/z ∈ N0, cf. [9, Section 3]. Hence a model which is built on the canonical factorization of Q(z) does not produce a minimal model for Q directly. 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Langer, “A characterization of generalized zeros of negative type of functions of the class Nκ”, Oper. Theory Adv. Appl., 17 (1986), 201–212. Vladimir Derkach Department of Mathematical Analysis Donetsk National University Universitetskaya str. 24 83055 Donetsk, Ukraine e-mail: [email protected] Seppo Hassi Department of Mathematics and Statistics University of Vaasa P.O. Box 700 65101 Vaasa, Finland e-mail: [email protected] Henk de Snoo Department of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen, Nederland e-mail: [email protected] Operator Theory: Advances and Applications, Vol. 175, 89–94 c 2007 Birkh¨auser Verlag Basel/Switzerland A Necessary Aspect of the Generalized Beals Condition for the Riesz Basis Property of Indefinite Sturm-Liouville Problems Andreas Fleige Abstract. For the Sturm-Liouville eigenvalue problem −f = λrf on [−1, 1] with Dirichlet boundary conditions and with an indefinite weight function r changing it’s sign at 0 we discuss the question whether the eigenfunctions 2 form a Riesz basis of the Hilbert space L|r|[−1, 1]. In the nineties the sufficient so called generalized one hand Beals condition was found for this Riesz basis property. Now using a new criterion of Parfyonov we show that already the old approach gives rise to a necessary and sufficient condition for the Riesz basis property under certain additional assumptions. Mathematics Subject Classification (2000). Primary 34B4; Secondary 34L10. Keywords. Indefinite Sturm-Liouville problem, Riesz basis, definitizable oper- ator. 1. Introduction We consider the regular indefinite Sturm-Liouville eigenvalue problem −f = λrf on [−1, 1],f(−1) = f(1) = 0 (1.1) where the real weight function r ∈ L1[−1, 1] has a single turning point, i.e., sign change, at 0. More precisely we assume r(x) < 0a.e.on[−1, 0),r(x) > 0a.e.on(0, 1] (1.2) It is well known that this eigenvalue problem has only real and simple eigenvalues ([5], [6], [7]). During the last two decades the question was intensively studied 2 − whether the eigenfunctions form a Riesz basis of the Hilbert space L|r|[ 1, 1] with the inner product 1 | | ∈ 2 − (f,g):= fg¯ r dx (f,g L|r|[ 1, 1]). −1 90 A. Fleige (e.g., [2], [5], [12], [6], [7], [13], [8], [1], [10]). This question is motivated by the fact that in case of a positive weight function r the eigenfunctions even form an orthonormal basis. However in the indefinite situation it took some time to find sufficient conditions for this Riesz basis property of the eigenvalue problem ([2], [5], [12], [6], [7], [13], [10]). In particular starting from a paper of Beals [2] a technique was developed which led to quite general sufficient conditions in the late eighties and nineties ([5], [6], [7], [13]). For this reason in [7] the author called his result the generalized one hand Beals condition. In our days in [4] this technique was still improved and applied to similar problems with λ-dependent boundary conditions. On the other hand there was a competition to find counterexamples, i.e., weight functions r such that the Riesz Basis property fails to hold. After in [13] Volkmer had proved the existence of such counterexamples the first concrete coun- terexample was presented in [8] and then in [1] counterexamples with continuously differentiable weight functions were given. Moreover, in [3] it was shown that coun- terexamples also depend on the type of the boundary conditions. Recently Parfyonov found a connection of the “positive” and the “negative” results in [10] where he gave a sufficient and necessary condition for the Riesz Basis property in the special situation of an odd weight function. We shall see here that already the previous approach using the generalized one hand Beals condition was not too far away from a necessary condition. More precisely we shall show that in the case of an odd weight function and under certain additional assumptions the generalized one hand Beals condition is equivalent to the condition of Parfyonov and hence to the Riesz Basis property of the eigenvalue problem. Moreover, this also leads to a sufficient condition of Parfyonov’s type for non odd weight functions. However, the question remains open whether in this general case we also get a necessary and sufficient condition for the Riesz Basis property in this way. In [5] it was detected that the Riesz Basis property is closely connected to the critical points of a certain definitizable operator. Therefore also here we shall make use of Langer’s theory of definitizable operators in Krein spaces outlined in [9]. 2. Two known conditions for the Riesz basis property ∈ 1 − 2 − For the real function r L [ 1, 1] with (1.2) the space0 Lr[ 1, 1] of all (equivalence − 1 | |2| | ∞ classes of) measurable functions f on [ 1, 1] with −1 f r dx < is a Krein space with the indefinite inner product 1 ∈ 2 − [f,g]:= fgrdx¯ (f,g Lr[ 1, 1]). −1 ∈ 2 − The operator J defined by Jf := sgn(r)f (f Lr[ 1, 1]) is a fundamental sym- metry and induces the Hilbert space inner product 1 | | ∈ 2 − (f,g)=[Jf,g]= fg¯ r dx (f,g Lr[ 1, 1]). −1 A Necessary Aspect of the Beals Condition 91 It is well known ([5], [6], [7]) that the operator A defined by 1 dom(A):={f ∈ L2[−1, 1] | f,f abs. cont., f ∈ L2[−1, 1], r r r f(−1) = f(1) = 0}, 1 Af := − f (f ∈ dom(A)) r is selfadjoint, nonnegative, boundedly invertible and hence definitizable in the 2 − · · Krein space (Lr[ 1, 1], [ , ]). Moreover the spectrum of A consists of a sequence of real and simple eigenvalues accumulating only at +∞ and −∞ and therefore ∞ is the only critical point of A. According to [7], Sections 3.2 and 3.3, we say that r satisfies the generalized 1 right-hand Beals condition if there exist numbers µ>0, 0 <ξ 3. A connection of the two conditions Throughout this section we additionally assume that the weight function r is continuously differentiable on (0,ξ]foranumber0<ξ≤ 1. Under some additional 92 A. Fleige assumptions we shall show that the Parfyonov condition implies the generalized right-hand Beals condition. To this end for µ ∈ (0, 1) we consider the function r(t) h (t):= (t ∈ (0,ξ]) µ µr(µt) with the derivative r(µt)r (t) − µr (µt)r(t) h (t)= (t ∈ (0,ξ]). µ µr(µt)2 Then if r satisfies the Parfyonov condition with µ ∈ (0, 1) it follows from (2.2) by l’Hˆopital’s rule that R1(t) R1(t) r(t) 2 ≤ lim = lim = lim = lim hµ(t) (3.1) 0 0 0 0 t Rµ(t) t Rµ(t) t µr(µt) t if the last limit exists. Therefore if r is continuously differentiable on [0,ξ]with r(0) = 0 then so is the function hµ with hµ(0) ≥ 2 and hence (2.1) is satisfied with 1 h = hµ. However, since hµ(0) = µ = 1 this is a trivial case of Theorem 3.1. Assume that r is continuously differentiable on (0,ξ] for a number 0 <ξ≤ 1 and r is odd, i.e., r(−t)=−r(t)(t ∈ (0, 1]). Moreover assume that for all µ ∈ (0, 1) the limits r(t) r(µt)r (t) − µr (µt)r(t) hµ,0 := lim ,hµ,0 := lim 2 (3.2) t0 µr(µt) t0 µr(µt) exist. Then the following statements are equivalent: (i) The function r satisfies the generalized right-hand Beals condition. (ii) ∞ is not a singular critical point of A. (iii) The eigenvalue problem (1.1) has the Riesz basis property. (iv) The function r satisfies the Parfyonov condition. (v) For at least one µ ∈ (0, 1) we have hµ,0 =1 . Proof. The proofs of the implications (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) were already men- tioned in section 2. The implication (iv) ⇒ (v) follows from (3.1). For the impli- cation (v) ⇒ (i) we must show that the continuous function h (t)(t ∈ (0,ξ]) h(t):= µ hµ,0 (t =0) is continuously differentiable. By the mean value theorem for each t ∈ (0,ξ]there is a number ξt ∈ (0,t) such that h(t) − h(0) = h (ξ ) −→ h (t −→ 0). t µ t µ,0 Therefore h is differentiable at 0 and the derivative is continuous in 0. Now from Theorem 3.1 we easily obtain the following sufficient condition of Parfyonov’s type without the restriction to odd weight functions. Note that this wasalreadyshownin[11]. A Necessary Aspect of the Beals Condition 93 Corollary 3.2. Assume that r is continuously differentiable on (0,ξ] for a number 0 <ξ≤ 1 and for all µ ∈ (0, 1) the limits (3.2) exist. Then the eigenvalue problem (1.1) has the Riesz basis property if the function r satisfies the Parfyonov condition. Proof. Consider the “odd extension”r ˜ of r restricted to (0, 1] r(t)(t ∈ (0, 1]) r˜(t):= −r(−t)(t ∈ [−1, 0)). Thenr ˜ satisfies the Parfyonov condition and by Theorem 3.1 also the generalized right-hand Beals condition. Since this is then also true for r the Riesz basis property follows from Theorem 2.1. It is well known that the eigenvalue problem (1.1) has the Riesz basis property if r ∈ C2[0,ξ]withr(0) = 0 and r (0) = 0 (e.g., [7], Corollary 3.8). This also serves as an example for the existence of the limits (3.2) since in this case we have by l’Hospital’s rule r (t) 1 hµ,0 = lim hµ(t) = lim 2 = 2 (=1), t0 t0 µ r (µt) µ r (t) − r (µt) r (0) − r (0) hµ,0 = lim hµ(t) = lim 2 µhµ(t) = 2 . t0 t0 2µ r (µt) 2r (µt) 2µ r (0) 2µr (0) References [1] N.L. Abasheeva, S.G. Pyatkov, Counterexamples in indefinite Sturm-Liouville prob- lems. Siberian Adv. Math. 7, No. 4 (1997), 1–8. [2] R. Beals, Indefinite Sturm-Liouville problems and half range completeness. J. Differ- ential Equations 56 (1985), 391–407. [3] P. Binding, B. Curgus, A counterexample in Sturm-Liouville completeness theory. Proc. Roy. Soc. Edinburgh Sect. A 134 (2004), 244–248. [4] P. Binding, B. Curgus, Riesz Bases of Root Vectors of Indefinite Sturm-Liouville Problems with Eigenparameter Dependent Boundary Conditions, I. To appear. [5]B.Curgus,H.Langer,A Krein space approach to symmetric ordinary differential operators with an indefinite weight function. J. Differential Equations 79 (1989), 31–61. [6] A. Fleige, The “turning point condition” of Beals for indefinite Sturm-Liouville prob- lems. Math. Nachr. 172 (1995), 109–112. [7] A. Fleige, Spectral Theory of Indefinite Krein-Feller Differential Operators. Mathe- matical Research 98. Akademie Verlag, Berlin, 1996. [8] A. Fleige, A counterexample to completeness properties for indefinite Sturm-Liouville problems. Math. Nachr. 190 (1998), 123–128. [9] H. Langer, Spectral functions of definitizable operators in Krein spaces In: D. Butkovic, H. Kraljevic and S. Kurepa (eds.): Functional analysis. Conf. held at Dubrovnik, Yugoslavia, November 2–14, 1981. Lecture Notes in Mathematics, Vol. 948, Springer-Verlag, Berlin, Heidelberg, New York, 1982, 1–46. 94 A. Fleige [10] A.I. Parfyonov, On an Embedding Criterion for Interpolation Spaces and Application to Indefinite Spectral Problems. Siberian Mathematical Journal, Vol. 44, No. 4 (2003), 638–644. [11] A.I. Parfyonov, On Curgus’ condition in indefinite Sturm-Liouville problems. Matem. Trudy., Vol. 7, No. 1 (2004), 153–188. (In Russian, translated in Sib. Adv. Math.) [12] S.G. Pyatkov, Elliptic eigenvalue problems with an indefinite weight function. Siberian Adv. Math. 4, No. 2 (1994), 87–121. [13] H. Volkmer, Sturm-Liouville problems with indefinite weights and Everitt’s inequal- ity. Proc. Roy. Soc. Edinburgh Sect. A 126, No. 5, (1996), 1097–1112. Andreas Fleige Baroper Schulstraße 27a D-44225 Dortmund, Germany e-mail: [email protected] Operator Theory: Advances and Applications, Vol. 175, 95–109 c 2007 Birkh¨auser Verlag Basel/Switzerland On Reducible Nonmonic Matrix Polynomials with General and Nonnegative Coefficients K.-H. F¨orster and B. Nagy Abstract. We consider nonmonic quadratic polynomials acting on a general or on a finite-dimensional linear space as a continuation of our work in [7,8]. Conditions are given for the existence of right roots, if the coefficient operators have lower block triangular representations. In the finite-dimensional case we consider (in a certain sense, entrywise) nonnegative coefficient matrices in the general (reducible) case, and extend several earlier results from the case of irreducible coefficients. In particular, we generalize results of Gail, Hantler and Taylor [9]. We show that our general methods are sufficiently strong to prove a remarkable result by Butler, Johnson and Wolkowicz [3], proved there by ingenious ad hoc methods. Mathematics Subject Classification (2000). Primary 15A22; Secondary 15A48, 47A56, 15A18. Keywords. Reducible matrix polynomials, block triangular operator coeffi- cients, (entrywise) nonnegative matrix coefficients, nonnegative matrix roots. 1. Introduction We consider quadratic (in general nonmonic) polynomials Q(·)oftheform λ2M + λL + K. In Section 2 we study such polynomials when the coefficients M, L and K are lower block triangular linear maps (with the same block size). Such polynomials can have (right) roots, but none of these roots need to be lower block triangular; see Example 2.2. In Theorem 2.6 we give a necessary and sufficient condition for the existence of a lower block triangular (of the same block size as the coefficients) right root of this matrix polynomial. This work was completed with partial support of the Hungarian National Science Grant OTKA No T-047276 and partial support of the DAAD, the Technical University of Berlin and the Budapest University of Technology and Economics. 96 K.-H. F¨orster and B. Nagy The main topic of this paper is the matrix polynomial 2 Q(λ)=λIn×n − (λ A + λB + C) with entrywise nonnegative n × n-matrices A, B and C. This class of matrix polynomials play an important role in the theory of quasi-birth-and-death Markov chains (see [2, 5.6]) and has been studied very intensively; for a rather extensive review on this topic see [2] and [14]. In [7] the spectral properties of such matrix polynomials were studied. In the case when S = A + B + C is irreducible a complete set of necessary and sufficient conditions for the existence of a nonnegative right root W of Q(·) were given and the spectral properties of the matrices W and V such that Q(λ)=(V − λA)(λIn×n − W )andW ≥ 0n×n were studied; see [7, Section 4]. Results in this direction for the case when S is reducible can be found in [2, 4.7], [9, Theorem 5] and [3, Theorem 2.3]. In Section 3 we study systematically the existence of nonnegative right roots of Q(·) when the coefficients are nonnegative and reducible. In contrast to the example in Section 2 mentioned above here Q(·) has nonnegative lower triangular right roots if and only if it has nonnegative right roots (Theorem 3.1). With the help of the Frobenius normal form of S and the function 0 ≤ ρ → spectral radius of S(ρ)=ρ2A+ρB +C we give several sufficient conditions for the existence of nonnegative roots of Q(·), and we generalize the results mentioned above. In particular, we give a completely different proof of [3, Theorem 2.3]. Our terminology is mostly traditional. For standard facts in the theory of linear bounded operators in Banach spaces we refer to [4], concerning (matrix or operator) polynomials to [10] or [15], concerning nonnegative matrices to [11], [1] or [16]. 2. Roots of polynomials with lower triangular block operator coefficients Let E be a (finite- or infinite-dimensional complex) vector space. We consider the polynomial Q(λ)=λ2M + λL + K, λ ∈ C, (2.1) where M, L and K are linear maps in E. We assume that E = E1 × E2 and that the coefficients have lower block triangular representations; for example M11 0 M = , (2.2) M21 M22 where Mrs is a linear map form Es into Er for 1 ≤ s ≤ r ≤ 2and0isthezero map from E2 into E1. On Reducible Nonmonic Matrix Polynomials 97 Then Q(λ) has the lower block triangular representation Q11(λ)0 Q(λ)= . (2.3) Q21(λ) Q22(λ) Note that if the dimension of the vector space E is finite, and the Q s are ma- trices, then the form (2.3) shows that Q(·) is reducible in the sense of [11].Note also that the irreducible case (for nonnegative coefficients) was treated in [7, Section 4]. Assume that Q(·) has a right root W , i.e., W is a linear map in E such that Q(W )=MW2 + LW + K =0,andletV be a linear map in E such that Q(λ)=(λM − V )(λIE − W ), (2.4) where IE denotes the identity operator in E. If V and W have the block representation V11 V12 W11 W12 V = and W = , (2.5) V21 V22 W21 W22 direct computations show that (2.4) implies the following system of matrix equa- tions M11W12 + V12 =0 V11W12 + V12W22 =0, (2.6) V21W11 + V22W21 = K21, M21W11 + V21 + M22W21 = −L21. From the first two equations in (2.6) it follows immediately: If W is lower block triangular, then V is lower block triangular (in this case the second equation is automatically fulfilled), the diagonal polynomials Qrr(·)havetherightroots − − Wrr,andQrr(λ)=(λMrr Vrr)(λIEr Wrr)forr =1, 2. Let the linear map X of E1 into E2 be a solution of the equation V22X − M22XW11 = K21 +(L21 + M21W11)W11, (2.7) and define W21 := X and V21 := −L21 − M21W11 − M22X. (2.8) Then the last two equations of (2.6) are fulfilled. On the other hand, from the last two equations of (2.6) it follows: if we define X := W21, then (2.7) and (2.8) are satisfied. Therefore we have proved that 1. implies 2. in the following proposition. Proposition 2.1. With the assumptions, definitions and notations above the follow- ing assertions are equivalent: 1. The polynomial Q(·) has a lower block triangular right root (of the same block size as the coefficients). 2. For r =1, 2 the polynomial Qrr(·) has a right root Wrr,andequation(2.7) has a solution X, which is a linear map of E1 into E2,whereV22 satisfies − − Q22(λ)=(λM22 V22)(λIE2 W22). 98 K.-H. F¨orster and B. Nagy − − Proof. 2. implies 1.: Let Qrr(λ)=(λMrr Vrr)(λIEr Wrr)forr =1, 2, let X be a solution of (2.7), and W21 and V21 be defined as in (2.8). Then these maps are the lower left blocks, and Wrr and Vrr are the diagonal blocks of a lower block triangular right root W of Q(·) and of a lower block diagonal V , respectively, such that Q(λ)=(λM − V )(λIE − W ). The next example exhibits a reducible matrix polynomial which has right roots, but none of its right roots is lower triangular. Example 2.2. Consider the matrix polynomial 00 00 Q(λ)=λ2M + K = λ2 + 10 −2 −2 00 00 11 = λ − λI2×2 − . 10 −1 −1 11 Let Q(λ)=(λM − V )(λI − W )with2× 2-matrices V and W =(wrs). Then 00 V = −MW = − and w11 w12 00 00 = K = VW = − 2 , −2 −2 w11 + w12w21 w12(w11 + w22) and this implies w12 =0. Remark. Assume that the coefficients M,L,K of Q(·) are lower triangular, that the pair (V,W) factorizes Q(·) (as in (2.4)), and let (V0,W0) denote the pair obtained by substituting the blocks V12,W12 in (2.5) by zero blocks. We have seen that if the pair (V0,W0) also factorizes Q(·), then the diagonal blocks (Vrr,Wrr) factorize the corresponding diagonal block Qrr(·). We show that if the diagonal blocks (Vrr,Wrr) factorize the corresponding diagonal block Qrr(·)andtheblock M21 is an injective operator, then (V0,W0)=(V,W), hence it factorizes Q(·). Indeed, if the diagonal blocks factorize as well as the pair (V,W), then we have VjjWjj = Kjj,MjjWjj + Vjj = −Ljj (j =1, 2), Vj1W1j + Vj2W2j = Kjj,Mj1W1j + Mj2W2j + Vjj = −Ljj (j =1, 2). Hence we obtain M21W12 = 0, and the injectivity of M21 implies W12 =0.Itmeans that W = W0, and we have seen that then V12 = 0 follows. Hence (V0,W0)= (V,W) factorizes the polynomial Q(·). Corollary 2.3. Let the coefficients of Q(·) be block diagonal (i.e., M12 = L12 = K12 =0and M21 = L21 = K21 =0).ThenQ(·) has a (block diagonal) right root if and only if all diagonal entries of Q(·) have right roots. Assume that V22 is invertible. Then equation (2.7) is equivalent to −1 −1 X − V22 M22XW11 = V22 (K21 +(L21 + M21W11)W11). (2.9) On Reducible Nonmonic Matrix Polynomials 99 Let L(E1,E2) denote the vector space of all linear maps from E1 into E2.We define the linear map −1 −1 3(V22 M22,W11): L(E1,E2) → L(E1,E2),withX→ V22 M22XW11. (2.10) − − Corollary 2.4. Assume that Qrr(λ)=(λMrr Vrr)(λIEr Wrr) for r =1, 2.IfV22 − 3 −1 · and IL(E1,E2) (V22 M22,W11) are invertible, then Q( ) has a lower triangular right root. Now let E1 and E2 be Banach spaces, E their Banach product space and the coefficients of Q(·) and their entries in the representation (2.2) be linear and bounded operators between the corresponding Banach spaces. By σ(T ) we denote the spectrum of a bounded linear operator T in a Banach space. Then we have: − − Corollary 2.5. Assume that Qrr(λ)=(λMrr Vrr)(λIEr Wrr) for r =1, 2.If −1 −1 1 ∈/ σ(V22 M22) · σ(W11)={λ · µ : λ ∈ σ(V22 M22) and µ ∈ σ(W11)}, (2.11) then Q(·) has a lower triangular right root. The proof follows from the last corollary and the equality σ(3(T,S)) = σ(T ) · σ(S), see [20, Theorem]. The next theorem follows from Proposition 2.1 by constructing inductively the blocks in the subdiagonals of W and V . Theorem 2.6. Let the polynomial Q(·) be as in (2.1), E = E1 × E2 ×···×Em,and assume that the coefficients M, L and K have lower block triangular representa- tions, for example ⎛ ⎞ M11 00··· 0 ⎜ ⎟ ⎜ . ⎟ ⎜ M21 M22 0 . 0 ⎟ ⎜ ⎟ M = ⎜ ...... ⎟ , ⎜ . . . . . ⎟ ⎝ ⎠ Mm−1,1 ··· ··· Mm−1,m−1 0 Mm1 ··· ··· ··· Mmm where Mpq are linear maps form Eq into Ep for 1 ≤ q ≤ p ≤ m. Then the following assertions are equivalent: 1. The polynomial Q(·) has a right root which is lower block triangular (of the same block size as the coefficients). 2. For r = 1,2, ...,m the polynomial Qrr(·) has a right root Wrr and for 1 ≤ q has a solution X (of the same size as Mpq) which is a linear map from Eq into − − Ep.HereVpp is a linear map in Ep such that Qpp(λ)=(λMpp Vpp)(λIEp Wpp). 100 K.-H. F¨orster and B. Nagy − − For the proof we note the following: Let Qrr(λ)=(λMrr Vrr)(λIEr Wrr) for r =1, 2,...,m, and let Xpq be a solution of (2.12) for 1 ≤ q Then the first of these maps are the lower left blocks, and Wrr are the diagonal blocks of a lower block triangular right root W of Q(·). The second of these maps are the lower left blocks, and Vrr are the diagonal blocks of a lower block triangular V such that Q(λ)=(λM − V )(λIE − W ). We can generalize Corollaries 2.3–2.4 in an obvious way to the more general case of the last theorem. 3. Nonnegative roots of nonmonic matrix polynomials with nonnegative coefficients In this section we consider the matrix polynomial 2 Q(λ)=λIn×n − (λ A + λB + C) (3.1) with entrywise nonnegative n × n-matrices A,B and C. Here and in the remainder In×n and 0n×n denote the n × n identity matrix and the n × n zero matrix, respectively; sometimes we write for short I and 0 for an appropriate (square) identity matrix and a (not necessarily square) zero matrix, respectively. As mentioned in the introduction, we are interested in the existence of non- negative right roots of Q(·) in the case when S = A + B + C is reducible;forthe simpler case when S is irreducible see [7, Sections 3 and 4]. The nonnegative matrix S is cogredient to its (lower triangular) Frobenius normal form (see [18, Section 2]), i.e., there exists a permutation matrix P such that ⎛ ⎞ S11 00··· 0 ⎜ ⎟ ⎜ . ⎟ ⎜ S21 S22 0 .0⎟ t ⎜ ⎟ PSP = ⎜ ...... ⎟ (3.2) ⎜ . . . . . ⎟ ⎝ ⎠ Sm−1,1 ··· ··· Sm−1,m−1 0 Sm1 ··· ··· ··· Smm where the diagonal blocks Srr are irreducible square matrices. For our purpose we can and will assume, that S is in its Frobenius form. Since 0n×n ≤ A ≤ S, and a corresponding inequality for the other coefficients B and C holds, all coefficients and Q(·) are lower block triangular matrices of the same block size as S. If W is a right root of Q(·), then there exists a (unique) matrix V such that Q(λ)=(V − λA)(λIn×n − W ),λ∈ C, (3.3) On Reducible Nonmonic Matrix Polynomials 101 and (3.3) is equivalent to V = In×n − B − AW and C = VW (3.4) In the next two theorems it is not essential that (3.2) is the Frobenius normal form of S; only the lower block triangular form of S is needed, and the fact that the nonnegative coefficients A, B and C have a corresponding lower block triangular form. If Q(·) has a nonnegative right root, we know from [7, Proposition 3.1] that Q(·) has a smallest (= minimal) nonnegative right root. 2 Theorem 3.1. Let Q(λ)=λIn×n −(λ A+λB+C) with nonnegative n × n-matrices A, B and C and let S = A+ B + C be in lower block triangular form (3.2).Then the following assertions are equivalent: 1. Q(·) has a nonnegative right root. 2. Q(·) has a nonnegative right root which is lower block triangular (of the same block sizes as S); in particular, the smallest nonnegative right root of Q(·) is lower block triangular. 3. For r =1, 2,...,m the polynomial Qrr(·) has a nonnegative right root Wrr and for 1 ≤ q has a nonnegative matrix X (of the same size as Spq) as a solution. Here Vpp is a Z-matrix such that Qpp(λ)=(Vpp − λApp)(λI − Wpp). Proof. 1. implies 2.: From [7, Proposition 3.1] we know that the fixed point itera- tion 2 ≤ ≤ Wl+1 = AWl + BWl + C with 0n×n W0 C, l =0, 1,..., converges to the smallest nonnegative right root of Q(·). In our case W0, and then all matrices of the sequence (Wl) are lower block triangular (of the same block sizes as S). Therefore its limit is also lower block triangular (of the same block sizes as S). That 2. implies 1. is clear. For the equivalence of 2. and 3. apply Theorem 2.5 (note the sign change for V between the first factors in (2.4) and (3.3)!) and results of [7, Section 3]. Note that in the case of Theorem 3.1 the matrices Vpq for 1 ≤ q 2 Theorem 3.2. Let Q(λ)=λIn×n −(λ A+λB+C) with nonnegative n × n-matrices A, B and C and let S = A + B + C be in the lower block triangular form (3.2). Assume that for k =1, 2,...,m there exists a positive ρk such that 2 r(Skk(ρk)) = r( ρk Akk + ρkBkk + Ckk ) <ρk, (3.7) and ρ1 ≤ ρ2 ≤···≤ρm. (3.8) Then Q(·) has a nonnegative right root; more precisely, there exist lower block triangular matrices W and V (with the same block sizes as S) such that Q(λ)= (λM − V )(λIn×n − W ),whereW is a nonnegative matrix and V is a nonsingular M-matrix. Proof. By [7, Theorem 3.4] there exists for k =1, 2,...,m a nonnegative matrix −1 Wkk with r(Wkk) <ρk and a nonsingular M-matrix Vkk with r(Vkk Akk) < 1/ρk 2 such that Qkk(λ)=λI − (λ Akk + λBkk + Ckk)=(λMkk − Vkk)(λIkk − Wkk). −1 This and (3.8) imply that r(3(Vpp App,Wqq)) < 1for1≤ q By Theorem 5.1 of [8] it is possible to generalize both theorems above to polynomials with coefficients which are more general than matrices, for example operators in ordered Banach spaces. Note that the assumptions of Theorem 3.2 about the spectral radii do not imply that r( ρ2A + ρB + C ) <ρfor some positive ρ (see [7, Example 3.3]). Therefore one cannot directly apply [7, Theorem 3.4]. The following example shows that the above theorem does not hold if we assume only (3.7), but not (3.8). Example 3.3. Let a be a positive number and define 2 00 00 10 Q(λ)=λI2×2 − λ + λ + . 0 a a 0 10 Then equation (3.5) becomes (1 − a)x =1+a. Theorem 3.1 shows that Q(·)has no nonnegative (triangular) right root for a>1. For a>1andpositiveρ1 and ρ2 2 1=r(S11(ρ1)) <ρ1 and aρ2 = r(S22(ρ2)) <ρ2 imply ρ1 > 1 > 1/a > ρ2. As a preparation for the next theorem we introduce the following notation. 2 Let Q(λ)=λIn×n − (λ A + λB + C) with nonnegative n × n-matrices A, B and On Reducible Nonmonic Matrix Polynomials 103 C and let S = A + B + C be in (lower triangular) Frobenius normal form (3.2). Assume that r(S(ˆρ)) = r(ˆρ2A +ˆρB + C ) ≤ ρˆ for some positiveρ. ˆ (3.9) Since for k =1, 2,...,m the matrix Skk(1) is irreducible, the function ]0, ∞[→ [0, ∞[,ρ→ r(Skk(ρ)) is real analytic; see [7, Section 3]. By r (Skk(·)) we denote its derivative. Now we define the following sets: M := {k ∈{1, 2,...,m} : r(Skk(ˆρ)) =ρ ˆ}, M− := {k ∈ M : r (Skk(ˆρ)) < 1}, M1 := {k ∈ M : r (Skk(ˆρ)) = 1} and M+ := {k ∈ M : r (Skk(ˆρ)) > 1}. For k =1, 2,...,mwe denote by dk the index of phase imprimitivity (see [7]) of the graph G(Ckk]Bkk[Akk). We recall (see [7, Proposition 4.5]): dk = 0 implies that for a positive ρ with r(Skk(ρ)) = ρ and for a θ ∈ [0, 2π[ the complex number ρeiθ is an eigenvalue of Q (·) if and only if θ ∈{0, 2π , 2 · 2π ,...,(d − 1) 2π }. kk dk dk k dk The following theorem generalizes [9, Theorem 5] for quadratic matrix poly- nomials considered in this paper. 2 Theorem 3.4. Let Q(λ)=λIn×n −(λ A+λB+C) with nonnegative n × n-matrices A, B and C and let S = A+B+C be in its Frobenius normal form (3.2). Assume that there exists a positive ρˆ such that r(S(ˆρ)) = r(ˆρ2A +ˆρB + C ) ≤ ρ.ˆ (3.10) With the notation above the following assertions hold: 1. The set of the eigenvalues of Q(·) is the whole plane C if and only if there exists a k ∈ M such that r(Skk(ρ)) = ρ for all nonnegative ρ. 2. Assume that the set of the eigenvalues of Q(·) is not the whole plane C.Then · − (a) Q( ) has exactly n k∈M− ∪ M1 dk eigenvalues (counting algebraic mul- D { ∈ C | | } tiplicities) in the open disc ρˆ = λ : λ < ρˆ . · (b) Q( ) has exactly k∈M− ∪ M+ dk +2 k∈M1 dk eigenvalues (counting al- gebraic multiplicities) on the circle Tρˆ = {λ ∈ C : |λ| =ˆρ }. Proof. The polynomial Q(·) is block triangular, therefore the set of its eigenval- ues is the union of the sets of the eigenvalues of its diagonal blocks Qkk(·),k = 1, 2,...,m. Each diagonal block in the Frobenius normal form is irreducible. There- fore we can apply to each of them [7, Theorem 4.10] and will obtain the asser- tions. In [9, Theorem 5] the main assumption is that S is substochastic. If r(S)=1, this implies (see [9, Equations (55) and (56)]) that there exists a p ∈{1, 2,...,m} 104 K.-H. F¨orster and B. Nagy such that M = {1, 2,...,p} and the Frobenius normal form of S(ρ)is ⎛ ⎞ S11(ρ)0··· 00··· 0 ⎜ ⎟ ⎜ .. .. ⎟ ⎜ 0 . . 00··· 0 ⎟ ⎜ ⎟ ⎜ ...... ⎟ ⎜ . . . . . ··· . ⎟ ⎜ ⎟ ⎜ 0000⎟ ⎜ ⎟ ⎜ 0 ··· 0 Spp(ρ)0··· 0 ⎟ . (3.11) ⎜ ⎟ ⎜ Sp+1,1(ρ) ··· ··· Sp+1,p(ρ) Sp+1,p+1(ρ)0··· 0 ⎟ ⎜ ⎟ ⎜ ...... ⎟ ⎜ ...... ⎟ ⎜ ⎟ ⎝ . . . . ⎠ . . . .. 0 Sm1(ρ) ··· ··· Smp(ρ) Sm,p+1(ρ) ··· ··· Smm(ρ) 2 Corollary 3.5. Let Q(λ)=λIn×n−(λ A+λB+C) with nonnegative n × n-matrices A, B and C. Assume that there exists a positive ρˆ such that r(S(ˆρ)) ≤ ρˆ.Then Q(·) has a nonnegative right root if one of the following assumptions is satisfied: 1. M := {k ∈{1, 2,...,m} : r(Skk(ˆρ)) =ρ ˆ} is empty. 2. There exists a p ∈{1, 2,...,m} such that M := {k ∈{1, 2,...,m} : r(Skk(ˆρ)) =ρ ˆ} = {1, 2,...,p}, and (3.11) is the Frobenius normal form of S(ρ). Proof. Assumption 1. is equivalent to r(S(ˆρ)) < ρˆ. The assertion follows from [7, Theorem 3.4] or Theorem 3.2 with m =1. Let Assumption 2. hold. Then r(S(ˆρ)) =ρ ˆ.Fork =1, 2,...,p each Qkk(·)hasa nonnegative right root with spectral radius equal toρ ˆ (see [7, Proposition 4.6]). Since the upper left corner of Q(·)(ofthefirstp × p blocks) is diagonal, it has a nonnegative right root with spectral radius equal toρ ˆ (see Corollary 2.3). The right lowercornerofS(·) has a spectral radius strictly less thanρ ˆ. Now an analogous argument as in the proof of Theorem 3.2 proves the assertion. As a completion to [9, Theorem 5] we obtain in the case of a quadratic matrix polynomial, the following: 2 Corollary 3.6. Let Q(λ)=λIn×n−(λ A+λB+C) with nonnegative n × n-matrices A, B and C.IfS = A + B + C is substochastic, then Q(·) has a nonnegative right root. The proof follows from the preceding corollary as the caseρ ˆ =1. In [3] the authors consider the polynomial −Q(·)where 2 Q(λ)=λIn×n − (λ In×n + C). (3.12) and C is a nonnegative n × n-matrix. In [3, Theorem 2.3] they give necessary and sufficient conditions for the existence of nonnegative roots of Q(·). Their proof uses 1 analytical methods, for example the convergence of the binomial series of 4 I − C. On Reducible Nonmonic Matrix Polynomials 105 We will give a different proof of their results by methods in the spirit of this paper (see Theorem 3.10 below). We need some preparations. Let Q(·) be as in (3.12). Then r(S(ρ)) = ρ2 +r(C)forpositiveρ. This implies that both equalities r(S(ˆρ)) =ρ ˆ and r (S(ˆρ)) = 1 hold if and only if both equalities r(C)=1/4andˆρ =1/2hold. From [7, Theorem 3.4] it follows Proposition 3.7. Let Q(·) be as in (3.12) with a nonnegative square matrix C with r(C) < 1/4.ThenQ(·) has a nonnegative right root. In addition, there exist a nonnegative matrix W and a nonsingular M-matrix V such that 2 Q(λ)=λIn×n − (λ In×n + C)=(V − λI)(λI − W ), (3.13) r(W ) < 1/2 and r(V −1) < 2. From [7, Theorems 3.4 and 4.10] it follows Proposition 3.8. Let Q(·) be as in (3.12) with a nonnegative irreducible square matrix C.ThenQ(·) has a nonnegative right root if and only if r(C) ≤ 1/4. In addition, there exist a nonnegative matrix W and a nonsingular M-matrix V such that (3.13) holds, and 1. r(W )=1/2 and r(V −1)=2,ifr(C)=1/4, 2. r(W ) < 1/2 and r(V −1) < 2,ifr(C) < 1/4. Note that for a nonnegative irreducible matrix its spectral radius is a simple root of its characteristic polynomial (see [16, Theorem 1.4.3]). This implies that it is in the terminology of [3] an eigenvalue value of degree 1, i.e., it is a semisimple eigenvalue. Therefore the first part of the last proposition is exactly [3, Theorem 2.3] for the case of an irreducible matrix C. Lemma 3.9. 1. Let Q(·) be as in (3.12) with a nonnegative irreducible square matrix C and 2 Q(λ)=λIn×n − (λ In×n + C)=(V − λI)(λI − W ), with nonnegative W and a nonsingular M-matrix V .ThenW is irreducible and V −1 =(I − W )−1 is strictly positive, i.e., all entries of V −1 are positive. 2. Let W be nonnegative and irreducible, V have a strictly positive inverse, −1 −1 r(W )=r(V ) and B ≥ 0n×n. Then the equation VX− XW = B has a nonnegative solution if and only if B =0n×n;hereW, V, B, 0n×n and X are matrices of the size n × n. Proof. 1. The irreducibility of W is easily proved by contradiction using C = VW and V = I − W . The second assertion follows from [16, Theorem 2.3]. 2. The equation VX − XW = B is equivalent to X − 3(V −1,W)X = X − V −1XW = V −1B. By the description of the spectrum of 3(V −1,W) following Corollary 2.5, we obtain r(3(V −1,W)) = r(V −1)r(W )=1.Nowwewillusesome results from the local spectral theory of nonnegative operators in finite-dimensional spaces, see [5, 6]. In the following we write for short 3 instead of 3(V −1,W). 106 K.-H. F¨orster and B. Nagy ♦ 3j ♦ 3 Set = 0≤j≤n . Then the linear maps and (acting in the linear space of all complex n × n-matrices) commute. Since W is irreducible and V −1 is strictly positive, it follows that ♦V −1B is strictly positive when B is nonnegative −1 −1 and nonzero. This implies that the local spectral radius r (♦V B)of♦V B with respect to 3 is equal to the spectral radius of 3,see[6,Proposition1].Now −1 −1 1=r (♦V B) ≤ r (V B) ≤ r(3) = 1. We are in finite-dimensional spaces, therefore, by [5, Theorem 12] or [19, Theorem 3.1], the equation X − 3X = X − V −1XW = V −1B with nonnegative B has a nonnegative solution if and only if B =0n×n. The following theorem was proved in [3, Theorem 2.3] by completely different methods. Theorem 3.10. Let 2 Q(λ)=λIn×n − (λ In×n + C). (3.14) where C is a nonnegative n × n-matrix. Then 1. r(C) < 1/4 implies that there is a nonnegative right root of Q(·). 2. r(C) > 1/4 implies that there is no nonnegative right root of Q(·). 3. If r(C)=1/4,thenQ(·) has a nonnegative right root if and only if 1/4 is a semisimple eigenvalue of C. 2 2 Proof. For S(λ)=λ In×n + C we have r(S(ρ)) = ρ + r(C) for nonnegative ρ. Then r(C) < 1/4 implies that r(S(1/2)) < 1/2. From [7, Theorem 3.4] we obtain that Q(·) has a nonnegative right root, which proves 1. Let r(C) > 1/4. In the Frobenius normal form of C there exists an irre- 2 ducible diagonal block Ckk with r(Ckk) > 1/4. Then r(Skk(ρ)) = ρ + r(Ckk) ≥ ρ + r(Ckk) − 1/4 for all nonnegative ρ. By [7, Theorem 4.10(i)] Qkk(·)hasno 2 nonnegative right root, here Qkk(λ)=λ − (λ + Ckk). By Theorem 3.1, Q(·)has no nonnegative right root, this proves 2. Now assume that r(C)=1/4 and 1/4 is an eigenvalue of C of degree 1. By [17, Theorem 3.1] or [13, Proposition II.1.], C is (cogredient to) ⎛ ⎞ C11 00 ⎝ ⎠ C˜ = C21 C22 0 , C31 C32 C33 where 1. C11 are C33 are possibly empty, but r(Ckk) < 1/4fork =1, 3ifthisisnot the case, 2. C22 is block diagonal, C22 = Diag(C1,C2,...,Cp) where p is the dimension of the eigenspace Ker(1/4I − C), Cl is irreducible and r(Cl)=1/4forl =1, 2,...,p. On Reducible Nonmonic Matrix Polynomials 107 From Proposition 3.8.1. we obtain 2 Q22(λ)=λI − (λ I + C22)=(V22 − λI)(λI − W22) with nonnegative W22 = Diag(W1,W2,...,Wp), an invertible M-matrix V2 = −1 −1 −1 −1 Diag(V1,V2,...,Vp)andr(W22)=r(Wl)=1/2=r(V22 ) = r(Vl ) for l =1, 2,...,p. From Proposition 3.8.2. we obtain for k =1, 3 2 Qkk(λ)=λI − (λ I + Ckk)=(Vkk − λI)(λI − Wkk) −1 with nonnegative Wkk,invertibleM-matrices Vkk, r(Wkk)<1/2andr(Vkk )<2. −1 This implies r(3(Vpp ,Wqq)) < 1for1≤ p 1. C11 and C33 are irreducible, 2. r(Ckk)=1/4fork =1, 3, 3. C22 may be empty (then C21 and C32 are also empty), but r(C22) < 1/4if this is not the case, 4. C31 + C32C21 > 0, (3.16) i.e., the matrix of the left-hand side is nonnegative and nonzero. Then Q˜(λ)=λI − (λ2I + C˜)=(V˜ − λI)(λI − W˜ ), where W˜ and V˜ are the corresponding principal submatrices of W and V , respectively. Further W˜ and V˜ have the same block structure as C˜ in (3.15). By Theorem 3.1, W31 is a nonnegative solution of V33X − XW11 = C31 − V32W21. Now Vrs = −Wrs are nonpositive for 1 ≤ s From C˜ = V˜ W˜ = W˜ V˜ , W˜ ≥ 0andV˜ = I − W˜ we obtain 0 ≤ C21 = W21V11 + W22V21 ≤ W21V11.NowtheinverseofV11 exists and is nonnegative, therefore 0 ≤ −1 C21V11 ≤ W21; in a similar way we obtain from 0 ≤ C32 = V32W22 +V33W32 that −1 −1 −1 0 ≤ V33 C32 ≤ W32 = −V32.Butthen0≤ V33 C32C21V11 ≤−V32W21 =0, which is equivalent to C32C21 = 0. Together with C31 = 0 from above, we get a contradiction to (3.16). Therefore 1/4 is not an eigenvalue of C of degree greater than 1. Acknowledgment The authors wish to thank the referee for his/her careful reading and for several suggestions. References [1] Berman, A., Plemmons, R.: Nonnegative Matrices in the Mathematical Sciences. SIAM: Philadelphia, 1994. [2] Bini, D.A., Latouche, G., Meini, B.: Numerical Methods for Structured Markov Chains, Oxford University Press, Oxford-New York, 2005. [3] Butler, G.J., Johnson, C.R., Wolkowicz, H.: Nonnegative Solutions of a Quadratic Matrix Equation Arising from Comparison Theorems in Ordinary Differential Equa- tions. SIAM J. Alg. Disc. Meth. 6, 47–53 (1985). [4] Conway, J.B.: A Course in Functional Analysis, Springer: New York-Berlin-Heidel- berg-Tokyo, 1985. [5] F¨orster, K.-H., Nagy, B.: On the Local Spectral Theory of Positive Operators. Oper- ator Theory, Advances and Applications 28, 71–81, Birkh¨auser: Basel-Boston-Berlin, 1988. [6] F¨orster, K.-H., Nagy, B.: On the Local Spectral Radius of a Nonnegative Element with Respect to an Irreducible Operator, Acta Sci. Math. 55, 155–166 (1991). [7] F¨orster, K.-H., Nagy, B.: On Nonmonic Quadratic Matrix Polynomials with Non- negative Coefficients. Operator Theory, Advances and Applications 162, 145–163, Birkh¨auser: Basel-Boston-Berlin, 2005. [8] F¨orster, K.-H., Nagy, B.: Spectral Properties of Operator Polynomials with Non- negative Coefficients. Operator Theory, Advances and Applications 163, 147–162, Birkh¨auser: Basel-Boston-Berlin, 2005. [9] Gail, H.R., Hantler, S.L., Taylor, B.A.: Spectral Analysis of M/G/1andG/M/1 Type Markov Chains, Adv. Appl. Prob. 28, 114–165 (1996). [10] Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials, Academic Press: New York, 1982. [11] Horn, R.A., Johnson, C.R.: Matrix Analysis, Cambridge University Press: Cam- bridge, 1985. [12] Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis, Cambridge University Press: Cambridge, 1991. [13] Jang, R., Victory, Jr., H.D.: On the Ideal Structure of Positive, Eventually Compact Operators on Banach Lattices, Pacific J. Math. 157, 57–85 (1993). On Reducible Nonmonic Matrix Polynomials 109 [14] Latouche, G., Ramaswami, S.: Introduction to Matrix Analytic Methods in Sto- chastic Modeling. ASA-SIAM Series on Stochastics and Applied Probability. SIAM: Philadelphia, 1999. [15] Marcus, A.S.: Introduction to the Spectral Theory of Polynomial Operator Pencils. Translation of Mathematical Monographs, Vol. 71, Amer. Math. Soc., Providence, 1988. [16] Minc, H.: Nonnegative Matrices, Wiley: New York, 1988. [17] Rothblum, U.G.: Algebraic Eigenspaces of Nonnegative Matrices, Linear Algebra Appl. 12, 281–291 (1975). [18] Schneider, H.: The Influence of the Marked Reduced Graph of a Nonnegative Matrix on the Jordan Form and on Related Properties: A Survey. Linear Algebra Appl. 84, 169–189 (1986). [19] Tam, Bit-Shun, Schneider, H. Linear Equations over Cones and Collatz-Wielandt Numbers. Linear Algebra Appl. 363, 295–332 (2003). [20] Trampus, A.: A Spectral Mapping Theorem for Functions of Two Commuting Linear Operators, Proc. Amer. Math. Soc. 14, 893–895 (1963). K.-H. F¨orster Technische Universit¨at Berlin Institut f¨ur Mathematik, MA 6-4 D-10623 Berlin, Germany e-mail: [email protected] B. Nagy University of Technology and Economics Department of Analysis Institute of Mathematics H-1521 Budapest, Hungary e-mail: [email protected] Operator Theory: Advances and Applications, Vol. 175, 111–120 c 2007 Birkh¨auser Verlag Basel/Switzerland On Exceptional Extensions Close to the Generalized Friedrichs Extension of Symmetric Operators Seppo Hassi, Henk de Snoo and Henrik Winkler Abstract. If the Q-function Q corresponding to a closed symmetric operator S with defect numbers (1, 1) and one of its selfadjoint extensions belongs to the Kac class N1 then it is known that all except one of the Q-functions of S belong to N1, too. In this note the situation that the given Q-function does not belong to the class N1 is considered. If Q ∈ Np, i.e., if the restriction of the spectral measure of Q on the positive or the negative axis corresponds to an N1-function, then Q itself is the Q-function of the exceptional extension, and, hence, it is associated with the generalized Friedrichs extension of S.IfQ or, equivalently, the spectral measure of Q is symmetric, or if the difference of Q and a symmetric Nevanlinna function belongs to the class N1 or Np,then Q is still exceptional in a wider sense. Similar results hold for the generalized Kre˘ın-von Neumann extension of the symmetric operator. Mathematics Subject Classification (2000). Primary 47A06, 47B25; Secondary 47A57 . Keywords. Q-function, generalized Friedrichs extension, generalized Kre˘ın-von Neumann extension, Kac class. 1. Introduction Let S be a not necessarily densely defined closed symmetric relation in a Hilbert space (H, (·, ·)) with defect numbers (1, 1) and let A be some canonical (i.e., dom A ⊂ H) selfadjoint extension of S with resolvent set ρ(A). Let χ(µ), µ ∈ C\R, be a nontrivial element of ker(S∗ − µ), where S∗ denotes the adjoint of S.Then the element χ(z):=(I +(z − µ)(A − z)−1)χ(µ),z∈ ρ(A), The research was partially supported by the Research Institute for Technology at the University of Vaasa. H. Winkler was supported by the “Fond zur F¨orderung der wissenschaftlichen Forschung” (FWF, Austria), grant number P15540-N05. 112 S. Hassi, H.S.V. de Snoo and H. Winkler belongs to ker(S∗ − z)andforeachz ∈ ρ(A) the symmetric relation S can be recovered from A and χ(z)by S = {{f,g}∈A :(g − zf,χ(¯z)) = 0 }. Let N be the set of Nevanlinna functions, i.e., the set of all functions which are analytic on C\R,satisfyQ(z)=Q(¯z), and map the open upper half-plane C+ into C+ ∪ R. Recall that the Q-function associated with S and A is a solution (unique up to a real constant) of the equation Q(z) − Q(µ) =(χ(z),χ(µ)),z,µ∈ ρ(A), z − µ¯ which implies that Q ∈ N. Each Nevanlinna function Q ∈ N which is not equal to a real constant is the Q-function of a closed symmetric relation S with defect numbers (1, 1) and a canonical selfadjoint extension A of S, see, e.g., [4]. Moreover, for a given Q ∈ N the relations S and A are uniquely determined up to isometric isomorphisms if S is completely nonselfadjoint, i.e., if there exists no nontrivial orthogonal decomposition of S such that one of the summands is selfadjoint. Note that a completely nonselfadjoint closed symmetric relation is automatically an operator. The canonical selfadjoint extensions of S can be uniquely parametrized by α ∈ (−π/2,π/2] : 1 (A(α) − z)−1 =(A − z)−1 − χ(z) (·,χ(¯z)), (1.1) Q(z)+tanα where Q is the Q-function corresponding to S and A = A(π/2). If S is nonnegative, among all selfadjoint extensions of S there are two ex- tremal nonnegative selfadjoint extensions, the so-called Friedrichs extension and the Kre˘ın-von Neumann extension. In [4] a generalization of the Friedrichs ex- tension was introduced for a class of non-semibounded symmetric operators with defect numbers (1,1), which was further studied in, e.g., [1], cf. [2]. If one of the Q-functions of S belongs to the so-called Kac class N1 (see below), then all but one of the Q-functions belong to N1. The Q-function which does not belong to N1 corresponds to the generalized Friedrichs extension. A corresponding characteri- zation of the Kre˘ın-von Neumann extension is investigated in [3]. For an extension of such results to a class of Nevanlinna functions wider than N1 which is related to Sturm-Liouville equations, see [5]. A generalization of the above results in a different direction can be found in [9] and will be considered in the present paper. In the above papers it is usually assumed that the Q-function corresponding to a given extension belongs to the class N1 or N1, respectively. In this note some results are presented for the cases that the given Q-function does not belong to the class N1 or N1 (the precise definitions of various classes are given in Section 2). If Q belongs to the class Np or Np then it corresponds to the generalized Friedrichs or to the generalized Kre˘ın-von Neumann extension, respectively; see Theorems 4.2 and 5.2 below. A new situation occurs if Q is symmetric or if the difference between Q and a symmetric Nevanlinna function is a function from one of the Exceptional extensions 113 above classes. Then there are at most two extensions which do not belong to the classes N1+ or N1+, see Theorems 4.3 and 5.3 below, cf. [9]. Nevertheless, it is also possible that there is no exceptional extension. 2. Preliminaries Let A be a canonical selfadjoint extension of S with a corresponding Q-function denoted by Q and let A(α) be the canonical selfadjoint extension of S which is determined by the identity (1.1). For α ∈ (−π/2,π/2) the Q-function Qα corre- sponding to A(α)andS satisfies Q(z)tanα − 1 1+tan2 α Q (z)= =tanα − ,z∈ C \ R, (2.1) α Q(z)+tanα Q(z)+tanα (see [6], Proposition 4.4 and what follows). It follows from (2.1) that 1+tan2 α Im Qα(z)= 2 Im Q(z). (2.2) |Q(z)+tanα| Each function Q ∈ N has an integral representation of the form 1 − λ Q(z)=bz + a + 2 dσ(λ), (2.3) R λ − z 1+λ with a ∈ R, b ≥ 0, and a measure σ with the property that dσ(λ) ∞ 2 < + . R 1+λ The following identities are immediate from (2.3) dσ(λ) Im Q(iy)=by + y 2 2 ,y>0, (2.4) R λ + y and − 2 λdσ(λ) Re Q(iy)=a +(1 y ) 2 2 2 ,y>0. (2.5) R (λ + y )(1 + λ ) Let Nγ , γ ∈ [0, 2), be the class of all functions Q ∈ N for which b =0inthe representation (2.3) and for which the inequality dσ(λ) ∞ γ < + (2.6) R 1+|λ| is satisfied (see [7]). If Q ∈ N1, the so-called Kac class, it has an integral repre- sentation of the form dσ(λ) − λ Q(z)=a1 + with a1 = a 2 dσ(λ). (2.7) R λ − z R 1+λ Let Nγ, γ ∈ [0, 2), be the class of all functions Q ∈ N with the property that 1 dσ(λ) ∞ 2−γ < + . (2.8) −1 |λ| 114 S. Hassi, H.S.V. de Snoo and H. Winkler If Q ∈ N1, it has an integral representation of the form (see [3]) z 1 Q(z)=bz +ˆa1 + dσ(λ)withˆa1 = a + 2 dσ(λ). (2.9) R λ(λ − z) R λ(1 + λ ) ≤ N ⊂ N N ⊂ N Clearly, if 0 γ1 <γ2 < 2then γ1 γ2 and γ1 γ2 .Moreover,with γ = 0 one has Q ∈ N0 if and only if limy→∞ yIm Q(iy) < ∞ and Q ∈ N0 if and −1 only if limy→0+ y Im Q(iy) < ∞. The remaining subclasses with γ ∈ (0, 2) can be characterized as follows (see [7], [8]): Proposition 2.1. Let γ ∈ (0, 2). A Nevanlinna function Q belongs to the class Nγ if and only if ∞ Im Q(iy) ∞ γ dy < + , (2.10) 1 y and it belongs to the class Nγ if and only if 1 Im Q(iy) ∞ 2−γ dy < + . (2.11) 0 y In [9] a generalization of results from [4] and [3] led to the following two propositions: Proposition 2.2. Let Q ∈ Nγ for some γ ∈ (0, 1],andletQα be given by the identity (2.1).Letα0 ∈ (−π/2,π/2) be the solution of tan α0 + a1 =0,wherea1 is the corresponding constant in the representation (2.7) of Q.ThenQα ∈ Nγ for ∈ − \{ } each α ( π/2,π/2) α0 , whereas Qα0 , which corresponds to the generalized N ∈ N ∈ N Friedrichs extension, does not belong to the class 2−γ : Qα0 2−γ.IfQ γ˜ for some γ˜ ∈ (0,γ) then Qα ∈ Nγ˜ for each α ∈ (−π/2,π/2) \{α0}. Proposition 2.3. Let Q ∈ Nγ for some γ ∈ (0, 1],andletQα be given by the identity (2.1).Letα0 ∈ (−π/2,π/2) be the solution of tan α0 +ˆa1 =0,whereaˆ1 is the corresponding constant in the representation (2.9) of Q.ThenQα ∈ Nγ for ∈ − \{ } each α ( π/2,π/2) α0 , whereas Qα0 , which corresponds to the generalized N ∈ N Kre˘ın-von Neumann extension, does not belong to the class 2−γ : Qα0 2−γ. If Q ∈ Nγ˜ for some γ˜ ∈ (0,γ),thenQα ∈ Nγ˜ for each α ∈ (−π/2,π/2) \{α0}. 3. Some new subclasses of Nevanlinna functions For the sequel of the paper it is necessary to introduce some more subclasses of the class of all Nevanlinna functions N. First the classes N1+ and N1+ will be defined: N1+ := Nγ and N1+ := Nγ. 1<γ<2 1<γ<2 Then Q ∈ N1+ if and only if ∞ Im Q(iy) ∞ 1+δ dy < + (3.1) 1 y Exceptional extensions 115 for each δ>0, and Q ∈ N1+ if and only if 1 Im Q(iy) ∞ 1−δ dy < + (3.2) 0 y for each δ>0. In particular, N1 ⊂ N1+ and N1 ⊂ N1+, and these inclusions are proper. Now the subclasses Np and Np will be introduced. Let Q ∈ N have the integral representation (2.3). Then there is a decomposition of Q in terms of the corresponding spectral function: define the functions Q+ and Q− by 1 λ Q+(z):=bz + a + − dσ(λ) − 2 [0,∞) λ z 1+λ and Q−(z):=Q(z) − Q+(z). Observe that Q ∈ N \ N1 if and only if either Q+ ∈ N \ N1 or Q− ∈ N \ N1. The class of all Nevanlinna functions Q with the property that either Q+ ∈ N1 and Q− ∈ N \ N1 or Q− ∈ N1 and Q+ ∈ N \ N1 is denoted by Np.Notethat Np ⊂ N\N1. Similarly, observe that Q ∈ N\N1 if and only if either Q+ ∈ N\N1 or Q− ∈ N \ N1. The class of all Nevanlinna functions Q with the property that either Q+ ∈ N1 and Q− ∈ N\N1 or Q− ∈ N1 and Q+ ∈ N\N1 is denoted by Np. Finally, Ns stands for the class of all symmetric Nevanlinna functions Q,that is, functions Q ∈ N for which the identity Q(z)=−Q(−z)holdsforallz ∈ C \ R. In particular, if Q ∈ Ns,thenReQ(iy)=0,y>0. 4. The case that Q ∈ N \ N1 In this section some functions in N \ N1 will be identified as exceptional functions for a family of Nevanlinna functions of the form (2.1). A first observation concerns the limiting behavior of functions in N \ N1. Lemma 4.1. Let Q ∈ N.IfQ+ ∈ N \ N1 then limy→+∞ Re Q+(iy)=−∞,and if Q− ∈ N \ N1 then limy→+∞ Re Q−(iy)=+∞. Proof. Assume that Q+ ∈ N \ N1 and define λ dσ1(λ):= dσ(λ),λ>0. 1+λ2 0 ∞ Then the identity dσ1(λ)=+∞ holds. Choose K>0 and a corresponding 0 0 l ≥ ≥ l>0 such that 0 dσ1(λ) K.Thenfory l one has ∞ 2 l 2 y ≥ y ≥ K 2 2 dσ1(λ) 2 2 dσ1(λ) . 0 λ + y 0 l + y 2 116 S. Hassi, H.S.V. de Snoo and H. Winkler √ It follows from the identity (2.5) that for y ≥ max{ 2,l} the inequality ∞ 2 ≤ − 1 y ≤ − K Re Q+(iy) a 2 2 dσ1(λ) a 2 0 λ + y 4 holds. Hence limy→∞ Re Q+(iy)=−∞,sinceK can be chosen arbitrary large. If Q− ∈ N \ N1, the statement of the lemma can be shown in a similar way. Theorem 4.2. Let Q ∈ Np and let Qα be given by (2.1).ThenQα ∈ N1 for each α ∈ (−π/2,π/2). Proof. Assume that Q− ∈ N1 and Q ∈ N \ N1.Let 1+tan2 α Q+α(z):=tanα − (4.1) Q+(z)+tanα As in [4], for x<−1 one has 1 1 Q+(x) − Q+(−1) = − dσ(λ) < 0. − [0,∞) λ x λ +1 −1 It follows that the Nevanlinna function Q1(z):=−(Q+(z)−Q+(−1)) is positive −1 −1 on (−∞, −1). Since limy→+∞ y Q1(iy) = limy→+∞(yQ+(iy)) = 0 one can assume that Q1 has a representation of the form ∞ 1 λ − Q1(z)=a + 2 dσ1(λ). −1 λ − z 1+λ −1 As in [7], for each λ ≥−1 the function hλ(x):=(λ − x) is increasing on (−∞, −1). By monotone convergence one concludes ∞ − λ ≥ lim Q1(x)=a 2 dσ1(λ) 0, x→−∞ −1 1+λ ∈ N − − ∈ N hence Q1 1.Lettanα1 := Q+( 1). It follows that Qα1+ 1,which implies by Proposition 2.2 that Q+α ∈ N1 for each α ∈ (−π/2,π/2) (see also [4]). Let c := limy→∞ Re Q−(iy)andlettanβ := tan α + c +1 forα ∈ (−π/2,π/2). Since limy→+∞ Re Q+(iy)=−∞ by Lemma 4.1, it follows from (2.2) that for α ∈ (−π/2,π/2) and sufficiently large y Im Qα(iy) Im Q+(iy)+ImQ−(iy) 2 = 2 2 1+tan α (Re Q+(iy)+ReQ−(iy)+tanα) +(ImQ−(iy)+ImQ+(iy)) Im Q+(iy) ≤ 2 2 +ImQ−(iy) (Re Q+(iy)+c +1+tanα) +(ImQ+(iy)) Im Q+β(iy) < +ImQ−(iy). (4.2) 1+tan2 β Hence Qα ∈ N1,sinceQ+β ∈ N1 and Q− ∈ N1.IfQ+ ∈ N1, the statement of the theorem can be shown in a similar way. Exceptional extensions 117 Theorem 4.3. Assume that Q ∈ N \ N1 is of the form Q = Qs + Qp such that Qp ∈ Np ∪ N1 and that for some xq ∈ R the shifted function Qˆs(z):=Qs(z − xq) belongs to Ns.LetQα be given by the identity (2.1).Then: (i) if Qp ∈ N1, there is at most one value α0 ∈ (−π/2,π/2) such that Qα ∈ N1+ for each α ∈ (−π/2,π/2) \{α0}; (ii) if Qp ∈ Np,thenQα ∈ N1+ for each α ∈ (−π/2,π/2). Proof. Let Qˆ(z):=Q(z − xq)andQˆp(z):=Qp(z − xq). Then also Qˆp ∈ Np ∪ N1. Let Qˆα be given by (2.1) with Qˆ instead of Q.SinceQˆs ∈ Ns, one has Re Qˆs(iy)= 0 and the identity (2.2) gives (1 + tan2 α)Im Qˆ(iy) Im Qˆα(iy)= (Im Qˆ(iy))2 + (tan α +ReQˆ (iy))2 p (4.3) 1+tan2 α ≤ . 2| tan α +ReQˆp(iy)| (i) Assume Qp ∈ N1 or equivalently that Qˆp ∈ N1.Let a1 := lim Re Qˆp(iy), y→+∞ and let α0 ∈ (−π/2,π/2) be the solution of tan α0 + a1 =0.Forα = α0 it follows from (4.3) that Im Qˆα is uniformly bounded on the interval (i, i∞), hence Qˆα ∈ N1+. (ii) Assume that Qp ∈ Np or equivalently Qˆp ∈ Np. Then Lemma 4.1 shows that lim |Re Qˆp(iy)| =+∞, y→+∞ and it follows from (4.3) that Im Qˆα is uniformly bounded on the interval (i, i∞). Hence Qˆα ∈ N1+ for each α ∈ (−π/2,π/2). Finally, observe that Qα(z − xq)=Qˆα and therefore Qα ∈ Nγ if and only if Qˆα ∈ Nγ . This completes the proof. In particular, if Q ∈ N \ N1 is symmetric, i.e., if Q ∈ Ns then assertion (i) of Theorem 4.3 holds. However, it should be observed that there are functions in N \ N1 which do not serve as exceptional functions for a family of Nevanlinna functions in N1+ of the form (2.1). Consider, for example, Q = i.ThenQ ∈ Ns and it is easy to see that Q ∈ N1+ \ N1.SinceQα = i for each α ∈ (−π/2,π/2), there is no exceptional extension. 5. The case that Q ∈ N \ N1 In this section some functions in N \ N1 will be identified as exceptional functions for a family of Nevanlinna functions of the form (2.1). Again, a first observation concerns the limiting behavior of functions in N \ N1. 118 S. Hassi, H.S.V. de Snoo and H. Winkler Lemma 5.1. Let Q ∈ N.IfQ+ ∈ N \ N1 then limy→+0 Re Q+(iy)=+∞,andif Q− ∈ N \ N1 then limy→+0 Re Q−(iy)=−∞. ∈ N \ N Proof. Assume√ that Q+ 0 1.ChooseK>0 and a corresponding real l with 1 ≥ ≤ ≤ 0 Hence limy→+0 Re Q+(iy)=+∞,sinceK can be chosen arbitrary large. In case Q− ∈ N \ N1, the statement of the lemma can be shown in a similar way. Of course, the result in Lemma 5.1 can be derived also from Lemma 4.1 by means of the transform Q˜(z)=−Q(1/z), which connects the classes N1 and N1, see [3]. Theorem 5.2. Let Q ∈ Np and let Qα be given by (2.1).ThenQα ∈ N1 for each α ∈ (−π/2,π/2). Proof. Assume that Q− ∈ N1 and Q ∈ N \ N1.LetQ+α be given by (4.1). For −1 limy→+0 Re Q−(iy)andlettanβ := tan α + c +1 for α ∈ (−π/2,π/2). Since limy→+0 Re Q+(iy)=+∞ by Lemma 5.1, it is seen as in (4.2) that Im Qα(iy) Im Q+β(iy) < +ImQ−(iy) 1+tan2 α 1+tan2 β for α ∈ (−π/2,π/2) and sufficiently small y. It follows that Qα ∈ N1,since Q+β ∈ N1 and Q− ∈ N1. Again if Q+ ∈ N1, the statement of the theorem can be shown in a similar way. Theorem 5.3. Assume that Q ∈ N \ N1 is of the form Q = Qs + Qp such that Qs ∈ Ns and Qp ∈ Np ∪ N1.LetQα be given by the identity (2.1).Then: (i) if Qp ∈ N1, then there is at most one value α0 ∈ (−π/2,π/2) such that Qα ∈ N1+ for each α ∈ (−π/2,π/2) \{α0}; (ii) if Qp ∈ Np,thenQα ∈ N1+ for each α ∈ (−π/2,π/2). Proof. Since Re Qs(iy) = 0, the estimate (4.3) still holds: 1+tan2 α Im Qˆα(iy) ≤ . (5.1) 2| tan α +ReQˆp(iy)| Hence, if Qp ∈ N1,ˆa1 := limy→+0 Re Qp(iy), and α0 ∈ (−π/2,π/2) is the solution of tan α0 +ˆa1 =0,thenwithα = α0 (5.1) implies that Im Qα is uniformly bounded on the interval (0,i), so that Qα ∈ N1+. On the other hand, if Qp ∈ Np then Lemma 5.1 implies that limy→+0 |Re Qp(iy)| =+∞.NowImQˆα is uniformly bounded on the interval (0,i)andQα ∈ N1+ for each α ∈ (−π/2,π/2) by (5.1). Here again one observes that assertion (i) of Theorem 5.3 holds for symmetric functions in N \ N1. Moreover, there are functions in N \ N1 which do not serve as exceptional functions for a family of Nevanlinna functions in N1+ of the form (2.1). References [1]S.Hassi,M.Kaltenb¨ack, and H.S.V. de Snoo, “Triplets of Hilbert spaces and Friedrichs extensions associated with the subclass N1 of Nevanlinna functions”, J. Operator Theory, 37 (1997), 155–181. [2] S. Hassi, M. Kaltenb¨ack, and H.S.V. de Snoo, “A characterization of semibounded selfadjoint operators”, Proc. Amer. Math. Soc., 125 (1997), 2681–2692. [3] S. Hassi, M. Kaltenb¨ack, and H.S.V. de Snoo, “Generalized Kre˘ın-von Neumann ex- tensions and associated operator models”, Acta Sci. Math. (Szeged), 64 (1998), 627– 655. [4] S. Hassi, H. Langer, and H.S.V. de Snoo, “Selfadjoint extensions for a class of symmet- ric operators with defect numbers (1,1)”, 15th OT Conference Proceedings, (1995), 115–145. 120 S. Hassi, H.S.V. de Snoo and H. Winkler [5] S. Hassi, M. M¨oller, and H.S.V. de Snoo, “A class of Nevanlinna functions associated with Sturm-Liouville operators”, Proc. Amer. Math. Soc., 134 (2006), 2885–2893. [6] S. Hassi, H.S.V. de Snoo, and H. Winkler, “Boundary-value problems for two- dimensional canonical systems”, Integral Equations Operator Theory, 36 (2000), 445– 479. [7] I.S. Kac and M.G. Kre˘ın, “R-functions-analytic functions mapping the upper half- plane into itself”, Amer. Math. Soc. Transl. (2), 103 (1974), 1–18. [8] H. Winkler, “Spectral estimations for canonical systems”, Math. Nachr., 220 (2000), 115–141. [9] H. Winkler, “On generalized Friedrichs and Kre˘ın-von Neumann extensions and canonical systems”, Math. Nachr., 236 (2002), 175–191. Seppo Hassi Department of Mathematics and Statistics University of Vaasa P.O. Box 700 65101 Vaasa, Finland e-mail: [email protected] Henk de Snoo Department of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen, Nederland e-mail: [email protected] Henrik Winkler Institut f¨ur Mathematik, MA 6-4 Technische Universit¨at Berlin Strasse des 17. Juni 136 D-10623 Berlin, Germany e-mail: [email protected] Operator Theory: Advances and Applications, Vol. 175, 121–158 c 2007 Birkh¨auser Verlag Basel/Switzerland On the Spectrum of the Self-adjoint Extensions of a Nonnegative Linear Relation of Defect One in a Krein Space P. Jonas and H. Langer Abstract. A nonnegative symmetric linear relation A0 with defect one in a Krein space H has self-adjoint extensions which are not nonnegative. If the resolvent set of such an extension A is not empty, A has a so-called exceptional eigenvalue α.Forα =0 , ∞ this means that α is an eigenvalue in the open upper half-plane, or a positive eigenvalue with a nonpositive eigenvector, or a negative eigenvalue with a nonnegative eigenvector. In this paper we study these exceptional eigenvalues and their dependence on a parameter if the self- adjoint extensions of A0 are parametrized according to M. G. Krein’s resolvent formula. An essential tool is a family of generalized Nevanlinna functions of the class N1 and their zeros or generalized zeros of nonpositive type. Mathematics Subject Classification (2000). Primary 47B50; Secondary 47A20, 47A55. Keywords. Linear relations in Krein spaces, nonnegative operators, operators with one negative square, selfadjoint extensions, generalized Nevanlinna func- tions. 1. Introduction It is well known that a nonnegative symmetric operator A0 in a Hilbert space H with defect one has self-adjoint extensions in H. They can be described, e.g., by Krein’s formula by means of a real parameter γ ∈ R (= R ∪{∞}). If a suitable such parametrization is fixed and the self-adjoint extensions of A0 are denoted by ∈ R ⊂ R A(γ),γ , then there exists a nonempty open interval (γ−,γ+) , such that for γ ∈ (γ−,γ+) the symmetric form A(γ)·, · on dom A(γ) has one negative square or, equivalently, the operator A(γ) has one negative eigenvalue, and for γ ∈ R\(γ−,γ+) The first author was supported by the Hochschul- und Wissenschaftsprogramm des Bundes und der L¨ander of Germany. 122 P. Jonas and H. Langer the operator A(γ) is nonnegative. If for γ ∈ (γ−,γ+) the negative eigenvalue of A(γ) is denoted by α(γ), then α(γ) is a strictly monotonous function of γ on (γ−,γ+) and tends to zero or −∞ if γ tends to the boundary points γ− or γ+. We shall illustrate this for the simple example of the nonnegative symmet- 2 2 − d ric operator A0 in L (0, 1) generated by 2 and the boundary conditions dx y(0) = y (0) = 0,y(1) = 0. Its self-adjoint extensions A(γ) are given by the same differential expression, and the boundary conditions y(1) = 0,y(0) − γy (0) = 0 with γ ∈ R;forγ = ∞ this boundary condition reads as y (0) = 0. Then the extension A(γ) has a negative eigenvalue α(γ) if and only if γ ∈ (−1, 0), and α(γ) is a strictly decreasing function on (−1, 0). Now let A be a self-adjoint relation in some Krein space H, [·, ·] with the property that the Hermitian sesquilinear form [·, ·] on A, defined by A f g f g ∈ , := [f ,g], , A, (1.1) f g A f g has one negative square on A. Under some regularity assumptions this is equivalent to the fact that the self-adjoint relation A has exactly one so-called exceptional eigenvalue α; this is an eigenvalue of A in the open upper half-plane C+ or a negative eigenvalue with a nonnegative eigenelement or a positive eigenvalue with a nonpositive eigenelement, or α ∈{0, ∞} andinthiscasetoα there corresponds a neutral eigenelement with an ‘approximate associated vector’ with some sign property (see (3.3)). The main object of our studies is a nonnegative symmetric relation A0 of defect one in some Krein space H and its self-adjoint extensions in H.Wemake use of the parametrization A(γ) of the self-adjoint extensions of A0 by a parameter γ ∈ R considered in [JL3]. In this paper there was found an interval (γ−,γ+)with the property that γ belongs to this interval if and only if the Hermitian sesquilin- · · ear form [ , ]A(γ) has one negative square. Under an additional assumption all extensions have non-empty resolvent sets and then, according to what was said above, for γ ∈ (γ−,γ+) the self-adjoint relation A(γ) has one exceptional eigenvalue α(γ). The main results of this paper concern the dependence of this exceptional eigenvalue α(γ)onγ: we show that the function γ → α(γ) is continuous and that there exist two more numbers γ0,γ∞ such that γ− ≤ γ0 ≤ γ∞ ≤ γ+, (1.2) and α(γ)=0forγ ∈ (γ−,γ0), α(γ)=∞ for γ ∈ (γ∞,γ+),andthatontheinterval (γ0,γ∞) the function γ →|α(γ)| is nondecreasing and there is no subinterval of (γ0,γ∞) on which the function γ → α(γ) is constant. We also describe the root subspace of A(γ) at a real exceptional eigenvalue =0 , ∞. Our study of the exceptional eigenvalues of the operators A(γ) is based on the study of the zeros of nonpositive type and their dependence on γ of the generalized Nevanlinna functions 2 Fγ (z)=γz + G1(z)+z G2(z),γ∈ R, (1.3) Self-adjoint Extensions of a Nonnegative Linear Relation 123 where G1 and G2 are Nevanlinna functions. These results may be of independent interest. For particular cases of families of generalized Nevanlinna functions the same questions were studied in [JL2] and [DaL2]. In the first mentioned paper the results were applied to the description of the eigenvalues of nonpositive type of a family of self-adjoint operators in Pontryagin spaces with negative index one. In [DaL2] a periodic Sturm-Liouville problem with an indefinite weight was studied. There a family of self-adjoint operators A(γ) in some Krein space as above arises from a self-adjoint boundary condition, which depends on some parameter, and the dependence of the exceptional eigenvalue on this parameter was described in the same way as in the more general abstract framework of the present paper. A brief synopsis is as follows. In the next section we consider the family of functions Fγ from (1.3). In Section 3 the exceptional eigenvalue of a self-adjoint relation A with the property that the Hermitian sesquilinear form [·, ·]A has one negative square is characterized in different ways. Section 4 contains the main result about the existence of the four numbers γ−,γ0,γ∞,γ+, the monotonicity of the modulus of the exceptional eigenvalue α(γ)ontheinterval(γ0,γ∞), and the further properties of the curve γ → α(γ) as mentioned above. Finally, we describe the root subspace of A(γ) at a real exceptional eigenvalue, and we give some criteria for the coincidence of the numbers γ− and γ0 as well as of γ∞ and γ+. 2. A family of holomorphic functions For κ ∈ N0 we denote by Nκ the set of all complex functions Q which are meromor- phic in C \ R, such that Q(z)=Q(z),z∈ hol Q (here hol Q denotes the domain of holomorphy of Q), and for which the kernel Q(z) − Q(ζ) ,z,ζ∈ hol Q, z = ζ, z − ζ has κ negative squares. Now let Q be a function of the class N1. Recall that z0 ∈ R := R ∪{∞} is called a generalized zero (generalized pole, respectively) of nonpositive type of Q if for each neighborhood U of z0 in C there exists a δU > 0 such that for 0 <ν<δU (δU <ν<∞, respectively) the equation Q(z)=−iν has a solution z ∈U∩C+. If the function Q is holomorphic in a deleted neighborhood of its generalized zero (generalized pole, respectively) z0 of nonpositive type, then z0 is a zero (pole, respectively) of Q. Every function Q ∈N1 has either a generalized zero of nonpositive type in R orasimplezerointheopen upper half-plane C+, see [KL]. This point, which is uniquely determined, is denoted by ζQ. Similarly, Q ∈N1 has either a generalized pole of nonpositive type in R or a simple pole in the open upper half-plane C+. This point, which is also uniquely determined, is denoted by ζQ; clearly, ζQ = ζ−Q−1 . We mention that if ζQ ∈ R then it coincides with the uniquely determined point ζ ∈ R for which there exists a + sequence (ζn) ⊂ C converging to ζ in C such that lim infn→∞ Im Q(ζn) < 0. This 124 P. Jonas and H. Langer is a direct consequence of the definition of ζQ mentioned above and well-known decomposition properties of functions of the class N1. ∞ By N1 we denote the class of all functions Q ∈N1 such that ζQ = ∞.This class will play a special role in this section. It was studied (with one replaced by any natural number) also in the papers [DLShZ], [DLSh], [DHdS]. The first lemma is an easy consequence of results from [DaL1] and [L2], therefore the proof is omitted. ∞ Lemma 2.1. Let the function Q be holomorphic at zero. Then Q ∈N1 if and only if it has a representation of the form - 2ρ 1+tz ν Q(z)=z dσ(t)+ aν z , (2.1) R t − z ν=0 where ρ ∈{0, 1}, σ is a bounded positive measure on R with 0 ∈/ supp σ and 2 t dσ(t)=∞ if ρ =1, ∈{0, 1, 2, 3},andaν ∈ R, ν =0,...,, a =0 if >0, R such that one of the following conditions is fulfilled: (a) ρ =0,=1,a1 < 0; (b) ρ =0,=2; (c) ρ ∈{0, 1},=3,a3 > 0; (d) ρ =1,∈{0, 1, 2}. ∞ The representation of a function Q ∈N1 in the form (2.1) is unique. In the following, let G1,G2 ∈N0,andforj =1, 2, we write Gj in the form 1+tz Gj(z)=αj + βjz + dσj(t), (2.2) R t − z where αj ∈ R, βj ≥ 0, and σj is a bounded positive measure on R (cf., e.g., [KK]). ∞ Theorem 2.2. The union N0 ∪N1 is the set of all functions F of the form 2 F (z)=γz + G1(z)+z G2(z) with γ ∈ R,G1,G2 ∈N0. (2.3) This function F belongs to the class N0 if and only if 2 2 β2 =0, 1+t dσ2(t) < ∞,α2 = tdσ2(t),γ≥ 1+t dσ2(t) − β1. R R R Proof. Denote the set of functions F of the form (2.3) by N . Evidently, N0 ⊂ N . ∞ Every function F ∈N1 canbewrittenasasumF = F0 +F1 where F0 ∈N0,F1 ∈ ∞ N1 ,andF1 is holomorphic at zero ([L2, §2]). By Lemma 2.1, the function F1 ∞ belongs to N , hence N0 ∪N1 ⊂ N . 2 Now let F ∈ N , i.e., F (z)=γz+G1(z)+z G2(z)whereγ ∈ R and G1,G2 ∈ N0 with integral representations (2.2). Choose ∆0 := (−1, 1), ∆∞ := R \ ∆0 and define for j =1, 2 1+tz 1+tz G 0(z):=α + dσ (t),G∞(z):=β z + dσ (t). j, j − j j, j − j ∆0 t z ∆∞ t z Self-adjoint Extensions of a Nonnegative Linear Relation 125 Then 2 2 z G2,0(z)=− tdσ2(t) − z 1+t dσ2(t) ∆0 ∆0 2 1+tz 2 +z α2 − tdσ2(t) + t dσ2(t) − ∆0 ∆0 t z and 2 2 F (z)=γz + G1 0(z)+z G2 0(z)+G1 ∞(z)+z G2 ∞(z) , , , , 2 2 = − tdσ2(t)+z γ − 1+t dσ2(t) + z α2 − tdσ2(t) ∆0 ∆0 ∆0 1+tz 2 2 + G1 0(z)+ t dσ2(t)+G1 ∞(z)+z G2 ∞(z). , − , , ∆0 t z ∞ Hence F belongs to N0 or N1 if and only if the following function F∞ belongs to ∞ N0 or N1 , respectively: 2 2 F∞(z): = z γ − 1+t dσ2(t) + z α2 − tdσ2(t) ∆0 ∆0 2 +G1,∞(z)+z G2,∞(z) 2 2 = z γ + β1 − 1+t dσ2(t) + z α2 − tdσ2(t) ∆0 ∆0 3 1+tz 2 1+tz +z β2 + dσ1(t)+z dσ2(t). (2.4) ∆ t − z ∆ t − z ∞ ∞ 2 If 1+t dσ2(t)=∞ then we replace the second last integral in (2.4): R 1+tz −1 −2 2 dσ1(t)= t dσ1(t)+z t 1+t dσ1(t) ∆ t − z ∆ ∆ ∞ ∞ ∞ 2 −1 2 1+tz −2 +z t dσ1(t)+z t dσ1(t). − ∆∞ ∆∞ t z ∞ It follows that F∞ is of the form (2.1) with ρ =1anda3 ≥ 0, therefore F∞ ∈N1 . 2 If 1+t dσ2(t) < ∞,then R 2 1+tz 2 z dσ2(t)=− tdσ2(t) − z 1+t dσ2(t) − ∆∞ t z ∆∞ ∆∞ 2 2 t (1 + tz) −z tdσ2(t)+ dσ2(t). − ∆∞ ∆∞ t z Inserting this expression for the last integral into (2.4) we see that F∞ has the ∞ form (2.1) with ρ = 0. Then, by Lemma 2.1, F∞ ∈N1 if β2 =0. 126 P. Jonas and H. Langer ∞ It remains to consider the case β2 =0.ThenF∞ ∈N1 if α2 = tdσ2(t). R ∞ 2 If β2 =0andα2 = tdσ2(t)thenF∞ ∈N1 if γ + β1 − 1+t dσ2(t) < 0. If R R 2 β2 =0,α2 = tdσ2(t)andγ + β1 − 1+t dσ2(t) ≥ 0, then F∞ ∈N0. R R For the rest of this section we fix two functions G1,G2 ∈N0 and consider the family of functions Fγ ,γ∈ R, as on the right-hand side of (2.3): 2 Fγ (z)=γz + G1(z)+z G2(z). Define ⎧ ⎪ 2 2 ⎪ (1 + t ) dσ2(t) − β1 if (1 + t ) dσ2(t) < ∞,β2 =0, ⎨ R R γ∞ := ⎪ and α2 = tdσ2(t), ⎩⎪ R ∞ otherwise. Then it follows that N1 if γ ∈ (−∞,γ∞), Fγ ∈ N0 if γ ∈ [γ∞, ∞). + If γ ∈ (−∞,γ∞) we denote the zero of nonpositive type of Fγ in R ∪ C by ∞ ζ(γ): ζ(γ):=ζFγ .Observethatγ∞ < holds if and only if 2 2 1+t 1+t dσ2(t) < ∞ and G2(z)= dσ2(t), R R t − z which in the notation of [KK, 4] means that G2 belongs to the class (R0). In the following, if σ is a measure with σ({ω}) > 0forsomeω ∈ R then we set (t − ω)−2 dσ(t)=∞. R Lemma 2.3. If γ ∈ (−∞,γ∞),thenζ(γ) = ∞ and the following statements hold: (i) ζ(γ)=0if and only if −2 2 −2 2 t 1+t dσ1(t) < ∞,γ≤ σ2({0}) − β1 − t 1+t dσ1(t), R R −1 α1 = − t dσ1(t). (2.5) R (ii) ζ(γ)=ω ∈ R \{0} if and only if −2 −2 |t − ω| dσ1(t) < ∞, |t − ω| dσ2(t) < ∞ (2.6) R R Self-adjoint Extensions of a Nonnegative Linear Relation 127 and 1+tω γω + α1 + β1ω + dσ1(t) t − ω R (2.7) 2 1+tω + ω α2 + β2ω + dσ2(t) =0, R t − ω 0 2 | − |−2 2 γ + β1 +2ωα2 +3ω β2 + R t ω 1+t dσ1(t) − 3 − 2 2 2 (2.8) 2tω 2tω ω +3t ω ≤ + 2 dσ2(t) 0. R (t − ω) ≤ The inequality (2.8) means that Fγ (ω) 0, where the derivative is to be understood in the Carath´eodory sense; the relation (2.7) means that Fγ (ω)=0if Fγ (ω) is the nontangential boundary value of Fγ at ω. Proof. By [L2, Corollary 2.1] the point ∞ cannot be a zero of nonpositive type of Fγ . It follows from [L2, Remark 3.2 and proof of Theorem 3.1] that ζ(γ)=ω ∈ R is a generalized zero of nonpositive type of Fγ if and only if −1 lim η Im Fγ (ω + iη) exists and is nonpositive (2.9) η↓0 and −1 lim η Re Fγ (ω + iη)=0. (2.10) η↓0 For arbitrary z = ω + iη we find −1 2 2 −2 2 η Im Fγ (z)=γ + β1 +2ωα2 +(3ω − η )β2 + |t − z| 1+t dσ1(t) R 2 2 2 3 −2 + 2ωt − ω +3ω t − 2tω |t − z| dσ2(t) (2.11) R 2 2 −2 − η 1+t − 2ωt |t − z| dσ2(t). R If ω = 0 and (2.9) holds then the last term on the right-hand side of (2.11) converges to −σ2({0})ifη → 0. Moreover, by Fatou’s lemma, −2 −2 2 σ1({0})=0, t dσ1(t) < ∞,γ+ β1 + t 1+t dσ1(t) − σ2({0}) ≤ 0. R R Now it is easy to see that the relation (2.10) can be written as (2.5), that the inequalities in (i) imply (2.9) and that (2.5) implies (2.10). If ζ(γ)=ω ∈ R \{0}, then the last term in (2.11) converges for η → 0to 2 ω − 1 σ2({ω}), and we conclude from (2.9) and Fatou’s lemma that −2 −2 |t − ω| dσ1(t) < ∞, |t − ω| dσ2(t) < ∞, R R in particular, σ1({ω})=σ2({ω}) = 0. Now it is easy to see that (2.9) implies (2.8) and that (2.10) implies (2.7), and also that, conversely, (2.6), (2.7) and (2.8) imply (2.9) and (2.10). 128 P. Jonas and H. Langer We define γ0 as follows: ⎧ 2 ⎨ { } − 1+t − dσ1(t) ∞ dσ1(t) σ2( 0 ) 2 dσ1(t) β1 if 2 < ,α1 + =0, γ0 := ⎩ R t R t R t −∞ otherwise. The numbers γ0,γ∞ satisfy the following inequalities: −∞ ≤ γ0 ≤ γ∞ ≤∞,γ0 < ∞, −∞ <γ∞, and Lemma 2.3 implies that for γ ∈ (−∞,γ∞)therelationζ(γ) = 0 holds if and only if γ ≤ γ0. Evidently, if Fγ (z)=γz − cz with some c ∈ R then Fγ belongs to N1 if γ Theorem 2.4. The equality γ0 = γ∞ holds if and only if Fγ (z)=γz−cz with some c ∈ R; in this case γ0 = γ∞ = c. If γ0 = γ∞ then the function ζ on (−∞,γ∞) has the following properties: (1) ζ is a continuous function, ζ(γ)=0if and only if γ ∈ (−∞,γ0],andζ(γ1) = ζ(γ2) if γ1 = γ2,γ1,γ2 ∈ (γ0,γ∞). ∞ (2) limγ↓γ0 ζ(γ)=0, limγ↑γ∞ ζ(γ)= . (3) ζ is real analytic at points γ ∈ (−∞,γ∞) with ζ(γ) ∈/ (supp σ1 ∪ supp σ2) ≤ ∈ R with possible exception of branching points of order 3 for which ζ(γ) . (4) σj {ζ(γ):γ ∈ (γ0,γ∞)}∩R =0,j=1, 2. (5) The function |ζ|: |ζ|(γ):= |ζ(γ)| is nondecreasing; it is increasing on (γ0,γ∞) if and only if Fγ is not of the form 2 Fγ (z)=γz + α1 + β1z + α2z with α1α2 > 0. 2 Proof. If γ0 = γ∞, then the numbers γ0 and γ∞ are finite, β2 =0, 1+t dσ2(t) R −2 2 −1 and t 1+t dσ1(t) are finite and α1 = − t dσ1(t),α2 = tdσ2(t). R R R Moreover, 2 −2 2 2 1+t dσ2(t)=σ2({0})− t 1+t dσ1(t) ≤ σ2({0}) ≤ 1+t dσ2(t), R R R hence σ1 =0,σ2 = c0δ0,wherec0 ∈ R and δ0 is the unit mass at 0. It follows that Fγ (z)=γz + β1z − c0z. Conversely, evidently, Fγ (z)=γz − cz, c ∈ R, implies γ0 = γ∞. For the rest of the proof we assume that γ0 = γ∞. The continuous dependence of ζ(γ)onγ is an easy consequence of [J1, Corollary 3.2 and Proposition 3.5]. The Self-adjoint Extensions of a Nonnegative Linear Relation 129 second assertion of (1) follows from Lemma 2.3: Suppose that for γ1,γ2 ∈ (γ0,γ∞) we have ζ(γ1)=ζ(γ2)=:ζ0.Thenζ0 = 0 and the relations 2 γjζ0 + G1(ζ0)+ζ0 G2(ζ0)=0 j =1, 2, (understood as in (2.7)) imply γ1 = γ2. In order to prove the first relation of (2) we can suppose that γ0 = −∞ and consider only γ<0. Then ζ(γ) is the generalized zero of nonpositive type of the 1 function F , and the first relation of (2) is again a consequence of [J1, Corollary |γ| γ 3.2 and Proposition 3.5]. In order to prove the second relation of (2) we introduce the function 2 −1 2 Fγ (z):=z F−γ − z = γz + G1(z)+z G2(z) −1 −1 with G1(z):=G2 − z , G2(z):=G1 − z , and we write −1 Gj(z)=αj + βjz + (1 + tz)(t − z) dσj(t),j=1, 2, R where αj, βj and σj have the same properties as αj,βj and σj. Define γ0, γ∞ for Fγ inthesamewayasγ0 and γ∞ were defined for Fγ . Then [ADo, Section 2] yields the relations −1 α1 = α2, β1 = σ2({0}), σ1({0})=β2,dσ1(t)=dσ2(−t )onR \{0}, and also the same relations with indices 1 and 2 interchanged. Further, an easy computation shows that γ0 = −γ∞, γ∞ = −γ0,andifγ ∈ (γ0,γ∞) the generalized −1 zero ζ(γ) of nonpositive type of Fγ is ζ(γ)=−ζ(−γ) . Therefore, the relation ∞ limγ↓γ0 ζ(γ) = 0 implies limγ↑γ∞ ζ(γ)=+ . Statement (3) follows from (2.7), (4) is a consequence of the relations (2.6) and [JL2, Lemma 2.1]. It remains to prove (5). We first show that the function |ζ| is nondecreasing in (γ0,γ∞). To this end we observe that there exist a sequence of real numbers bn and a sequence of compactly supported positive measures τn on R, such that the sequence of functions H : n −1 Hn(z):=bnz + (t − z) dτn(t),n=1, 2,..., R + 2 converges for n →∞locally uniformly on C to G1(z)+z G2(z). Indeed, it is easy to see that the functions n n 1+tz 2 1+tz α1 + β1z + dσ1(t)+z α2 + β2z + dσ2(t) ,n=1, 2,..., − − −n t z −n t z + 2 converge for m →∞locally uniformly on C to G1(z)+z G2(z)forn →∞. Since for arbitrary α ∈ R and β ≥ 0 the functions −1 β |α| m (sign α)m − z + m2 (m − z)−1 +(−m − z)−1 ,m=1, 2,..., 2 130 P. Jonas and H. Langer converge locally uniformly to the function α + βz, there exist sequences of com- pactly supported positive measures σ1,n(t),σ2,n(t), such that the functions − −1 2 − −1 (t z) dσ1,n(t)+z (t z) dσ2,n(t) R R − −1 − − − −1 2 = (t z) dσ1,n(t) z dσ2,n(t) tdσ2,n(t)+ (t z) t dσ2,n(t) R R R R 2 converge locally uniformly to G1(z)+z G2(z). If we approximate the constants as above by Nevanlinna functions we get a sequence of functions Hn with the desired properties. Consider γ ∈ (γ0,γ∞). Since Fγ ∈N1, from the definition of the class N1 it follows that there exist a natural number n0 and an ε>0 with ∆ := [γ −ε, γ +ε] ⊂ (γ0,γ∞), such that for γ ∈ ∆andn>n0 the functions Fγ,n: Fγ,n(z):=γz + Hn(z) belong to N1. Denote the zero of nonpositive type of Fγ,n by ζn(γ). Then, according to [JL2, Theorem 3.1], for n>n0 the functions |ζn| are nondecreasing on ∆. By [J1, Corollary 3.2 and Proposition 3.5], limn→∞ ζn(γ)=ζ(γ),γ∈ ∆, therefore also the function |ζ| is nondecreasing on ∆. For the proof of the second statement of (5) it is sufficient to consider the + function |ζ| on an interval (γ1,γ2) such that ζ(γ) ∈ C if γ ∈ [γ1,γ2]. Suppose first that σ1 = 0 or that (supp σ2) \{0} = ∅. Then, by the above construction of Hn, we may assume that for each bounded interval ∆ there exists an n1 such that for n ≥ n1 the measure τn, restricted to ∆, is not zero and does not depend on n. Applying [JL2, (3.6)] and the reasoning which follows that relation, we see that there exists an a>0 such that d|ζn(γ)| inf ≥ a if n ≥ n1, γ∈[γ1,γ2] dγ which implies that the function |ζ| is increasing. Let now σ1 = 0 and supp σ2 ⊂{0}.Then 2 3 Fγ (z)=γz + α1 + β1 − σ2({0}) z + α2z + β2z . Set β := β1 − σ2({0}), suppose β2 > 0andlet(γ1,γ2)beasabove.Thenasimple computation, starting from Fγ (ζ(γ)) = 0, yields d log |ζ(γ)| dζ(γ) =Im ζ(γ)−1 dγ dγ −2 = β2 β2Im ζ(γ) − i α2 +3β2Re ζ(γ) > 0 for all γ ∈ (γ1,γ2). Therefore the function |ζ| is strictly increasing on (γ0,γ∞). Finally, the case 2 Fγ (z)=γ(z)+α1 + β z + α2z can be considered by straightforward calculations. Self-adjoint Extensions of a Nonnegative Linear Relation 131 For γ1 ∈ (γ0,γ∞)and ζ1 = ζ(γ1) ∈ R \ (supp σ1 ∪ supp σ2 ∪{0}), in a neighborhood of ζ1 the graph of the function ζ can be described similarly to [JL2, Proposition 3.2]. To this end we denote by a± the directions of the (one-sided) tangents of the curve γ → ζ(γ)atγ1: ζ(γ) − ζ(γ1) a± := ± lim . (2.12) γ→γ1±0 |ζ(γ) − ζ(γ1)| Then one of the following four cases prevails for some ε>0: (rr) ζ(γ) ∈ R if |γ − γ1| <ε. + (cr) ζ(γ) ∈ C if γ ∈ (γ1 − ε, γ1), ζ(γ) ∈ R if γ ∈ (γ1,γ1 + ε), and a− = −i. + (rc) ζ(γ) ∈ R if γ ∈ (γ1 − ε, γ1), ζ(γ) ∈ C if γ ∈ (γ1,γ1 + ε), and a+ = i. + (cc) ζ(γ) ∈ C if 0 < |γ − γ1| <εand a− =(signγ0)exp(−(sign γ0)iπ/3) ,a+ =(signγ0) exp ((sign γ0)iπ/3) . If we denote the inverse function of ζ by γ,thatis,γ(ζ(γ)) = γ for γ ∈ (γ0,γ∞), then −1 2 γ(ζ)=−ζ G1(ζ)+ζ G2(ζ) . The case (rr) holds if and only if γ (ζ1) = 0, in the other cases we have γ (ζ1)=0 and, additionally, γ (ζ1) > 0,<0, = 0, respectively, in the case (cr), (rc) and (cc), respectively. The graph of the function ζ ‘visits’ an open interval I with 0 ∈/ I and such that G1 and G2 are holomorphic on I at most once. This is a consequence of the following theorem. Theorem 2.5. If I is a component of the open set R \ (supp σ1 ∪ supp σ2 ∪{0}), then the set I ∩ ζ (γ0,γ∞) is empty or connected. Proof. If 2 Fγ (z)=γz + α1 + β z + α2z ,β:= β1 − σ2({0}), (2.13) the theorem can be proved by a straightforward computation. Suppose that Fγ is not of the form (2.13). Then the third derivative of the function 2 F (ζ):=−ζγ(ζ)=G1(ζ)+ζ G2(ζ) is positive in real points of holomorphy of F . We consider the case I ⊂ (0, ∞), the case I ⊂ (−∞, 0) can be treated similarly. Assume that I ∩ ζ (γ0,γ∞) is not connected. Then there exist points ζ1,ζ2 ∈ I ∩ ζ (γ0,γ∞) , ζ1 <ζ2, such that γ (ζ1)=γ (ζ2)=0,γ(ζ1) ≤ 0,γ(ζ2) ≥ 0. therefore F (ζ1) ≥ 0,F (ζ2) ≤ 0. ButthisisimpossiblesinceF (ζ) is positive on (ζ1,ζ2). 132 P. Jonas and H. Langer 3. The exceptional eigenvalue of a self-adjoint relation withonenegativesquare Let H, [ · , · ] be a Krein space, and let · be a Hilbert norm on H such that [·, ·]is · -continuous. A closed linear relation A in H is a closed linear subspace 2 of H . In the sequel a closed linear operator A in H is identified with the closed f linear relation given by its graph : f ∈D(A) . For the definitions of the Af domain D(A), the kernel ker A, the range ran A, the adjoint A+, the resolvent set ρ(A), the spectrum σ(A) and the point spectrum σp(A) we refer to [DdS1], [DdS2], [JL3]. The extended spectrum σ(A) is defined as follows: σ(A)if0∈/ σ A−1 , σ(A):= σ(A) ∪{∞} if 0 ∈ σ A−1 . 0 If A is not an operator, that is A(0) = g : ∈ A = {0},then∞ is called an g eigenvalue of A with the nonzero elements of A(0) as corresponding eigenvectors. The set σp(A)ifA(0) = {0}, σp(A):= σp(A) ∪{∞} if A(0) = {0} is called the extended point spectrum of A. The linear relation A in the Krein space H is called symmetric if the sesquilin- ear form [·, ·] (see (1.1)) is Hermitian, that is A f g [f ,g]=[f,g ], , ∈ A, f g and A is called self-adjoint if for elements g,g ∈Hthe relation f [f ,g]=[f,g ] for all ∈ A f g yields ∈ A, see, e.g., [DdS1], [DdS2]. The symmetric linear relation A in H g is called nonnegative if f f ≡ ≥ f ∈ , [f ,f] 0, A, f f A f and it is said to have one negative square if the Hermitian sesquilinear form f g ≡ f g ∈ , [f ,g], , A, (3.1) f g A f g has this property. Let A be a self-adjoint relation with ρ(A) = ∅,letµ ∈ ρ(A) and denote by {·, ·} the following Hermitian sesquilinear form on H: {f,g} := I + µ(A − µ)−1 f,(A − µ)−1g ,f,g∈H. (3.2) Self-adjoint Extensions of a Nonnegative Linear Relation 133 For κ ∈ N0 andaKreinspaceH,byNκ(H) we denote the set of all functions Q with values in L(H) (the set of bounded linear operators in H), which are meromorphic in C \ R and such that Q(z)=Q(z)+ for all z ∈ hol (Q)andthat the kernel Q(z) − Q(ζ)+ ,z,ζ∈ hol (Q), z − ζ has κ negative squares; here + denotes the Krein space adjoint. Then the following statements are equivalent: (a) A is nonnegative (has one negative square, respectively). (b) For some (and hence for all) µ ∈ ρ(A) the Hermitian sesquilinear form {·, ·} on H is nonnegative (has one negative square, respectively). −1 (c) The Krein space operator function z → z(A−z) belongs to N0(H)(N1(H), respectively). In particular, the fact that for a nonnegative self-adjoint relation A with ρ(A) = ∅ the operator function in (c) is a Nevanlinna function implies that in this case C \ R ⊂ ρ(A). The main result of this section is the following theorem. Theorem 3.1. Let A be a self-adjoint relation with one negative square in the Krein space H such that ρ(A) = ∅. Then exactly one of the following two statements holds true: (i) The non-real spectrum of A consists of one pair of complex conjugate points α ∈ C+ and α which are simple normal eigenvalues and such that the two- dimensional linear span of the corresponding eigenvectors is a non-degenerate indefinite subspace. In the orthogonal complement of this span the relation A induces a nonnegative self-adjoint linear relation with non-empty resolvent set. (ii) The spectrum of A is real and there exists a unique point α ∈ R with the following properties: If α =0 , ∞ then it is an eigenvalue of A with an eigenelement e0 such that (sgn α)[e0,e0] ≤ 0; if α =0then α is an eigenvalue of A with a neutral eigenelement e0 and en ⊂ − → there exists a sequence A such that en e0 0 and en [e − e ,e − e ] → 0, lim [e ,e ] ≤ 0ifm, n →∞; (3.3) n m n m →∞ n n n ∞ en ⊂ − → if α = there exists a sequence A such that en e0 0 if en n →∞for some nonzero neutral element e0 ∈ A(0) and (3.3) holds. In the proof of this theorem we need the notion of a definitizable self-adjoint relation in a Krein space and its spectral function. The self-adjoint relation A in the Krein space H is called definitizable if ρ(A) = ∅ and if there exists a real 134 P. Jonas and H. Langer polynomial p such that the self-adjoint relation p(A) is nonnegative (see [DdS2]). Obviously, a nonnegative self-adjoint relation A with ρ(A) = ∅ is definitizable. The self-adjoint relation A with µ ∈ ρ(A),µ= µ, is definitizable if and only if the unitary operator U in the Krein space H defined by U := −I +(µ − µ)(A − µ)−1, (3.4) i.e., the Cayley transform of A, is definitizable in the sense of [L1]. The spectrum of a definitizable unitary operator U lies on the unit circle T with possible exception of a finite set of eigenvalues, which is symmetric with respect to T.OnT the operator U has a spectral function with possibly a finite number of critical points, see [L1]; the set of these critical points is denoted by c(U). The Cayley transform U of the definitizable self-adjoint relation A with µ ∈ ρ(A) can be written as U = ψ(A)with ψ(z):=−(z − µ)(z − µ)−1. (3.5) According to [DdS2, Proposition 3.2] we have ψ(σ(A)) = σ(U) and hence σ(A) lies on R with possible exception of a finite set of complex eigenvalues, which is symmetric with respect to R.IfEU denotes the spectral function of U then the spectral function EA of A is defined by the relation EA(∆) = EU (ψ(∆)) , where ∆ ∈B(A). Here B(A) denotes the Boolean algebra generated by the con- nected subsets of R (R is considered to be homeomorphic to the unit circle) for which the endpoints are not in c(A):=ψ−1(c(U)); this set c(A) is called the set of critical points of A. The spectral function EA defines a homomorphism from B(A) into the set of self-adjoint projections in H such that for all ∆ ∈B(A)the following holds: 1. EA R =1− E0,whereE0 is the Riesz–Dunford projection corresponding to the set of nonreal eigenvalues of A. 2 2 2. A is the direct sum of the three subspaces A∩(E0H) ,A∩ E (∆)H ,and A ∩ R \ H 2 H2 A EA ∆ of . 2 3. σ A ∩ EA(∆)H ⊂ ∆. The point λ ∈ σ(A)issaidtobeaspectral point of positive (negative,re- spectively) type of the definitizable self-adjoint relation A if for some open set ∆ ∈B(A) such that λ ∈ ∆thesetEA(∆)H\{0} consists of positive (negative, re- spectively) elements or, equivalently, the space (EA(∆)H, [ · , · ]) is a Hilbert space (anti-Hilbert space, respectively). The spectral function EA has also the following properties: 4. If λ ∈ σ(A) and for a definitizing polynomial p of A we have p(λ) > 0(< 0, respectively) then λ is a spectral point of positive (negative, respectively) type of A. 5. λ ∈ c(A) if and only if for all ∆ ∈B(A)withλ ∈ ∆ the range EA(∆)H contains positive as well as negative elements. Self-adjoint Extensions of a Nonnegative Linear Relation 135 In the following lemma we set ϕ(U):=(2Imµ)−2 µ(1 + U)+µ(1 + U −1) =(A − µ)−1A(A − µ)−1. (3.6) Then the Hermitian sesquilinear form in (3.2) becomes {f,g} =[ϕ(U)f,g],f,g∈H. Lemma 3.2. Let A be a self-adjoint relation with one negative square in the Krein space H, suppose that ρ(A) = ∅,andchooseµ ∈ ρ(A) ∩ C+.ThenA and also its Cayley transform U = ψ(A)(see (3.4)) are definitizable. There exists a unique point β ∈ C, 0 < |β|≤1, such that with the operator ϕ(U) from (3.6) the relation ϕ(U)(U − β)(U −1 − β)x, x ≥ 0 for all x ∈H (3.7) holds. The point β is an eigenvalue of U with a [ϕ(U) · , · ]-nonpositive eigenelement e,andα = ψ−1(β) is an eigenvalue of the self-adjoint relation A with the same eigenelement e. −1 If |β| < 1, that is α ∈ C+,thenσ(U) \ T = β,β and σ(A) \ R = {α, α}, β and α are simple eigenvalues of U and A, respectively, the corresponding eigenvectors of U and A coincide and the two-dimensional subspace −1 ker (U − β)+ker U − β =ker(A − α)+ker(A − α) is non-degenerated and indefinite. If |β| =1, that is if α ∈ R then σ(U) ⊂ T and σ(A) ⊂ R.For∆ ∈B(A) it holds α ∈ ∆(α/∈ ∆, respectively) if and only if the Hermitian sesquilinear form [·, ·]A has one negative square (is positive semi-definite, respectively) on A ∩ 2 EA(∆)H . Proof. We consider the factor space H := H/ ker ϕ(U). Since ker ϕ(U)isthe isotropic subspace of the Hermitian sesquilinear form (3.2), the space H, {·, ·} is a non-degenerated inner product space with negative index one. We complete it to a Pontryagin space H, {·, ·} with one negative square. The operator U induces a bounded operator U in H, which maps H onto itself and leaves the inner product {·, ·} invariant. The closure U of U in H is a unitary operator. ∈ | |≤ According to Pontryagin’s Theorem there exists a point β σp(U), β 1, with a nonpositive eigenelement e.Therelation ran U −1 − β , e = {0} implies U − β U −1 − β x, x ≥ 0 for all x ∈ H, consequently, the relation (3.7) holds and U is a definitizable operator. The point β is an eigenvalue of U since otherwise the range of U − β would be dense in H and hence [ϕ(U)x, x] ≥ 0 for all x ∈H, which is impossible since this form has one negative square. On ker (U − β)wehave[ϕ(U) · , · ]=ϕ(β)[ · , · ]. If this form [⊥] would be positive definite there, then ker (U −β)+(ker (U − β)) would be dense 136 P. Jonas and H. Langer [⊥] in H. Since, by (3.7), [ϕ(U) · , · ] is positive semi-definite on (ker (U − β)) there exists an x ∈ ker (U − β)with[ϕ(U)x, x] < 0, a contradiction. Therefore there exists a [ϕ(U) · , · ]-nonpositive nonzero element e ∈ ker (U − β). The point β, |β|≤1, with the property (3.7) is uniquely determined. Indeed, if there would be a point β = β with the same properties as β, then also ker(U −β ) would contain a nonpositive element and this element is orthogonal to e,whichis impossible since the inner product [ϕ(U) · , · ] has only one negative square. This argument also proves that β with |β| < 1 is a simple eigenvalue of U. According to the spectral mapping theorem α = ψ−1(β) is an eigenvalue of A with the same root subspace as that of U at β. For β ∈ T or α ∈ R it follows as above that σ(U) ⊂ T and σ(A) ⊂ R.If∆∈ B(A) such that α/∈ ∆, then β = ψ(α) ∈/ ψ(∆), EU (ψ(∆))H =(U −β)EU (ψ(∆))H and we get from (3.7) −1 −1 (A − µ) A(A − µ) EA(∆)x, EA(∆)x = ϕ(U)EU (ψ(∆))x, EU (ψ(∆))x ≥ 0. From this relation the last claim of the lemma follows easily. Proof of Theorem 3.1. First we shall show that the point α of Lemma 3.2 has the properties stated in the theorem. Evidently, if α ∈ C+ then the case (i) prevails. If α ∈ R\{0}, it follows from Lemma 3.2 that the corresponding point β = ψ(α)isan eigenvalue of U and α is an eigenvalue of A with the same [ϕ(U) · , · ]-nonpositive eigenelement e, and hence 0 ≥ [ψ(U)e, e]= (A − µ)−1A(A − µ)e, e = |α − µ|2 α [e, e]. It follows that [e, e]isnonpositiveifα>0 and nonnegative if α<0. It remains to consider the cases α =0andα = ∞. We restrict ourselves to the first case, for α = ∞ a similar reasoning applies. If α = 0 we choose a bounded open interval ∆ ∈B(A)with0∈ ∆and consider in the Krein space H∆ := E(∆)H the restriction A∆ := A|H∆ ,whichis a bounded self-adjoint operator. Repeating in fact considerations in the proof of H H Lemma 3.2, we equip the factor space ∆ := ∆/ ker A∆ with the inner product ∈ ∈ f,g ∆ := A∆f,g ,f f,g g, which has one negative square. The completion of H∆, which is a Pontryagin space with negative index one, is denoted by H∆. The operator A∆ induces bounded self-adjoint operators A∆ and A∆ in H∆ and H∆, respectively, and since in H∆ all the nonzero spectral points of A∆ are of positive type, there exists a nonzero element x ∈ ker A∆ with {x, x}∆ ≤ 0. For the space H∆ this means that there exists a Cauchy sequence (xn) ⊂ H∆ which does not converge to zero and is such that A∆xn → 0inH∆ and lim {xn, xn}≤0 if n →∞.That(xn) is a Cauchy sequence means that {xn − xm, xn − xm}∆ −→ 0, {xn − xm, y}∆ −→ 0 for m, n →∞and all y ∈ H∆, or, equivalently, A∆(xn − xm),xn − xm −→ 0, A∆(xn − xm),y −→ 0 (3.8) Self-adjoint Extensions of a Nonnegative Linear Relation 137 for m, n →∞and all y ∈H∆,xn ∈ xn. By the second relation in (3.8), the elements A∆xn converge weakly to some e0 ∈H∆, and limn→∞ [A∆xn,y0] =0for 2 some y0 ∈H∆ implies e0 = 0, and limn→∞ [A∆xn,y] = 0 for all y ∈H∆ implies A∆e0 = 0. Finally, limn→∞ [A∆xn,xn] ≤ 0. Now we replace the sequence (xn) by the sequence (en) of arithmetic means of a subsequence of (xn) such that A∆en converges strongly to e0. Then the sequence (en) has all the properties of the sequence (xn) mentioned above, and therefore − → →∞ ∈ ∩ en e0 0ifn and the relations in (3.3) hold. Since e0 ker A∆ ran A∆, the element e0 is neutral. It remains to be shown that any α ∈ R, which is different from α in Lemma 3.2, does not satisfy the condition (ii) of Theorem 3.1 with α replaced by α .By Lemma 3.2 there exists a connected open set ∆ ∈B(A)withα ∈ ∆ such that 2 the form [·, ·]A is positive semi-definite on A ∩ (EA(∆)H) .Letα =0 , ∞ and assume, in addition, that 0, ∞ ∈/ ∆. Then for any f0 ∈ ker (A − α ),f0 =0,we have α [f0,f0] > 0 and hence (ii) with α replaced by α does not hold. Now let α = 0 and assume in addition that ∆ is bounded. Suppose that there ∈ fn ⊂ − → exist an f0 ker A, f0 = 0, and a sequence A such that fn f0 0 fn for n →∞and the following limit exists and is nonpositive: ≤ lim fn,fn 0. (3.9) n→∞ We claim that lim (I − EA(∆)) f ,fn =0. (3.10) n→∞ n u Indeed, consider y ∈H.Theny := (I − E (∆)) y ∈ ran A,andif ∈ A the A y relation − y,(I EA(∆)) fn = y, fn = u, fn ,n=1, 2,..., and the boundedness of the sequence fn imply that sup I − EA(∆) fn : n =1, 2,... < ∞. From this inequality and lim I − EA(∆) f = I − EA(∆) f0 =0 n→∞ n the relation (3.10) follows. Now (3.9) and (3.10) imply lim [EA(∆)f ,EA(∆)fn] = lim [EA(∆)f ,fn] ≤ 0, n→∞ n n→∞ n 2 and, since [·, ·]A is positive semi-definite on A ∩ EA(∆)H , we find lim [EA(∆)fn,EA(∆)fn]=0. (3.11) n→∞ 138 P. Jonas and H. Langer If J∆ is a fundamental symmetry of the Krein space (EA(∆)H, [·, ·]) then for n =1, 2,... EA(∆)fn,J∆EA(∆)fn 1/2 1/2 ≤ EA(∆)fn,EA(∆)fn AJ∆EA(∆)fn,J∆EA(∆)fn . → This relation and (3.11) imply that EA(∆)fn 0 and hence f0 = 0, a contradic- tion. For α = ∞ a similar reasoning applies.. Remark 3.3. The proof of Theorem 3.1 shows that this theorem remains true if ≤ (3.3) is replaced by limn→∞ en,en 0. In order to explain Theorem 3.1 we first mention that for a nonnegative self- adjoint relation A with nonempty resolvent set the positive spectral points are of positive type and the negative spectral points are of negative type. In partic- ular, all the eigenvectors corresponding to positive eigenvalues are positive and all eigenvectors corresponding to negative eigenvalues are negative, and there is no non-real spectrum. If A has one negative square, it has either a pair α, α of simple non-real complex conjugate eigenvalues, or on the real axis there is a posi- tive eigenvalue α with a nonpositive eigenvector, or a negative eigenvalue α with a nonnegative eigenvector, or at 0 or ∞ there is a neutral eigenvector to which there corresponds an ‘approximate associated vector’ with a certain sign property, see (3.3). The following definition of the exceptional eigenvalue will play an important role in this paper. Definition 3.4. Let A be a self-adjoint relation with ρ(A) = ∅ and one negative square in the Krein space H. The point α ∈ C+ ∪ R from Theorem 3.1 is called the exceptional eigenvalue of A and it is denoted in the following by α(A). In the following theorem we give some equivalent descriptions of the excep- tional eigenvalue α(A). For the definition of a subset of R of positive type with respect to an operator function in the class N1(H), which will be used in the formulation and the proof of this theorem, we refer the reader to [J2, § 3.1]. Theorem 3.5. Consider the self-adjoint relation A with one negative square and ρ(A) = ∅ in the Krein space H,andletµ ∈ ρ(A) \ R,α∈ C+ ∪ R. Then each of the following conditions (i–iv) is equivalent to the fact that α is the exceptional eigenvalue of A. (i) If A has an eigenvalue in C+ then this eigenvalue is α.Ifσ(A) is real then α R ∈B ∈ istheuniquepointin such that for all open sets ∆ (A) with α ∆ the ∩ H 2 H self-adjoint linear relation A EA(∆) is not nonnegative in EA(∆) . (ii) For the rational function r, r(t):= ψ(t) − ψ(α) ψ(t)−1 − ψ(α) , we have (A − µ)−1A(A − µ)−1r(A)x, x ≥ 0,x∈H. Self-adjoint Extensions of a Nonnegative Linear Relation 139 (iii) α is a pole in C+ of the operator function z → z(A − z)−1,orα ∈ R and + there exists an x ∈H andasequence(zn) ⊂ C converging to α (in C) such −1 that lim infn→∞ Im zn[(A − zn) x, x] < 0. (iv) α is a pole in C+ of the operator function z → z(A − z)−1,orα ∈ R and α does not belong to an open subset of R of positive type with respect to the operator function z → z(A − z)−1. Proof. It was proved in Lemma 3.2 that (i) and (ii) are equivalent to the fact that α is the exceptional eigenvalue of A. For the proof that (i), (iii), and (iv) are equivalent it is sufficient to consider only the case α ∈ R. Therefore in the rest of this proof we assume that σ(A) ⊂ R. Suppose that (i) holds. Then there exists an x ∈Hsuch that the function −1 Gx : z → z (A − z) x, x belongs to the class N1.Ifα ∈ R is the generalized pole of nonpositive type of + Gx then there exists a sequence (zn) ⊂ C which converges to α (in C)andfor which lim inf Im Gx(zn) < 0. (3.12) n→∞ Suppose that α = α .Let∆beanopensubsetofR with ∆ ∈B(A) such that α ∈ ∆andα ∈/ ∆. Then −1 −1 Gx(z)=z EA(∆)(A − z) x, x + z EA(R \ ∆)(A − z) x, x , (3.13) −1 and by (i) the function z → EA(R \ ∆)(A − z) x, x belongs to the class N0. This is in contradiction with (3.12) since the first term on the right-hand side of (3.13) is holomorphic at α . Therefore α = α and (iii) holds. Assume now that (iii) holds. Then for every open ∆ ∈B(A) containing α and for the element x ∈Hfrom (iii) the function Gx belongs to N1. Therefore 2 A ∩ EA(∆)H is not nonnegative. If the point α in (i) would not be unique then A would have more than one negative square, which is not the case. This shows that (i) holds. By assumption of Theorem 3.5 the function G : z → z(A − z)−1 belongs to N1(H). Then on account of the additional assumption σ(A) ⊂ R there exists a unique point α0 ∈ R such that R \{α0} is of positive type with respect to G and every open subset ∆0 of R with α0 ∈ ∆0 is not of positive type with respect to G. With this point α0 the statement (iv) is equivalent to α = α0. Now [J3, Lemma 3.8, (a)⇔(c)] implies that (iii) and (iv) are equivalent. Remark 3.6. For α ∈ R the conditions (i)–(iv) are equivalent to the fact that α is the generalized pole of nonpositive type of the function G : z → z(A − z)−1 in the following sense: For each neighborhood U of α in C there exists a δU > 0 + such that for every µ with δU <µ<∞ there exists a z ∈U∩C such that G(z)+iµJ is not injective; here J is an arbitrary fixed fundamental symmetry of H. That this condition is equivalent to (iv) follows from the similar statement [J1, 140 P. Jonas and H. Langer Proposition 3.5] for generalized Carath´eodory functions by means of a fractional linear transformation of the independent variable. Finally we prove a continuity property of the exceptional eigenvalue. Theorem 3.7. Let A and An,n=1, 2,..., be self-adjoint relations with one nega- tive square in the Krein space H. Suppose that there exists a point µ ∈ C \ R and an open neighborhood U of µ such that U⊂ρ(An) for n =1, 2,..., U⊂ρ(A),and that −1 −1 −1 −1 lim (An − µ) =(A − µ) , lim (An − µ) =(A − µ) (3.14) n→∞ n→∞ in the strong operator topology. Then for the exceptional eigenvalues it holds lim α(An)=α(A). (3.15) n→∞ Proof. We consider the sequence of exceptional eigenvalues α(An). Let α Anν be a subsequence which converges in C+ ∪ R with respect to the topology of the closed complex plane; denote the limit by α0. From (3.7) we obtain − −1 − ≥ ∈H ϕ ψ(Anν ) ψ(Anν ) ψ(α(Anν )) ψ(Anν ) ψ(α(Anν ) f,f 0,f , where ψ is as in (3.5). If we pass to the limit ν →∞it follows easily from the → assumptions (3.14) and α(Anν ) α0 that −1 ϕ ψ(A) ψ(A) − ψ(α0) ψ(A) − ψ(α0) f,f ≥ 0 for all f ∈H. Since, on the other hand, the point β = ψ(α) in (3.7) is uniquely determined we must have α0 = α(A). Therefore each convergent subsequence of the sequence α(An) has the same limit and (3.15) follows. Remark 3.8. It is easy to see that Theorem 3.7 remains true if the assumption (3.14) is replaced by −k −k −k −k lim (An − µ) =(A − µ) , lim (An − µ) =(A − µ) for k =1, 2, n→∞ n→∞ with the limits existing only in the weak operator topology. 4. The exceptional eigenvalues of the self-adjoint extensions of a nonnegative closed linear relation of regular defect one 4.1. The family of self-adjoint extensions We recall some notions and results from [JL3]. Let A0 be a closed nonnegative linear relation in the Krein space (H, [·, ·]). By r(A0)wedenotethesetofall points of regular type of A0, i.e., the set of all z ∈ C for which there exists a cz > 0 such that f f − zf ≥c f , ∈ A0. z f Self-adjoint Extensions of a Nonnegative Linear Relation 141 A0 is said to be of regular defect one if there exist non-real points z0,¯z0 ∈ r(A0) [⊥] [⊥] with dim ran(A0 − z0) =dim ran (A0 − z0) =1.By[JL3,Theorem3.2] the closed nonnegative relations of regular defect one are precisely those closed nonnegative relations of defect one which have at least one self-adjoint extension with nonempty resolvent set; here and in the sequel we consider only self-adjoint extensions without exit, that is within the given Krein space H. The self-adjoint extensions of a closed nonnegative relation of regular defect one are either non- negative or have one negative square, and their resolvent sets are nonempty with the possible exception of one extension. In the rest of this paper, A0 is a closed nonnegative relation in (H, [·, ·]) of regular defect one. It is our aim to study the exceptional eigenvalue of the self- adjoint extensions of A0 which have one negative square. First we consider the special case where no nonnegative self-adjoint extension A of A0 with ρ(A) = ∅ exists. Then A0 has exactly one nonnegative self-adjoint extension Aˆ, this extension has the property ρ(Aˆ)=∅,andallself-adjointex- tensions of A0 except Aˆ have nonempty resolvent set and one negative square. The relations A0 with this property were characterized in [JL3, Theorem 8.1]. All self-adjoint extensions A of A0 different from Aˆ havethesamespectrumσ ,the same spectral function E and the same exceptional eigenvalue α ,whichisalso 2 an eigenvalue of A0.Ifα ∈ C\R then A∩ E (R)H ⊂ A0.Ifα ∈ R then σ ⊂ R ⊂ R \{ } and for every connected closed subset ∆ α such that E (∆) is defined we 2 have A ∩ E (∆)H ⊂ A0, see [JL3, Section 4 and Theorem 7.4]. In the rest of this section we exclude this special case, that is we assume that there exists a nonnegative self-adjoint extension A of A0 with ρ(A) = ∅.Weset −1 R(z):=(A − z) , z ∈ ρ(A), and denote by EA the spectral function of A in the [⊥] Krein space H. We fix a defect element g ∈ ran (A0 + i) and define g(z):= I +(z − i)R(z) g, Q(z):=z[g,g]+(z2 +1)[R(z)g,g]. (4.1) If Q is constant on C+ then, according to [JL3, Lemma 7.1] Q(z)=0forz ∈ C+. According to [JL3, Theorem 7.2] the self-adjoint extensions of A0 coincide with the linear relations A(γ), γ ∈ R, defined as follows: A(∞) = A, −1 [ · ,g(¯z)] g(z) A( ) − z = R(z) − ,γ∈ R \{0}, (4.2) γ γ + Q(z) for some (and then for all) z ∈ ρ(A)withγ + Q(z) =0,and A(0) := lim A(γ), γ→0,γ=0 where the limit is to be understood in the sense of the gap metric in H2. Here A(0) is the only self-adjoint extension of A0 which can have an empty resolvent set, see [JL3, Theorem 7.2]. If Q is not identically equal to zero then the relation 142 P. Jonas and H. Langer (4.2) holds also for γ = 0. For every γ ∈ R we have ρ(A(γ)) ∩ ρ(A)={z ∈ ρ(A): γ + Q(z) =0 }. (4.3) According to [JL3, Theorem 8.2] there exists a nonempty open interval (γ−,γ+), −∞ ≤ γ− ≤ 0 ≤ γ+ ≤∞, such that A(γ) has one negative square ⇐⇒ γ ∈ (γ−,γ+), A(γ) is nonnegative ⇐⇒ γ ∈ R \ (γ−,γ+). Here γ+ and γ− canbeexpressedintermsofA and g (see [JL3, Section 8]): If J is 1/2 a fundamental symmetry of H, x J := [Jx,x] , x ∈H, and if we denote by Top the operator part of the self-adjoint relation T in the Hilbert space (H, [J · , · ]), then 1/2 2 ∈D 1/2 (JA)op g J if g ((JA)op ), γ+ = +∞ otherwise, (4.4) − −1 1/2 2 ∈D −1 1/2 (JA )op g J if g ((JA )op ), γ− = −∞ otherwise. If γ ∈ (γ−,γ+)andρ(A(γ)) = ∅, the exceptional eigenvalue of A(γ) is denoted by α(γ): α(γ):=α(A(γ)), + see Definition 3.4. Recall that these values α(γ),γ∈ (γ−,γ+), belong to R ∪ C . In the sequel we shall study the function (γ−,γ+) γ → α(γ). 4.2. The family of functions z Q(z)+γ ,γ∈ R In this subsection we set for γ ∈ R 2 2 Fγ (z):=z(Q(z)+γ)=(1+z )z[R(z)g,g]+z [g,g]+γz. (4.5) A simple calculation shows that the function z → z[R(z)g,g] belongs to N0 since A is nonnegative. Therefore, the functions Fγ form a family as considered in Section 2: 2 Fγ (z)=γz+G1(z)+z G2(z),G1(z)=z [R(z)g,g],G2(z)=[g,g]+z [R(z)g,g]. We express the numbers αj ,βj, the measures σj,j=1, 2, and also γ0 and γ∞, which were introduced in Section 2, in terms of A and g. Evidently, α1 =ReG1(i)=−Im [R(i)g,g]=−[R(i)g,R(i)g], α2 =ReG2(i)=−[R(i)g,R(i)g]+[g,g], −1 β := β1 = β2 = lim η Im iη[R(iη)g,g] η↑∞ 1 = lim Re [R(iη)g,g] = lim (R(iη)+R(−iη))g,g . η↑∞ η↑∞ 2 Further, σ1 = σ2 =: σ, and an application of the Stieltjes-Livˇsic inversion formula to the relation 2 −1 2 −1 1+zt (1 + z ) z[R(z)g,g]=(1+z ) α1 + βz + dσ(t) R t − z Self-adjoint Extensions of a Nonnegative Linear Relation 143 yields, for every interval (a, b)witha, b = 0, the formula b−0 dσ(t)= R(−i)AR(i)EA((a, b))g,g . (4.6) a+0 On the other hand, [R(−i)AR(i)g,g]=Imi [R(i)g,g] 1+it =Imα + βi+ dσ(t) = β + dσ(t), R t − i R and hence, with ∆n := (−n, n),n=1, 2,..., β = lim [R(−i)AR(i)(1 − EA(∆n))g,g]. (4.7) n→∞ Lemma 4.1. Assume that t2 dσ(t) < ∞ or, equivalently, R sup AEA(∆n)g,g : n =1, 2,... < ∞. (4.8) 1/2 Then the sequence EA (∆n) g converges for n →∞in D (JA)op with respect to the graph norm of this domain. If g(∞) := limn→∞ EA(∆n)g and g∞ := g − g(∞), then g belongs to the root subspace of A−1 corresponding to 0 and we have ∞ −1 2 −1 β = A g∞,g∞ ,α2 − t 1+t dσ(t)=[g∞,g∞]. R In particular, under the assumption (4.8), γ∞ < ∞ if and only if −1 A g∞,g∞ = g∞,g∞ =0, and in this case γ∞ = lim AEA(∆n)g,g . (4.9) n→∞ 1/2 Proof. On span EA(∆n \ ∆1)H : n =2,... the norm [A · , · ] is equivalent to 1/2 the graph norm of (JA)op .Ifn1,n2 ∈ N,n2 ≥ n1,then − − 2 [A(EA(∆n2 ) EA(∆n1 ))g,(EA(∆n2 ) EA(∆n1 ))g]= 1+t dσ(t), ∆n2 \∆n1 which implies the first assertion. For every bounded interval ∆ we have EA(∆)g∞ = EA(∆)g − lim EA(∆)EA(∆n)g =0. n→∞ −1 −1 −1 Hence g∞ and A g∞ belong to the root space of A at zero, and [A g∞,g(∞)]= [g∞,g(∞)] = 0. In the subspace (I − EA(∆1))H the following identities hold: R(−i)R(i) = I − R(−i)R(i) A−2 , (4.10) (I−EA(∆1))H (I−EA(∆1))H R(−i)AR(i) =(A−1 −A−1R(−i)R(i)) . (4.11) (I−EA(∆1))H (I−EA(∆1))H 144 P. Jonas and H. Langer Since the operator A−1 is nonnegative, its root subspace at zero co- (I−EA(∆1))H −2 incides with the kernel of its square. Therefore A g∞ = 0 and, by (4.10) and (4.11), −1 R(−i)R(i)g∞ =0,R(−i)AR(i)g∞ = A g∞. (4.12) In view of (4.7), β = lim R(−i)AR(i)(I − EA(∆n))g,g n→∞ −1 −1 = R(−i)AR(i)g∞,g(∞) + g∞ = A g∞,g(∞) + g∞ = A g∞,g∞ . Moreover, α2 − tdσ(t)=− R(−i)R(i)g,g +[g,g] − lim tdσ(t) n→∞ R ∆n = (I − R(−i)R(i))g,g − lim (I − R(−i)R(i))EA(∆n)g,g n→∞ = lim (I − R(−i)R(i))(I − EA(∆n))g,g n→∞ = (I − R(−i)R(i))g∞,g∞ = g∞,g∞ . Here the last equality is a consequence of (4.12). Finally, the last assertion of the lemma follows from the definition of γ∞. Remark 4.2. In the terminology of [KK, Section 4], the inequality γ∞ < ∞ means that the function z −→ z[R(z)g,g] belongs to the class (R0). In this case (4.9) and [KK, Theorem S 1.4.2] give also other expressions for γ∞, e.g., 2 γ∞ =sup η Re [R(iη)g,g] . η>0 −1 −1 Set Γn := R \ [−n ,n ],n=1, 2,.... The following lemma can be proved similarly to Lemma 4.1. 0 −2 ∞ Lemma 4.3. Assume that R\{0} t dσ(t) < or, equivalently, −1 sup A EA(Γn)g,g : n =1, 2,... < ∞. (4.13) −1 1/2 Then the sequence EA(Γn)g converges for n →∞in D (JA )op with respect to the graph norm of this domain. If g(0) := limn→∞ EA(Γn)g and g0 := g − g(0), then g belongs to the root subspace of A at zero, 0 Ag0,g(0) = g0,g(0) =0, and we have −1 σ({0})= Ag0,g0 ,α1 + lim t dσ(t)=− g0,g0 . n→∞ Γn In particular, under the assumption (4.13), γ0 > −∞ if and only if [Ag0,g0]=[g0,g0]=0, and in this case −1 γ0 = − lim A EA(Γn)g,g . n→∞ Self-adjoint Extensions of a Nonnegative Linear Relation 145 From the above lemmas, with the definitions of g∞ and g0 we obtain the following relations: ⎧ ⎪ lim →∞ AE (∆ )g,g if sup AE (∆ )g,g < ∞, ⎨ n A n n A n −1 γ∞ = A g∞,g∞ = g∞,g∞ =0, ⎩⎪ ∞ otherwise, (4.14) ⎧ −1 −1 ⎪ − lim →∞ A E (Γ )g,g if sup A E (Γ )g,g < ∞, ⎨ n A n n A n γ0 = Ag0,g0 = g0,g0 =0, ⎩⎪ −∞ otherwise. Trivially, γ0 ≤ 0 ≤ γ∞, hence γ0 = γ∞ ⇐⇒ γ0 = γ∞ =0. (4.15) 4.3. The case ρ(A(0))=∅ We shall show that in this case the exceptional eigenvalue α(γ),γ∈ (γ−,γ+), is 0 or ∞. To prove this, we rely on the results of [JL3]. Theorem 4.4. Assume that there exists a nonnegative self-adjoint extension A of A0 with ρ(A) = ∅. Then the following conditions are equivalent. (i) ρ(A(0))=∅. (ii) The element g admits the unique representation g = g0 + g∞,g0 ∈ ker A0, 1 g∞ ∈ A0(0) ,and[g0,g0]=[g∞,g∞]=0. (iii) γ0 = γ∞. If these conditions are fulfilled, then γ+ =0⇐⇒ g∞ =0,γ− =0⇐⇒ g0 =0, (4.16) and 0 γ ∈ (γ−, 0), α(γ)= ∞ γ ∈ (0,γ+). [⊥] Proof. If (i) holds then A(0) = A. Then, by [JL3, Corollary 6.2], ran A0 + i is spanned by an element of the form g0 + g∞ where g0 ∈ ker A0,g∞ ∈ A0(0), and g0,g0 = g∞,g∞ = 0, that is, (ii) holds. Assume now that (ii) holds. Since g0,g∞ =0wehave 2 Q(z):=z g0 + g∞,g0 + g∞ + z +1 R(z)(g0 + g∞),g0 + g∞ =0 for every z ∈ ρ(A). Then, by [JL3, Lemma 7.1] there exists a self-adjoint exten- sion A of A0 with ρ(A)=∅. By [JL3, Theorem 7.2], A coincides with A(0) and, therefore, (i) holds. Moreover, (ii) implies sup AE (∆ )g,g : n =1, 2,... A n −1 =sup A EA(Γn)g,g : n =1, 2,... =0, 1 By [JL3, (3.1)] we have ker A0 ∩ A0(0) = {0}. 146 P. Jonas and H. Langer and by the definition of g∞ and g0 in Lemma 4.1 and Lemma 4.3 we have g0 = g0 and g∞ = g∞. Then, by (4.14), γ0 = γ∞ = 0, that is, (iii) is true. If (iii) holds, then, by (4.15), γ0 = γ∞ = 0. On account of (4.14), this implies for every bounded real interval ∆ with 0 ∈/ ∆that AEA(∆)g,g = 0 and hence E (∆)g = 0. By the definition of g∞ and g0 we obtain A g0 = EA (−1, 1) g, g∞ = I − EA (−1, 1) g. Therefore, g = g0 + g∞ and, in view of Ag0,g0 = 0 (see (4.14)), for every x ∈ EA (−1, 1) H, 1/2 1/2 Ag0,x ≤ Ag0,g0 Ax, x =0, ∈ ∈ which implies g0 ker A. Similarly, g∞ A(0). Again by (4.14) we have g0,g0 = g∞,g∞ = 0. Hence (ii) holds with A0 replaced by A. This yields [g(z),g(z)] = 0 for all z ∈ C \ R, and, by [JL3, Theorem 6.1], (i) is true. Assume now that the conditions (i–iii) are satisfied. By (4.4) the relation γ+ = 0 is equivalent to g ∈ ker A. In view of (ii) this is equivalent to g∞ ∈ (ker A) ∩ A(0). The latter holds, by the assumption on A, if and only if g∞ =0 (see [JL3, Theorem 1.3]). Similarly it follows that γ− = 0 is equivalent to g0 =0. It remains to prove the last assertion about the exceptional eigenvalues. Let γ ∈ (γ−, 0) ∪ (0,γ+). Then, by (ii) and [JL3, Theorem 7.2], −1 −1 −1 −1 A(γ) − z = R(z) − γ ·,iz g0 + g∞ iz g0 + g∞ , and the following function has one negative square: −1 −1 −1 z → z A( ) − z = zR(z) − γ z ·,g0 g0 (4.17) γ −1 −1 − γ z ·,g∞ g∞ − γ − i ·,g0 g∞ + i ·,g∞ g0 . If γ>0thenγ+ > 0 and, by (4.16), g∞ = 0. In this case, the terms on the right- hand side of (4.17) with the exception of the third term are Nevanlinna functions. −1 We choose some x ∈Hsuch that the function Gx : Gx(z):=z (A(γ) − z) x, x belongs to N1.Thenwehave[x, g∞] = 0 and the generalized pole of the function −1 2 z →−γ z|[x, g∞]| ,whichisthepoint∞, coincides with the generalized pole + of Gx. Therefore there exists a sequence (zn) ⊂ C converging to ∞ in C such that lim infn→∞ Im Gx(zn) < 0. Then, on account of Theorem 3.5, α(γ)=∞.If γ<0thenγ− < 0 and, by (4.16), g0 = 0. Now all the terms on the right-hand side of (4.17) with the exception of the second term are Nevanlinna functions and, as above, using Theorem 3.5 we find α(γ)=0. 4.4. The generic case Now we describe the exceptional eigenvalue α(γ),γ ∈ (γ−,γ+), if ρ(A(γ)) = ∅ for all γ ∈ R. To this end we apply the results of Section 2 to the functions Fγ from (4.5). Recall that the function Fγ belongs to the class N1 if and only if γ ∈ (−∞,γ∞), and as in Section 2, for these γ we denote the zero of nonpositive type of Fγ by ζ(γ). Self-adjoint Extensions of a Nonnegative Linear Relation 147 Lemma 4.5. Suppose that ρ A(γ) = ∅ for all γ ∈ R.Then −∞ ≤ γ− ≤ γ0 ≤ 0 ≤ γ∞ ≤ γ+ ≤ +∞, (4.18) + and α(γ)=ζ(γ) for γ ∈ (γ0,γ∞).Fortheseγ,ifα(γ) ∈ C then g(α(γ)) (see (4.1)) is an eigenelement of A(γ) at the eigenvalue α(γ),ifα(γ)=α(γ) =0 then the limit g(α(γ)) := limε→0 g(α(γ)+iε) exists and is an eigenelement of A(γ) at α(γ) and (sign α(γ)) [g(α(γ)),g(α(γ))] ≤ 0. (4.19) Proof. By Theorem 4.4 we have γ0 = γ∞. The relations (4.14) show that γ0 ≤ 0 ≤ γ∞ holds. Let γ ∈ (γ0,γ∞). Then, according to Lemma 2.3 and Theorem 2.4, ζ(γ) =0 , ∞.WeproveLemma4.5inthetwocasesζ(γ) ∈/ R and ζ(γ) ∈ R \{0} separately. If ζ(γ) ∈/ R then the second term on the right-hand side of (4.2) has a pole at + ζ(γ)andζ(γ) is an eigenvalue of A(γ) in C . It follows that A(γ) is not nonnegative, i.e., γ ∈ (γ−,γ+)andα(γ)=ζ(γ). Since g(α(γ)) is a defect element of A0 at the point α(γ) it is an eigenelement of the extension A(γ) of A0 to the eigenvalue α(γ). Let ω := ζ(γ) ∈ R \{0}. Then, by Lemma 2.3, |t − ω|−2 dσ(t) < ∞.Let R ∆ be a bounded open interval, such that ω ∈ ∆, 0 ∈/ ∆.¯ We denote by ι the linear mapping φ → EA(∆)φ(A)g defined for all rational functions φ with only non-real poles. Then ι is an isometry from the linear manifold of these elements 2 H − · · φ of L (∆; σ)into EA(∆) , [R( i)AR (i) , ] .Weextendι to an isometry from 2 L (∆; σ) onto a closed subspace of EA(∆)H, [R(−i)AR(i)·, ·] . For any funda- mental symmetry J the scalar products [J·, ·]and[R(−i)AR(i)·, ·]areequivalent on EA(∆)H. Since the function ( ·−ω)−1 belongs to L2(∆; σ) and the functions ( ·− ω − i )−1 converge in L2(∆; σ)to(·−ω)−1 for → 0, also lim R(ω + i )g =: gω (4.20) →0 exists. For arbitrary z ∈ ρ(A)wehave −1 R(z)gω =(z − ω) (R(z)g − gω). (4.21) One verifies without difficulty that for the function Fγ considered in the present section, ω ∈ R \{0} and under the assumption (4.20) the relation (2.9) holds if and only if 2 2 ω(ω +1)[gω,gω]+(3ω +1)[gω,g]+2ω[g,g]+γ ≤ 0, (4.22) and (2.10) holds if and only if 2 2 γω + ω [g,g]+ω(1 + ω )[gω,g]=0. (4.23) Therefore the relations (4.22) and (4.23) hold. Inserting (4.23) into (4.22) we obtain 2 2 ω(ω +1)[gω,gω]+2ω [gω,g]+ω[g,g] ≤ 0. (4.24) 148 P. Jonas and H. Langer Using the relations (4.21) and (4.23) it can be verified by an easy calculation that the element g(ω)=g +(ω − i)gω (4.25) is not the zero element and an eigenelement of A(γ) to the eigenvalue ω.Further, 2 [g(ω),g(ω)] = [g,g]+(ω − i)[gω,g]+(ω + i)[g,gω]+(ω +1)[gω,gω] (4.26) and, by (4.24), (sign ω)[g(ω),g(ω)] ≤ 0. (4.27) Hence A(γ) is not nonnegative, i.e., γ ∈ (γ−,γ+), and, by Theorem 3.1, ω = ζ(γ) coincides with α(γ). Lemma 4.6. If γ ∈ (γ−,γ+) and α(γ) =0 , ∞,thenγ ∈ (γ0,γ∞),α(γ)=ζ(γ),and we have the direct orthogonal decomposition ker A(γ) − α(γ) =span{g(α(γ))}+˙ ker(A − α(γ)). (4.28) + Proof. If α(γ) ∈ C then, by (4.2), Fγ has a pole at α(γ) and therefore ζ(γ)= α(γ). Then Theorem 2.4,(1) implies γ ∈ (γ0,γ∞). In this case (4.28) holds with ker (A − α(γ)) = {0}. Assume that ω := α(γ) ∈ R \{0}. Then there exists an x ∈H, x =0,such x that ∈ A( ) and ωx γ (sign ω)[x, x] ≤ 0. (4.29) By [JL3, Corollary 7.3], there exists an x0 ∈H, x0 = 0, such that x R(−i)x0 g =(γ + i[g,g]) − [x0,g] . (4.30) ωx x0 − iR(−i)x0 ig We have γ + i[g,g] = 0 since otherwise (4.30) would imply ω = i.Wealsohave x [x0,g] = 0. Indeed, [x0,g] = 0 would imply that ∈ A and that x is an ωx eigenvector of the bounded nonnegative operator R(−i)AR(i), corresponding to the eigenvalue ω(ω2 +1)−1, and hence that (sign ω)[x, x] > 0. This contradicts (4.29). From (4.30) we get −1 1 − (ω + i)R(−i) x0 = Cg with C := (i − ω)[x0,g](γ + i[g,g]) , (4.31) which implies EA({ω})g =0, (4.32) where EA denotes again the spectral function of A. If we set x0 := EA({ω})x0,˜x0 := x0 − x0,then[x0,g]=[˜x0,g], and we shall show that for − − x˜ := (γ + i[g,g])R( i)˜x0 [˜x0,g]g (4.33) x˜ it holds ∈ A( ) and ωx˜ γ (sign ω)[˜x, x˜] ≤ 0. (4.34) Self-adjoint Extensions of a Nonnegative Linear Relation 149 Indeed, −1 x˜ = x − γ + i[g,g] R(−i)x0 = x − γ + i[g,g] (ω + i) x0 (4.35) and x˜ x (ω + i)−1x = − (γ + i[g,g]) 0 ωx˜ ωx ω(ω + i)−1x 0 R(−i)x0 R(−i)x0 =(γ + i[g,g]) − (4.36) x0−iR(−i)x0 x − iR(−i)x 0 0 g R(−i)˜x0 g − [x0,g] =(γ + i[g,g]) − [˜x0,g] ∈ A(γ). ig x˜0 − iR(−i)˜x0 ig To verify (4.34) observe that because of [x0,x0]=[g,x0]=0wehave [x, x0]=0. (4.37) As A is nonnegative, (signω)[x0,x0] ≥ 0. (4.38) Using (4.29), (4.35), (4.37), and (4.38) we find 0 ≥ (sign ω)[x, x] −1 −1 =(signω) x +(γ + i[g,g])(ω + i) x0, x +(γ + i[g,g])(ω + i) x0 ≥ (sign ω)[x, x]. By (4.31) we have (I − (ω + i)R(−i))˜x0 = Cg. (4.39) Let ∆ be a bounded open interval such that ω ∈ ∆and0∈/ ∆.¯ Then, because of (4.6) and (4.39), for every ε>0, |C|2 ((t − ω)2 + 2)−1 dσ(t) ∆ 2 = |C| R(ω + iε)R(ω − iε)R(−i)AR(i)EA(∆)g,EA(∆)g 2 2 2 = |C| R(ω + iε)(A−ω)R(ω−iε)(A−ω)R(−i) AR(i) E (∆)x0, x0 A 2 2 2 2 −1 2 −2 = |C| (t − ω) ((t − ω) + ) (1 + t ) td[EA(t)x0, x0], ∆ and therefore, (t − ω)−2 dσ(t) < ∞, (4.40) ∆ i.e., the function t → (t − ω)−1 belongs to L2(∆; σ). Now we see as in the proof of Lemma 4.5 that limε↓0 R(ω + iε)g =: gω exists. Applying the operator I +(ω + i + iε)R(ω + iε) to (4.39) we find I + iεR(ω + iε) x0 = C g +(ω + i + iε)R(ω + iε)g , and for ε ↓ 0, on account of E ({ω})x0 =0, A x0 = C g +(ω + i)gω . (4.41) 150 P. Jonas and H. Langer Taking the inner product with g = I − EA({ω}) g on both sides of this relation and inserting C from (4.31) we obtain −1 [x0,g]=(i − ω)[x0,g] γ + i[g,g] g +(ω + i)gω , i.e., 2 γ + ω[g,g]+(ω +1)[gω,g]=0. (4.42) The relations (4.33) and (4.41) yield x =(i − ω) x0,g R(−i) g +(ω + i)g − x0,g g ω = − x0,g g − (i − ω) R(−i)g +(ω − i)R(−i)gω . This implies in view of (ω + i)R(−i)gω = gω − R(−i)g (see (4.21)) the relation x = − x0,g g − (i − ω)gω = −[x0,g] g(ω). (4.43) From (4.34) we obtain 0 ≥ ω g − (i − ω)gω,g− (i − ω)gω 2 2 = ω(ω +1) gω,gω +2ω gω,g + ω[g,g]. Adding (4.42) it follows that 2 2 0 ≥ ω(ω +1) gω,gω +(3ω +1) gω,g +2ω[g,g]+γ. (4.44) Now (4.44) and (4.42) imply ω = ζ(γ), hence γ ∈ (γ0,γ∞). The first equality in (4.36) and (4.43) give (4.28). Now we prove the main result about the curve γ → α(γ) of the exceptional eigenvalue. For the definition of the elements g0,g∞ see Lemma 4.1 and Lemma 4.3, the measure σ and the number β were defined in Subsection 4.2. Theorem 4.7. Suppose that all self-adjoint extensions of A0 have nonempty resol- vent set. Then the following statements hold. + (i) The function α :(γ−,γ+) → C ∪ R is continuous and lim α(γ)=0, lim α(γ)=∞. γ↓γ− γ↑γ+ (ii) We have −∞ ≤ γ− ≤ γ0 ≤ 0 ≤ γ∞ ≤ γ+ ≤∞, γ0 <γ∞,and 0ifγ ∈ (γ−,γ0), α(γ)= ∞ if γ ∈ (γ∞,γ+). (iii) The function |α| : |α|(γ):=|α(γ)|,γ ∈ (γ0,γ∞), is nondecreasing and γ1,γ2 ∈ (γ0,γ∞),γ1 = γ2 implies α(γ1) = α(γ2).Ifσ =0 or β =0 ,then this function is strictly increasing, if σ =0and β =0, then it is strictly increasing if and only if [g0,g0][g∞,g∞] ≥ 0. (iv) The function α is real analytic in γ ∈ (γ0,γ∞) if α(γ) ∈/ supp σ with possible exception of branch points of order ≤ 3 for which α(γ) ∈ R. Self-adjoint Extensions of a Nonnegative Linear Relation 151 (v) σ α(γ): γ ∈ (γ0,γ∞) ∩ R =0. (vi) For every component I of the open set R \ supp σ ∪{0} the set I ∩ α(γ): γ ∈ (γ0,γ∞) is empty or connected. The statement (v) means that α(γ) cannot move within the support of σ, (vi) means that every bounded component I as in the statement is ‘visited’ by the exceptional eigenvalue at most once. ProofofTheorem4.7. The first assertion of (i) follows from (4.2) and Theorem 3.7. By Theorem 4.4 we have γ0 <γ∞. Then, according to Lemma 4.5, α(γ)=ζ(γ) for γ ∈ (γ0,γ∞), and Theorem 2.4 implies lim α(γ)=0, lim α(γ)=∞. (4.45) γ↓γ0 γ↑γ∞ If γ ∈ (γ−,γ0)∪(γ∞,γ+) then by Lemma 4.6 either α(γ)=0orα(γ)=∞. Hence, on account of the continuity of γ → α(γ) and (4.45), α(γ)=0forγ ∈ (γ−,γ0) and α(γ)=∞ for γ ∈ (γ∞,γ+). This proves the assertions (i) and (ii). Lemma 4.6 and Theorem 2.4 imply the first assertion of (iii) and that the function γ →|α(γ)| is nondecreasing. The criterion on strict monotonicity follows from Theorem 2.4 in view of the fact that we have α1 = − g0,g0 ,α2 = − g0,g0 +[g,g]= g∞,g∞ . This proves assertion (iii). The assertions (iv), (v), and (vi) are consequences of Theorem 2.4 and Theorem 2.5. The following two examples show that intervals of constancy of the functions γ → α(γ)andγ →|α(γ)| may occur, even in finite-dimensional situations. 01 Example 4.8. In the Krein space C2, · , · we consider the nonnegative 10 00 2 operator T := , a>0. Define H0 := H∞ := C , H := H0 ×H∞ and the a a 0 × −1 T nonnegative relation A = Ta0 Ta∞ for some a0, a∞ > 0. Let g =(b1,b2,c1,c2) . With the help of Lemmas 4.1 and 4.3 we can easily compute the quantities γ+, γ∞, γ−, γ0: ∞ if c1 =0 ∞ if c1 =0 γ+ = 2 −1 2 ,γ∞ = 2 , a0|b1| + a∞ |c2| if c1 =0 a0|b1| if c1 =0 −∞ if b1 =0 −∞ if b1 =0 γ− = 2 −1 2 ,γ0 = 2 . −a∞|c1| − a0 |b2| if b1 =0 −a∞|c1| if b1 =0 Example 4.9. Let (H, [·, ·]) be the orthogonal direct sum of the Krein subspaces H · · H · · x ∈H ∈H ( , [ , ]) and ( , [ , ]). Then A := y : x ,y is a nonnegative self-adjoint relation. We choose some g = g + g with g ∈H , g ∈H and [g ,g ][g ,g ] < 0. Then (γ0,γ∞) γ →|α(γ)| is not strictly increasing. 152 P. Jonas and H. Langer In the following theorem we give some criteria for the coincidence of the numbers γ∞ and γ+ as well as of γo and γ−. Recall that ∆n =(−n, n), Γn = R \ [−n−1,n−1]. Theorem 4.10. (1) Each of the following conditions implies γ∞ = γ+: (i) limn→∞ [AE(∆n)g,g]=∞. (ii) A(0) is trivial or a definite subspace. (2) Each of the following conditions implies γ0 = γ−: −1 (i) limn→∞ [A E(Γn)g,g]=∞. (ii) ker A is trivial or a definite subspace. Proof. We prove assertion (1), assertion (2) can be proved by a similar reasoning. Condition (i) implies γ∞ = ∞ (see (4.14)), and therefore γ∞ = γ+ = ∞. Assume that (ii) holds. If limn→∞ [AE(∆n)g,g] < ∞, then, by Lemma 4.1, (E(∆n)g) 1/2 converges in D((JA)op )toanelementg(∞) with respect to the graph norm, and −1 g∞ := g − g(∞) belongs to the root space of A corresponding to 0. Then, by (ii), we have either [g∞,g∞] =0org∞ = 0. In the first case Lemma 4.1 implies 1/2 γ∞ = ∞. In the second case we have g = g(∞) ∈D (JA)op , and Lemma 4.1 and (4.4) give 1/2 2 γ∞ = lim AE(∆n)g,g = lim (JA)op E(∆n)g = γ+. n→∞ n→∞ J 4.5. The root subspace at a real nonzero exceptional eigenvalue We suppose as in Subsection 4.4 that all self-adjoint extensions of A0 have non- empty resolvent set. Let γ ∈ (γ0,γ∞) be such that ω := α(γ) ∈ R \{0}.Weshow that modulo ker(A − ω) the elements of the Jordan chain of A(γ) at ω can be expressed as limits of the defect elements g(ω + iε), ε ↓ 0. First we recall that by the Lemmas 4.5 and 4.6 with g(ω) = limε↓0 g(ω + iε) we have ker (A(γ) − ω)=span{g(ω)}+˙ ker (A − ω) and (sign ω)[g(ω),g(ω)] ≤ 0. Moreover, g(ω)=g +(ω − i)gω,wheregω := limε↓0 R(ω + iε)g (see (4.20)). If E(γ) denotes the spectral function of A(γ) and ∆ is a bounded open interval such that ω ∈ ∆, 0 ∈/ ∆, then E(γ)(∆)H, [A(γ) · , · ] is a Pontryagin space with negative index 1, and A(γ) is self-adjoint in this space. According to [JL2], the length of any Jordan chain of A(γ) at ω is ≤ 3, k+1 k dim ker(A(γ) − ω) ) ker(A(γ) − ω) ≤ 1,k=1, 2, and exactly one of the following cases holds: 2 (e) ker(A(γ) −ω)=ker(A(γ) −ω) and ω is not a singular critical point of A(γ). 2 (p1s)ker(A(γ) −ω)=ker(A(γ) −ω) and ω is a singular critical point of A(γ). 2 3 (p2r)ker(A(γ) −ω) =ker( A(γ) −ω) =ker(A(γ) −ω) and ω is a regular critical point of A(γ). Self-adjoint Extensions of a Nonnegative Linear Relation 153 2 3 (p2s)ker(A(γ) − ω) = ker (A(γ) − ω) = ker (A(γ) − ω) and ω is a singular critical point of A(γ). 2 3 (p3)ker(A(γ) − ω) =ker( A(γ) − ω) =ker( A(γ) − ω) . In the case (p3) ω is a regular critical point of A(γ). Theorem 4.11. Assume that all self-adjoint extensions of A0 have nonempty re- solvent set and let γ ∈ (γ0,γ∞) be such that ω := α(γ) ∈ R \{0}.Then A(γ) − ω g(ω)=0, and the following equivalences hold: (e) ⇐⇒ (sign ω)[g(ω),g(ω)] < 0. (p1s) ⇐⇒ [g(ω),g(ω)] = 0, lim R(ω + iε)g(ω + iε)doesnotexist. ε↓0 (p2r) ⇐⇒ [g(ω),g(ω)] = 0,f1 := lim R(ω + iε)g(ω + iε)exists, [g(ω),f1] =0 . ε↓0 (p2s) ⇐⇒ [g(ω),g(ω)] = 0,f1 := lim R(ω + iε)g(ω + iε)exists, [g(ω),f1]=0, ε↓0 lim R(ω + iε)2g(ω + iε)doesnotexist. ε↓0 (p3) ⇐⇒ [g(ω),g(ω)] = 0,f1 := lim R(ω + iε)g(ω + iε)exists, [g(ω),f1]=0, ε↓0 2 f2 := lim R(ω + iε) g(ω + iε)exists. ε↓0 Moreover, in the cases (p2r) and (p2s), the elements g(ω),f1 form a Jordan chain of A(γ) at ω and 2 ker (A(γ) − ω) =span{g(ω),f1} [+]˙ ker(A − ω); in the case (p3) the elements g(ω),f1,f2 form a Jordan chain of A(γ) at ω and 3 ker (A(γ) − ω) =span{g(ω),f1,f2} [+]˙ ker(A − ω). Proof. First we mention that a real eigenvalue of a self-adjoint relation B with non-empty resolvent set in a Pontryagin space is a regular critical point of B if and only if the corresponding root subspace is non-degenerated, see [KL1]. The statement of the theorem is now a consequence of the following two claims and of Lemma 4.6. Claim 1: If [g(ω),g(ω)] = 0 then there exists a Jordan chain of A(γ) at ω of length ≥ 2 if and only if f1 := lim R(ω + iε)g(ω + iε) (4.46) ε↓0 exists; in this case the elements g(ω),f1 form a Jordan chain of A(γ) at ω,andthis Jordan chain is orthogonal to ker(A − ω). 154 P. Jonas and H. Langer To prove this, assume first that f1 = limε↓0 R(ω + iε)g(ω + iε) exists. If z, ζ ∈ C \ R,z= ζ, we find from (4.2) by a straightforward calculation that −1 −1 (A(γ) − ζ) − (z − ζ) R(z)g(z) [g(z),g(z)] − (γ + Q(z))(z − ζ)−1) = −(z − ζ)−2g(z) − g(ζ). (z − ζ)(γ + Q(ζ)) If we set z = ω + iε,theninviewof lim [g(ω + iε),g(ω − iε)] = [g(ω),g(ω)] = 0 ε→0 and 2 lim γ + Q(ω + iε) = γ + ω[g,g]+ ω +1 [gω,g]=0 ε→0 (see (4.23)), we find −1 −1 −2 (A(γ) − ζ) − (ω − ζ) f1 = −(ω − ζ) g(ω). (4.47) This relation is equivalent to f1 ∈ A( ) − ω. (4.48) g(ω) γ By (4.32), for z = z we have E({ω})R(z)g(z)=R(z) I +(z − i)R(z) E({ω})g =0, ⊥ − ⊥ − therefore f1[ ] ker (A ω)andspan g(ω),f1 [ ] ker (A ω). ∈H ∈ − Conversely, assume that there exist elements x ,y ker A(γ) ω ,y= x 0, such that ∈ A( ) − ω. This implies [y,y] = 0, and by Lemma 4.6 and y γ [g(ω),g(ω)] = 0 we find y = αg(ω)forsomenumberα = 0. Hence x −1 ∈ A( ) − ω for x := α x . (4.49) g(ω) γ If ζ = ζ then x −1 −1 ∈ (A( ) − ζ) − (ω − ζ) , −(ω − ζ)−2g(ω) γ or R(ζ) − (ω − ζ)−1 x − (γ + Q(ζ))−1[x, g(ζ)]g(ζ) (4.50) −2 −2 −2 = −(ω − ζ) g(ω)=−(ω − ζ) g − (ω − ζ) (ω − i)gω (see (4.20), (4.25)). Let ∆ be a real bounded open interval such that ω ∈ ∆, 0 ∈/ ∆, and denote the spectral function of A again by E . Evidently, A −1 R(ζ) − (ω− ζ) EA(∆)x (4.51) =−(ω− ζ)−1E (∆)(A− ω)R(ζ)E (∆)x ∈ ran (A− ω) . A A EA(∆)H Self-adjoint Extensions of a Nonnegative Linear Relation 155 For ζ = ζ we have EA(∆)g(ζ)=EA(∆) I +(ζ − i)R(ζ) g (4.52) = E (∆) I+(ζ− i)R(ζ) (A− ω)E (∆)g ∈ ran (A− ω) . A A ω EA(∆)H We apply EA(∆) to the left- and the right-hand side of (4.50). Then (4.51) and (4.52) yield E (∆)g ∈ ran (A− ω) , A ω EA(∆)H and that the set R(ω + iε)EA(∆)gω : ε ∈ R \{0} is bounded in EA(∆)H. Making use of the isometry ι defined in the proof of Lemma 4.5 we conclude that the integrals −1 |t − ω|−2 (t − ω)2 + ε2 dσ(t),ε>0, ∆ are uniformly bounded. Therefore the function t → (t − ω)−4 is σ-integrable. This implies that the functions ∆ t → (t − i)(t − ω − iε)−2 converge in L2(∆; σ)for ε ↓ 0, which is equivalent to the fact that the limit (4.46) exists. Then, by (4.48) and (4.49), x − f1 ∈ ker A(γ) − ω , which on account of Lemma 4.6 gives 2 ker A(γ) − ω =span{f1} +ker A(γ) − ω =span{g(ω),f1}[+]˙ ker (A − ω). Claim 2: If [g(ω),g(ω)] = 0,f1 := limε↓0 R(ω + iε)g(ω + iε)existsand [f0,f1] = 0 then there exists a Jordan chain of A(γ) at ω of length 3 if and only if 2 f2 := limε↓0 R(ω + iε) g(ω + iε) exists; in this case the elements g(ω),f1,f2 form a Jordan chain of A(γ) at ω, and this chain is orthogonal to ker(A − ω). The proof of this claim is analogous to the proof of the Claim 1; e.g., the relation (4.47) is to be replaced by −1 −1 −2 −3 (A(γ) − ζ) − (ω − ζ) f2 = −(ω − ζ) f1 +(ω − ζ) g(ω). Remark 4.12. If the assumptions of Theorem 4.11 are fulfilled, it follows from the proof of Theorem 4.11 that limε↓0 R(ω+iε)g(ω+iε) exists if and only if the function −4 t → (t − ω) is σ-integrable. Analogously, the limits limε↓0 R(ω + iε)g(ω + iε) 2 −6 and limε↓0 R(ω + iε) g(ω + iε) exist if and only if the function t → (t − ω) is σ-integrable. If, in addition to the assumptions of Theorem 4.11, ω/∈ supp σ, then only the cases (e), (p2r)and(p3) can occur. With the reasoning at the end of Section 2, the behavior of the curve γ → α(γ) in a neighborhood of ω can be characterized by the structure of the root subspace of A(γ) at ω. Proposition 4.13. Let the assumptions of Theorem 4.11 be fulfilled with γ replaced by some γ1 ∈ (γ0,γ∞) such that ω := α(γ1) ∈/ supp σ.Ifby(rr), (cr), (rc),and (cc) we denote the properties of a curve in the neighborhood of some real point 156 P. Jonas and H. Langer as introduced at the end of Section 2(with ζ1 replaced by ω), then the following equivalences hold: (e) ⇐⇒ ω is of type (rr), (p2r)and[f1,g(ω)] < 0 ⇐⇒ ω is of type (cr), (p2r)and[f1,g(ω)] > 0 ⇐⇒ ω is of type (rc), (p3) ⇐⇒ ω is of type (cc). 4.6. Rank one perturbations of a nonnegative self-adjoint relation In the preceding considerations the starting point was a nonnegative relation A0 of regular defect one. After excluding some special cases a nonnegative self-adjoint extension A of A0 with ρ(A) = ∅ was fixed, and this self-adjoint extension played an essential role. This viewpoint may be changed by starting from a nonnegative self-adjoint relation A with ρ(A) = ∅ and a rank one perturbation B of A,either in resolvent sense or in the sense of generalized sums. In the first case, let A be a nonnegative self-adjoint relation A with ρ(A) = ∅ and let B be a self-adjoint relation with ρ(B) = ∅,B = A, such that for some (and hence for all) λ ∈ ρ(A) ∩ ρ(B) the operator (A − λ)−1 − (B − λ)−1 is of −1 −1 rank one, say (A − λ) −(B − λ) =[· ,h] h with some h, h ∈H. Setting g := I +(i − λ)(A − i)−1 h,thenA and B are self-adjoint extensions of the relation x x x x A0 := : ∈ A, [y − λx, h]=0 = : ∈ A, [y − ix, g]=0 . y y y y This relation A0 is nonnegative and of regular defect one. Hence the relation B belongs to the family of operators A(γ) as constructed above with the help of A0 and γ, and the results of this paper can be used in order to study the spectral properties of the relation B. To define rank one perturbations B of a nonnegative self-adjoint operator A with ρ(A) = ∅ in the sense of generalized sums we recall some definitions from [JL1]. Denote by J a fundamental symmetry of the Krein space H and by H 1 (A) 2 the linear space D(JA)1/2 provided with the graph norm 2 1 2 2 1/2 x 1 = x + (JA) / x ,x∈D(JA) . 2 Define the negative norm x 1 := sup |[x, y]| : y ∈H1 (A), y 1 ≤ 1 , − 2 2 2 and let H 1 (A) be the completion of H with respect to this norm. Then [ · , · ]can − 2 be extended by continuity to H 1 (A) ×H 1 (A). The operator A can be extended 2 − 2 −1 to a bounded operator A from H 1 (A)intoH 1 (A), and (A − z) ,z∈ ρ(A), 2 − 2 canbeextendedbycontinuitytoanisomorphismR(z)ofH 1 (A)ontoH 1 (A). − 2 2 If f ∈H− 1 (A)andα ∈ R we define the operator A α [ · , f]f as the restriction 2 of A + α [ · , f]f ∈L H 1 (A), H 1 (A) to those x ∈H1 (A)forwhichAx + 2 − 2 2 Self-adjoint Extensions of a Nonnegative Linear Relation 157 α [x, f]f ∈H. The operator A α [ · , f]f is self-adjoint and its resolvent set is not empty. Now we consider for some f ∈H 1 (A) the family of self-adjoint operators − 2 A[α] := A α [ · , f] f, α ∈ R. In particular, f may belong to H.Ifg := R(i)f,then[x, f]=[(A + i)x, g],x∈ D(A), and we set A0 := A|{x∈D(A): [(A+i)x,g]=0}. Starting from A and the defect element g, we define the self-adjoint extensions A(γ) of A0 as above. Then γ+ =[R(−i)AR(i)f,f] < ∞, and Theorem 4.10 implies that in this case γ∞ = γ+. It was shown in [JL1, 2.2] that −1 A[α] = A(γ), where γ = α + γ+,α∈ R. −1 For an expression of γ− − γ+ in terms of f see [JL1, Proposition 2.1]. References [ADo] Aronszajn, N., Donoghue, W.F.: On exponential representations of analytic functions in the upper half-plane with positive imaginary part, J. d’Analyse Math. 5 (1956-57), 321–388. [DaL1] Daho, K., Langer, H.: Matrix functions of the class Nκ,Math.Nachr.120 (1985), 275–294. [DaL2] Daho, K., Langer, H.: Sturm–Liouville operators with an indefinite weight func- tion: the periodic case, Radovi Matematiˇcki 2 (1986), 165–188. [DHdS] Derkach, V., Hassi, S., de Snoo, H.S.V.: Rank one perturbations in a Pontryagin space with one negative square, J. Funct. Anal. 188 (2002), 317–349. [DdS1] Dijksma, A., de Snoo, H.S.V.: Symmetric and selfadjoint relations in Krein spaces I, Operator Theory: Advances and Applications, vol. 24 (1987), Birk- h¨auser Verlag Basel, 145–166. [DdS2] Dijksma, A., de Snoo, H.S.V.: Symmetric and selfadjoint relations in Krein spacesII,Ann.Acad.Sci.Fenn.,Ser.A.I.Mathematica12 (1987), 199–216. [DLShZ] Dijksma, A., Langer, H., Shondin, Yu., Zeinstra, C.: Selfadjoint differential op- erators with inner singularities and Pontryagin spaces, Operator Theory: Adv. Appl., vol. 118,Birkh¨auser Verlag, Basel, 2000, 105–175. ∞ [DLSh] Dijksma, A., Luger, A., Shondin, Yu.: Minimal models for Nκ -functions, Op- erator Theory: Adv. Appl., vol. 163,Birkh¨auser Verlag, Basel, 2006, 97–134. [J1] Jonas, P.: A class of operator-valued meromorphic functions on the unit disc, Ann. Acad. Sci. Fenn. Ser. A. I: Mathematica 17 (1992), 257–284. [J2] Jonas, P.: Operator representations of definitizable functions, Ann. Acad. Sci. Fenn. Ser. A. I: Mathematica 25 (2000), 41–72. [J3] Jonas, P.: On locally definite operators in Krein spaces, in: Spectral Theory and Its Applications, Bucharest, Theta 2003, 95–127. 158 P. Jonas and H. Langer [JL1] Jonas, P., Langer, H.: Some questions in the perturbation theory of J- nonnegative operators in Krein spaces, Math. Nachr. 114 (1983), 205–226. [JL2] Jonas, P., Langer, H.: A model for π-self-adjoint operators and a special linear pencil, Integral Equations Operator Theory 8 (1985), 13–35. [JL3] Jonas, P., Langer, H.: Selfadjoint extensions of a closed linear relation of defect one in a Krein space, Operator Theory: Advances and Applications, vol. 80 (1995), Birkh¨auser Verlag Basel, 176–205. [KK] Kac, I.S., Krein, M.G.: R-functions - analytic functions mapping the upper half-plane into itself, Supplement I of the Russian translation of the book by F.V. Atkinson, Discrete and continuous boundary problems, Moscow 1968. Eng- lish translation: Amer. Math. Soc. Transl. (2) vol. 103 (1974), 1–18. [KL1] Krein, M.G., Langer, H.: On the spectral function of a selfadjoint operator in a space with an indefinite metric, Dokl. Akad. Nauk SSSR 152 (1963), 39–42 (Russian) [KL] Krein, M.G., Langer, H.: Some propositions on analytic matrix functions related to the theory of operators in the space Πκ, Acta Sci. Math. 43 (1981), 181–205. [L1] Langer, H.: Spektraltheorie linearer Operatoren in J-R¨aumen und einige An- wendungen auf die Schar L(λ)=λ2I + λB + C, Habilitationsschrift Technische Universit¨at Dresden, 1965. [L2] Langer, H.: Characterization of generalized zeros of negative type of functions of the class Nκ, Operator Theory: Advances and Applications, vol. 17 (1986), Birkh¨auser Verlag Basel, 202–212. P. Jonas Institut f¨ur Mathematik Technische Universit¨at Berlin Straße des 17. Juni 135 D-10623 Berlin, Germany e-mail: [email protected] H. Langer Institut f¨ur Analysis und Computational Mathematics Technische Universit¨at Wien Wiedner Hauptstrasse 8–10 A-1040 Vienna, Austria e-mail: [email protected] Operator Theory: Advances and Applications, Vol. 175, 159–168 c 2007 Birkh¨auser Verlag Basel/Switzerland Canonical Differential Equations of Hilbert-Schmidt Type Michael Kaltenb¨ack and Harald Woracek Abstract. A canonical system of differential equations, or Hamiltonian sys- tem, is a system of order two of the form Jy(x)=−zH(x)y(x), x ∈ R+.We characterize the property that the selfadjoint operators associated to a canon- ical system have resolvents of Hilbert-Schmidt type in terms of the Hamilton- ian H as well as in terms of the associated Titchmarsh-Weyl coefficient. Mathematics Subject Classification (2000). Primary 47B25, 47E05, 34B20; Sec- ondary 34A55, 34L05, 47A57. Keywords. Canonical differential equation, Hilbert-Schmidt. 1. Introduction A canonical system of differential equations is an equation of the form Jy (x)=−zH(x)y(x),x∈ R+, (1.1) where y(x)isaC2-valued function on R+, 0 −1 J = , 10 2×2 + 1 and H(x)isaR -valued function on R such that H(x) ≥ 0andH|(0,x] ∈ L for all x ∈ R+. The function H(x) is called the Hamiltonian corresponding to the canonical differential equation (1.1). We always assume that moreover tr(H(x)) = 1,x∈ R+. Canonical differential equations are usually studied with operator theoretic methods. A very good and detailed account on the operator model associated with the equation (1.1) can be found in [5]. Let us recall the basic notions: Denote by L2(H, R+) the Hilbert space of all measurable C2-valued functions f(x)onR+ such that +∞ f 2 = f(x)∗H(x)f(x)dx < +∞. 0 160 M. Kaltenb¨ack and H. Woracek 2 R+ 2 R+ ∈ One considers the closed subspace Ls(H, )ofL (H, ) consisting of all f L2(H, R+) such that H(x)f(x) is constant on H-indivisible intervals. Thereby an interval I ⊆ R+ is called H-indivisible if for some ϕ ∈ R cos ϕ H(x)=ξ ξT ,x∈ I a.e.,ξ = . ϕ ϕ ϕ sin ϕ Moreover one considers the linear relation { ∈ 2 R+ 2 − } Tmax,s = (f; g) Ls(H, ) : f is loc. abs. cont.,Jf = Hg , (1.2) and its restriction Tmin,s = {(f; g) ∈ Tmax,s : f(0+) = 0}, (1.3) which turns out to be a symmetric operator such that ∗ Tmin,s = Tmax,s. All the selfadjoint extensions of Tmin,s are given by A(ν)={(f; g) ∈ Tmax,s :sinνf1(0+) = cos νf2(0+)},ν∈ R , (1.4) and thereby we have A(ν1)=A(ν2) if and only if ν1 − ν2 ∈ πZ. For a detailed discussion of linear relations in Hilbert spaces see [2]. A fundamental notion in the theory of canonical systems is the Titchmarsh- Weyl coefficient associated with the equation (1.1). Let W (x,z)=(wij (x,z))i,j=1,2, x ∈ R+, z ∈ C,bethe2× 2-matrix-valued solution of the initial value problem dW(x, z) J = zW(x, z)H(x),x>0,W(0+,z)=I, dx and define w11(x, z)τ + w12(x, z) qH (z) = lim . (1.5) x→+∞ w21(x, z)τ + w22(x, z) This limit exists for z ∈ C \ R and does not depend on τ ∈ R. The function qH is called the Titchmarsh-Weyl coefficient of (1.1). It belongs to the Nevanlinna class N0, i.e., is holomorphic, satisfies qH (¯z)=qH (z),z∈ C \ R, and has the property that the kernel q (z) − q (w) L (w, z)= H H qH z − w¯ is positive semidefinite. The inverse spectral theorem, a deep result due to L. de Branges (see [1]), shows that (1.5) sets up a bijective correspondence between the set of all trace normed Hamiltonians and the Nevanlinna class N0. In a generalization to the indefinite setting one allows the Titchmarsh-Weyl coefficient q to belong to the generalized Nevanlinna class N<∞, i.e., allows the kernel Lq to have a finite number of negative squares, and asks for an equation of 2 R+ similar type as (1.1), or for an operator model similar to Ls(H, ), Tmax,s,such that again there is a bijective correspondence between N<∞ and the respective indefinite Hamiltonians. Canonical Differential Equations of Hilbert-Schmidt Type 161 It turns out that an indefinite Hamiltonian is composed out of a finite num- ber H1,...,Hn of positive Hamiltonians and a finite number of real parameters. Thereby the occurring Hamiltonians and parameters are subject to certain con- ditions. One among them is that selfadjoint extensions of the symmetry Tmin,s in 2 R+ Ls(Hi, ) have resolvents of Hilbert-Schmidt type. It is the aim of this note to characterize this property explicitly in terms of the Hamiltonian as well as in terms of the spectral measure of the associated Titchmarsh-Weyl coefficient. In §2 we prove our main result Theorem 2.4 which gives a necessary and sufficient integrability condition on H in order that selfad- joint extensions have resolvents of Hilbert-Schmidt type. To this end we reduce the problem to a specific boundary condition and apply classical criteria for integral operators which can be found, e.g., in [3]. It is the subject of §3toreformulate these conditions in terms of the spectral measure of the Titchmarsh-Weyl coeffi- cient qH , see Theorem 3.1. Thereby we use the theory of integral representations and operator models for Nevanlinna functions as developed, e.g., in [4]. In general our exposition relies on the material presented in [5] and the classical theory of integral operators, otherwise is fairly elementary. 2. Integral operators of Hilbert-Schmidt type In the subsequent lemma we are going to use a well-known criterion on whether an integral operator is of Hilbert-Schmidt type. Concerning this lemma note that the square root of H(t) exists because H(t) is assumed to be positive semidefinite. + Lemma 2.1. Let H(t)=(hij (t))i,j=1,2 be a trace normed Hamiltonian on R such that +∞ h11(t)dt < +∞. (2.1) 0 Consider the space L2(R+)2 of all C2-valued functions on R+ which are square integrable with respect to the Lebesgue measure, and consider the kernel 1 0 −χ{ } 1 K(x, y)=H(x) 2 y +∞ (Cg)(x)= K(x, y)g(y)dy (2.3) 0 with g ∈ dom(C) if the integral exists for almost every x and Cg ∈ L2(R+)2. Moreover, we set x M(x)=(mij (x))i,j=1,2 = H(t)dt. 0 162 M. Kaltenb¨ack and H. Woracek Then C is a continuous, everywhere defined operator which belongs to the Hilbert- Schmidt class if and only if +∞ 2 m22(t)h11(t)dt < +∞. (2.4) 0 In this case the Hilbert-Schmidt norm of C coincides with the square root of this number. Proof. It is well known (see for example [3]) that C is a continuous, everywhere defined operator which belongs to the Hilbert-Schmidt class if and only if +∞ +∞ tr(K(x, y)∗K(x, y))dxdy < +∞. 0 0 2 In this case C 2 coincides with the value of integral. We calculate K(x, y)∗K(x, y) 1 0 −χ{ } 0 −χ{ } 1 = H(y) 2 y>x H(x) y 1 Set (k (y)) =1 2 = H(y) 2 , and note that k12 = k21. We then rewrite (2.5) as ij i,j , k11(y)k11(y) k11(y)k12(y) χ{y>x}h22(x) k11(y)k12(y) k12(y)k12(y) k12(y)k12(y) k12(y)k22(y) +χ{y ∈ ∈ 2 R+ with f dom(B) if f Ls(H, ) such that (Bf)(x) exists for almost every ∈ R+ ∈ 2 R+ x and Bf Ls(H, ). Then B is a continuous, everywhere defined operator which belongs to the Hilbert-Schmidt class if and only if (2.4) holds true. Proof. The integral operator B can be written as +∞ (Bf)(x)= k(x, y)f(y)dy, 0 where k(x, y)isthekernel χ 0 k(x, y)= {y 1 φ(f)(t)=H(t) 2 f(t). It is straightforward to show that φ is an isometry. Let K(x, y) be as in (2.2) and the operator C defined as in (2.3). By Lemma 2.1 the operator C is a continuous, everywhere defined operator which belongs to the Hilbert-Schmidt class if and only if (2.4) holds true. We have Cφ = φB.Infact,φB ⊆ Cφ is obvious from the definitions of the respective kernels. For the other inclusion let f ∈ dom(Cφ). From (2.2) we see 1 2 + that C(φ(f))(x)isoftheformH(x) 2 g(x)forg ∈ L (H, R )with +∞ g(x)= k(x, y)f(y)dy 0 +∞ x 00 = JH(t)f(t)dt − JH(t)f(t)dt. 0 01 0 ∈ 2 R+ It remains to show that g Ls(H, ). So let (a, b) be an indivisible interval of type ϕ ∈ R and calculate for x ∈ (a, b) d ξT g(x)=ξT JH(x)f(x)=ξT Jξ ξT f(x)=0. dx ϕ ϕ ϕ ϕ ϕ ∈ 2 R+ Hence g Ls(H, ), and we proved that Cφ = φB. ∈ 2 R+ 2 2 R+ If g L ( ) φ(Ls(H, )), then & ' x +∞ T 1 1 0 1 1 (Cg)(x)=H(x) 2 JH(y) 2 g(y)dy − H(y) 2 g(y)dy . 0 1 0 0 The second integral vanishes because 1 1 2 + H(y) 2 ∈ φ(L (H, R )). 0 s 164 M. Kaltenb¨ack and H. Woracek If x is contained in an indivisible interval (a, b)oftypeϕ ∈ R, we assume that (a, b) is maximal, i.e., (a, b) is not contained in a larger indivisible interval. For a The previous lemma is of importance if we consider the operator theoret- ical background of canonical differential equations. Using the notation from the introduction we obtain the subsequent result. + Corollary 2.3. Let H(t)=(hij (t))i,j=1,2 be a trace normed Hamiltonian on R π π such that (2.1) holds true. Then the selfadjoint extensions A(ν),ν∈ (− 2 , 2 ] of Tmin,s have a resolvent (A(ν) − z)−1,z∈ ρ(A(ν)) which is of Hilbert-Schmidt type if and only if (2.4) holds. Proof. Because of the resolvent identity and since the product of a bounded op- erator and a Hilbert-Schmidt operator is of Hilbert-Schmidt type, the selfadjoint extension A(ν)ofTmin,s has a resolvent which is of Hilbert-Schmidt type if and only if (A(ν) − z)−1 is of Hilbert-Schmidt type for one z ∈ ρ(A(ν)). Moreover, by Krein’s formula (see Proposition 4.4 in [5]) the resolvents of different selfadjoint extensions are one-dimensional perturbations of each other. π π Thus, the resolvents of A(ν),ν∈ (− 2 , 2 ] being of Hilbert-Schmidt type is equiv- alent to the fact that π −1 A − z (2.6) 2 π is a Hilbert-Schmidt operator for some z ∈ ρ(A( 2 )). Note that by our assumption (2.1) 1 span =kerTmax , (2.7) 0 ,s π π −1 and hence ker A( 2 )={0}. This means that A( 2 ) is a densely defined selfadjoint 2 R+ ∈ π operator on Ls(H, ). Moreover, as the elements f dom A( 2 ) vanish at 0 in π −1 the upper entry we see that B ⊆ A( 2 ) ,whereB is defined as in Lemma 2.2. If (2.4) is satisfied, then B is an everywhere defined, bounded operator be- π −1 π longing to the Hilbert-Schmidt class. We conclude B = A( 2 ) ,and0∈ ρ(A( 2 )). Canonical Differential Equations of Hilbert-Schmidt Type 165 For the converse direction assume that (2.6) is of Hilbert-Schmidt type. We π −1 are going to show that A( 2 ) is an everywhere defined, bounded operator be- longing to the Hilbert-Schmidt class. If z = 0, we are done. Otherwise, besides zero π the spectrum of (2.6) consists of eigenvalues. Since ker A( 2 )={0},thenumber −z−1 cannot be an eigenvalue of (2.6). Because of −1 π −1 1 π −1 1 π −1 A = A − z + A − z , (2.8) 2 z 2 z 2 π −1 A( 2 ) is an everywhere defined, bounded operator belonging to the Hilbert- Schmidt class. ∈ 2 R+ π −1 If g Ls(H, )andf = A( 2 ) g,thenf = JHg. It follows from Lemma 7.8, [5] and from (2.7) that f(x) tends to zero in the lower component for x → +∞. Moreover, f(0+) is zero in the upper component. Thus Bg exists and coincides π −1 π −1 with f = A( 2 ) g, and we proved that B = A( 2 ) . By Lemma 2.2 condition (2.4) holds true. The above assertions yield a criterion for a canonical differential equation to have a compact resolvent of Hilbert-Schmidt type. Put cos ϕ ξ := . ϕ sin ϕ + Theorem 2.4. Let H(t)=(hij (t))i,j=1,2 be a trace normed Hamiltonian on R . π π Then the selfadjoint extensions A(ν),ν∈ (− 2 , 2 ] of Tmin,s have a resolvent (A(ν) − z)−1,z∈ ρ(A(ν)), which is of Hilbert-Schmidt type if and only if there exists a real ϕ such that +∞ T ∞ ξϕ H(t)ξϕdt < + , (2.9) 0 and +∞ T T π + π ∞ ξϕ+ M(t)ξϕ 2 ξϕ H(t)ξϕdt < + . (2.10) 0 2 Here M(x) is defined by x M(x)=(mij (x))i,j=1,2 = H(t)dt. 0 Proof. First we consider a transformation of the Hamiltonian H(t). For µ ∈ R set sin µ − cos µ N = , µ cos µ sin µ ∗ → ∗ and Hµ(t)=NµH(t)Nµ (see [5]). It is straightforward to verify that ψ : f Nµf 2 R+ 2 R+ is an isomorphism from Ls(H, )ontoLs(Hµ, ). 166 M. Kaltenb¨ack and H. Woracek µ µ If we denote by Tmin,s (Tmax,s) the respective differential operators on the 2 R+ µ µ space Ls(Hµ, )andbyA (ν) the corresponding selfadjoint extensions of Tmin,s, then µ µ ψTmin,s = Tmin,sψ, ψTmax,s = Tmax,sψ, µ π and ψA(ν)=A (ν +µ− 2 )ψ.Thus,theA(ν)’s have resolvents of Hilbert-Schmidt type if and only if the Aµ(ν)’s do so. Condition (2.9) for the given H(t)isequivalenttocondition(2.1)whenh11(t) π now denotes the left upper entry of Hµ(t)withµ = 2 − ϕ. Similarly condition (2.10) for H(t)isequivalenttocondition(2.4)whenh11(t),h22(t)(m11(t),m22(t)) π now denote the diagonal entries of Hµ(t) (the primitive of Hµ(t)) with µ = 2 − ϕ. If there exists a real ϕ such that (2.9) and (2.10) hold, then by Corollary µ π 2.3 the resolvents of the A (ν)’s are Hilbert-Schmidt type for µ = 2 − ϕ.As we mentioned above this is equivalent to the A(ν)’s having resolvents of Hilbert- Schmidt type. π π Conversely, assume that A(ν),ν∈ (− 2 , 2 ], have resolvents of Hilbert- ∈ R ∈ 2 R+ Schmidt type. If there exists a ϕ such that ξϕ Ls(H, ), then we set π µ = 2 − ϕ and see that (2.9) holds. We just mentioned that this is the same as saying that (2.1) holds true when h11(t) denotes the left upper entry of Hµ(t). Since also the Aµ(ν)’s have resolvents of Hilbert-Schmidt type, Corollary 2.3 shows that (2.4) holds for Hµ(t) which in turn yields (2.10). It remains to exclude the case that none of the constant functions ξϕ belong to 2 R+ Ls(H, ). In fact, if we were in this situation, then all of the selfadjoint relations A(ν) would have a trivial kernel because the scalar multiples of ξϕ are the only 2 R+ possible candidates for elements of Ls(H, )tobelongtokerA(ϕ). −1 π Thus all A(ν) are operators, and by (2.8) with A( 2 ) replaced by A(ν) obtain that the operators A(ν)−1 are everywhere defined, bounded and of Hil- −1 bert-Schmidt type. Hence the closed one-dimensional restriction (Tmin,s) is a bounded, symmetric operator with a closed domain of co-dimension one. It is easy to check that −1 −1 ⊥ D =(Tmin,s) ⊕ ({0}×dom((Tmin,s) ) ) −1 is a selfadjoint extension of (Tmin,s) . Its inverse would then be a selfadjoint extension of Tmin,s, and would, therefore, coincide with A(ν)forsomeν.Butthis contradicts the fact that D−1 has a non-trivial kernel. 3. Titchmarsh-Weyl coefficients and the Hilbert-Schmidt property Recall that a Nevanlinna function has a unique integral representation 1 − t a + bz + 2 dσ(t), (3.1) R t − z t +1 where a, b ∈ R,b≥ 0andσ is a non-negative Borel measure on R such that 1 ∞ 2 dσ(t) < + . R t +1 Canonical Differential Equations of Hilbert-Schmidt Type 167 + Theorem 3.1. Let H(t)=(hij (t))i,j=1,2 be a trace normed Hamiltonian on R ,and let qH (z) be the Titchmarsh-Weyl coefficient of the canonical differential equation (1.1). Assume that (3.1) is the integral representation of qH (z). Then H(x) satisfies (2.9) and (2.10) if and only of σ is a discrete measure - σ = αnδtn , (3.2) n such that - 1 < +∞. (3.3) t2 n n Proof. Corollary 4.2, [5] shows that Tmin,s is a completely non-selfadjoint symmet- ric relation with defect index (1, 1), and by Theorem 4.3, [5] the Titchmarsh-Weyl π coefficient qH (z)istheQ-function of A( 2 )andTmin,s. 2 R+ π Thus according to Theorem 2.5, [4] the triplet (Ls(H, ),Tmin,s,A( 2 )) is unitarily equivalent to (H,S,A), where H = L2(σ), A = {(ϕ(t); tϕ(t)) : ϕ(t),tϕ(t) ∈ L2(σ)}, S = {(ϕ(t); tϕ(t)) : ϕ(t),tϕ(t) ∈ L2(σ), ϕ(t)dσ(t)=0}, R in the case that b = 0 in the representation (3.1) of qH (z)and H = L2(σ) ⊕ C, ϕ(t) tϕ(t) A = ; : ϕ(t),tϕ(t) ∈ L2(σ),c∈ C , 0 c ϕ(t) tϕ(t) S = ; : ϕ(t),tϕ(t) ∈ L2(σ),bc+ ϕ(t)dσ(t)=0 , 0 c R π −1 otherwise. We conclude that A( 2 − z) is of Hilbert-Schmidt type if and only if the operator ϕ(t) ϕ(t) → , t − z on L2(σ) is of Hilbert-Schmidt type. This in turn is equivalent to the fact that σ is a discrete measure (3.2) such that (3.3) holds. References [1] L. de Branges, Hilbert spaces of entire functions Prentice-Hall, London 1968. [2] A. Dijksma, H. de Snoo, Self-adjoint extensions of symmetric subspaces Pacific J. Math. 54 (1974), 71–100. [3] I. Gohberg, S. Goldberg, M.A. Kaashoek, Classes of linear operators. Vol. I Operator Theory: Advances and Applications, 49, Birkh¨auser Verlag, Basel, 1990. 168 M. Kaltenb¨ack and H. Woracek [4] S. Hassi, H. Langer, H. de Snoo, Selfadjoint extensions for a class of symmetric operators with defect numbers (1, 1) Topics in operator theory, operator algebras and applications (Timi¸soara, 1994), 115–145, Rom. Acad., Bucharest, 1995. [5] S. Hassi, H. de Snoo, H. Winkler, Boundary-value problems for two-dimensional canonical systems Integral Equations Operator Theory 36 (2000), no. 4, 445–479. Michael Kaltenb¨ack and Harald Woracek Institut f¨ur Analysis und Scientific Computing Technische Universit¨at Wien Wiedner Hauptstraße 8-10 A 1040 Wien, Austria e-mail: [email protected] e-mail: [email protected] Operator Theory: Advances and Applications, Vol. 175, 169–191 c 2007 Birkh¨auser Verlag Basel/Switzerland Spectral Analysis of Differential Operators with Indefinite Weights and a Local Point Interaction Ilia Karabash and Aleksey Kostenko Abstract. We consider quasi-self-adjoint extensions of the symmetric operator A − x d2 A {f ∈ W 2 R f f } = (sgn ) dx2 ,dom( )= 2 ( ): (0) = (0) = 0 , in the Hilbert space L2(R). The main result is a criterion of similarity to a normal operator for operators of this class. The spectra and resolvents of these extensions are described. As an application we describe the main spectral properties of the x − d2 cδ x − d2 cδ operators (sgn ) dx2 + and (sgn ) dx2 + . Mathematics Subject Classification (2000). Primary 47A45; Secondary 47B50. Keywords. Symmetric operator, quasi-self-adjoint extensions, similarity prob- lem, boundary triplets, Weyl functions. Introduction Consider the symmetric operator A in the Hilbert space L2(R) defined by 2 dom(A)={f ∈ W2 (R):f(0) = f (0) = 0}, (Af)(x)=−(sgn x)f (x)forf ∈ dom A. (0.1) The object of investigation is the similarity of quasi-self-adjoint extensions of A (see [1]) to a normal operator. Let us recall that two operators T1 and T2 in a Hilbert space H are called similar if there exists a bounded operator C with −1 −1 bounded inverse C such that T1 = C T2C. Spectral problems (Ly)(x)=λr(x)y(x), (0.2) where L is an elliptic operator and the function r(x) change sign, occur in certain physical models (see [4] and references therein). The question whether the system of eigenfunctions of the problem (0.2) forms a Riesz basis was studied in [3], [4], 170 I.M. Karabash and A.S. Kostenko 1 [19], [32] [33], [34] (see also references in [34]). If the operator r L has a nonempty 1 continuous spectrum, then the corresponding problem is the similarity of r L to a self-adjoint (normal) operator. In [9], [6], [15], [7], [8], [16] the Krein–Langer spectral theory of definitizable operators (see [28]) was applied to similarity problems for quasi J-nonnegative 1 ´ operators (see [15]) of the form r L.Inparticular,B.CurgusandB.Najman[7] showed that the operator d2 A˜ = −(sgn x) , dom(A˜)=W 2(R), (0.3) dx2 2 is similar to a self-adjoint one. This result was proved by another method in [21]; the method is based on the Naboko–Malamud criterion of similarity to a self-adjoint operator [31], [29] (see also [5]). One more proof is presented in [20]. In the recent papers [22], [13], [14], [24] the Naboko-Malamud criterion was applied to different J-self-adjoint differential operators. Differential operators with an indefinite weight are of interest from one more · 1 point of view. The characteristic function W ( ) of the operator r L as well as the ∗ corresponding J-form J − W JW is unbounded in C+. Therefore known sufficient conditions of similarity to a self-adjoint operator cannot be applied here (see [30], [20] and bibliography therein). In the present paper we describe quasi-self-adjoint extensions AB of the sym- metric operator A in terms of boundary triplets (see [18], [11]). In Sections 3–4 we formulate a criterion of similarity of AB to a normal (self-adjoint) operator. In order to illustrate these results in Section 5 we obtain simple similarity criteria for operators with local point interactions at zero 2 2 d d A˜1 := sgn x − + c1δ ,c1 ∈ C, A˜2 := sgn x − + c2δ ,c2 ∈ C. dx2 dx2 (See definitions of the operators A˜1, A˜2 in [2] and also in Section 5 of the present paper.) The results of the paper were announced in [23]. Notation: By H, H we denote separable Hilbert spaces. The set of all bounded linear operators from H to H is denoted by [H, H]or[H]ifH = H. C(H) stands for the set of closed densely defined operators in H.LetT be a linear operator in a Hilbert space H. In what follows dom(T ), ker(T ), ran(T ) are the domain, kernel, range of T , respectively. We denote by σ(T ), σr(T ), σc(T ) the point, residual and continuous spectra of T .Byσp(T )thesetofeigenvaluesofT is indicated. −1 We denote the resolvent set by ρ(T ); RT (λ):=(T − λI) , λ ∈ ρ(T ), is the { ∈ \ − H} resolvent of T . Recall that σr(T )= λ σ(T ) σp(T ):ran(T λI) = , σc(T )=σ(T ) \ (σp(T ) σr(T )). We set C± := {λ ∈ C : ± Im λ>0}, R+ := (0, +∞), R− := (−∞, 0). By χI(t) we denote the characteristic function of the interval I, i.e., χI (t)=1for t ∈I, χI(t)=0fort ∈I. Finally, we set χ±(t):=χR± (t). Differential Operators with Indefinite Weights and a Point Interaction 171 1. Preliminaries 1.1. A similarity criterion Our approach is based on the concept of boundary triplets (see [18], [11]) and the resolvent similarity criterion obtained by S.N. Naboko [31] and M.M. Malamud [29] (in [5] this criterion was obtained under an additional assumption). Theorem 1.1 ([29, 31]). AclosedoperatorT in a Hilbert space H is similar to a self-adjoint one if and only if σ(A) ⊂ R and for all f ∈ H the inequalities +∞ 2 2 sup ε RT (µ + iε) f dµ ≤ C f , ε>0 −∞ (1.1) +∞ 2 ∗ 2 sup ε RT ∗ (µ + iε) f dµ ≤ C f , ε>0 −∞ are valid with constants C and C∗ independent of f. 1.2. Linear relations Definition 1.1. (i) A closed linear relation Θ in H is a closed subspace Θ of H⊕H. (ii) The closed linear relation Θ is symmetric if for all {f1,g1}, {f2,g2}∈Θ the condition (g1,f2) − (f1,g2)=0, (1.2) is satisfied. (iii) The closed linear relation Θ is self-adjoint if it is maximal symmetric, i.e., Θ is symmetric and there does not exist a closed symmetric relation Θ˜ such that Θ is properly contained in Θ˜ . Let us illustrate closed linear relations by simple examples. Example 1.1. (i) Let B beaclosedoperatorinH, not necessarily bounded. Then the graph G(B) of B isaclosedrelationinH. Moreover, if B = B∗ is a self-adjoint operator, then G(B) is a self-adjoint relation in H. (ii) The subspaces Θ0 := {0}×H, Θ1 := H×{0} of H×H are self-adjoint relations in H. Obviously, Θ0 is not the graph of any operator. 1.3. Boundary triplets Let A ∈C(H) be a closed symmetric operator with equal deficiency indices n+(A)= ∗ n−(A)(n±(T ):=dimN±i and by Nλ := ker(T − λ) the deficiency subspaces of A are indicated). Without loss of generality we can assume that A is simple. This means that A has no self-adjoint parts. 172 I.M. Karabash and A.S. Kostenko Definition 1.2 ([1]). (i) A closed extension A˜ of A is called a proper extension if A ⊂ A˜ ⊂ A∗.The set of all proper extensions is denoted by ExtA. (ii) AproperextensionA˜ is called a quasi-self-adjoint if dim(dom(A˜)/ dom(A)) = n±(A). (1.3) We recall the definition of a boundary triplet which may be considered as an abstract version of the second Green formula. Definition 1.3 ([18]). AtripletΠ={H, Γ0, Γ1} consisting of an auxiliary Hilbert space H and linear mappings ∗ Γj :dom(A ) −→ H ,j∈{0, 1}, (1.4) is called a boundary triplet for the adjoint operator A∗ of A if the following two conditions are satisfied: (i) The second Green formula ∗ ∗ ∗ (A f,g) − (f,A g)=(Γ1f,Γ0g)H − (Γ0f,Γ1g)H,f,g∈ dom(A ), (1.5) takes place and (ii) the mapping ∗ Γ:dom(A ) −→ H ⊕ H , Γf := {Γ0f,Γ1f}, (1.6) is surjective. The above definition allows one to describe the set ExtA in the following way (see [10, 11]). ∗ Proposition 1.1 ([10, 11]). Let Π={H, Γ0, Γ1} be a boundary triplet for A .Then the mapping Γ establishes a bijective correspondence A˜ → Θ:=Γ(dom(A˜)) between the set ExtA and the set of closed linear relations in H. By Proposition 1.1 the following definition is natural. ∗ Definition 1.4. Let Π={H, Γ0, Γ1} be a boundary triplet for the operator A . (i) Denote AΘ = A˜ if Θ=Γ(dom(A˜)), that is ∗ ∗ AΘ := A |DΘ, where DΘ := {f ∈ dom(A ):{Γ0f,Γ1f}∈Θ}. (1.7) (ii) If Θ=G(B) is the graph of B ∈C(H),thendom(AΘ) is determined by the equation dom(AB )=DB := DΘ =ker(Γ1 − BΓ0). We set AB := AΘ. Let us make the following remarks. Remark 1.1. 1) The deficiency indices n±(A) are equal to the dimension of H, i.e., dim(H)= n±(A). ∗ 2) There exist two self-adjoint extensions Aj := A | ker(Γj) which are naturally ∈ associated to a boundary triplet. According to Definition 1.4 Aj = AΘj ,j {0, 1},whereΘ0 = {0}×H, Θ1 = H×{0}. Conversely, if A0 is a self-adjoint Differential Operators with Indefinite Weights and a Point Interaction 173 extension of A, then there exists a boundary triplet Π={H, Γ0, Γ1} such that ∗ A0 = A | ker(Γ0). 3) Θ is the graph of an operator B ∈C(H) iff A˜ and A0 are disjoint, i.e., dom(A˜) ∩ dom(A0)=dom(A). 4) Θ = G(B) with B ∈ [H] iff A˜ and A0 are transversal, i.e., A˜ and A0 are ∗ disjoint and dom(A˜)+dom(A0)=dom(A ). Definition 1.5 ([12]). The proper extension A˜ ∈ ExtA is called almost solvable if there exists a boundary triplet Π={H, Γ0, Γ1} and an operator B ∈ [H] such that dom(A˜)=dom(AB ):=ker(Γ1 − BΓ0). (1.8) The set of almost solvable extensions is denoted by AsA. Note that the class AsA is sufficiently wide. Proper extensions having two regular points λ1,λ2 ∈ C such that Im λ1 · Im λ2 < 0belongtoAsA. All quasi-self-adjoint extensions are in AsA if n±(A) < ∞. 1.4. Weyl functions It is well known that Weyl functions are an important tool in the direct and inverse spectral theory of singular Sturm–Liouville operators. In [10, 11] the concept of Weyl function was generalized to an arbitrary symmetric operator A with infinite deficiency indices n+(A)=n−(A). In this subsection we recall basic facts about Weyl functions. Definition 1.6 ([10, 11]). Let Π={H, Γ0, Γ1} be a boundary triplet for the operator ∗ A . The Weyl function of A corresponding to the boundary triplet {H, Γ0, Γ1} is a unique mapping M(·):ρ(A0) −→ [H] (1.9) satisfying Γ1fλ = M(λ)Γ0fλ for all fλ ∈ Nλ,λ∈ ρ(A0), (1.10) ∗ where Nλ := ker(A − λI). It is well known (see [10, 11]) that the above implicit definition of the Weyl function is correct and M(·)isanR-function obeying 0 ∈ ρ(Im(M(i))). The Weyl function immediately provides some information about the “spectral properties” of proper extensions. We confine ourselves to the case of almost solvable extensions of the symmetric operator A. Proposition 1.2 ([11, 12]). Suppose that Π={H, Γ0, Γ1} is a boundary triplet for ∗ A ,M(·) is the corresponding Weyl function, λ ∈ ρ(A0) and B ∈ [H].Then: 1) λ ∈ ρ(AB) if and only if 0 ∈ ρ(B − M(λ)); 2) λ ∈ σi(AB ) if and only if 0 ∈ σi(B − M(λ)),i∈{p, r, c}. 174 I.M. Karabash and A.S. Kostenko 1.5. γ-fields With each boundary triplet we can associate a so-called γ-field. ∗ Definition 1.7 ([11]). Let Π={H, Γ0, Γ1} be a boundary triplet for A .Theγ-field γ(·) corresponding to Π is defined by −1 γ(λ):=(Γ0|Nλ) : H−→Nλ,λ∈ ρ(A0). (1.11) One can easily check that −1 γ(λ)=(A0 − λ0)(A0 − λ) γ(λ0),λ,λ0 ∈ ρ(A0), (1.12) and consequently γ(·)isaγ-field in the sense of [26]. It is shown in [11] that the γ-field γ(·) and the Weyl function M(·) are related by ∗ ∗ M(λ) − M(λ0) =(λ − λ¯0)γ(λ0) γ(λ),λ,λ0 ∈ ρ(A0). (1.13) The relation (1.13) means the M(·)isaQ-function in the sense of [26]. The following version of the Krein-Naimark formula for canonical resolvents (see for instance [26]) is based on the notion of boundary triplets. Theorem 1.2 ([10, 11]). Let A˜ be an almost solvable extension of A (A˜ ∈AsA), i.e., A˜ = AB with B ∈ [H] for some boundary triplet Π={H, Γ0, Γ1}.Then −1 −1 −1 ∗ (AB − λ) =(A0 − λ) + γ(λ)(B − M(λ)) γ (λ),λ∈ ρ(AB). (1.14) Here M(·) and γ(·) are the Weyl function and γ-field corresponding to the triplet Π. 2. Extensions of the minimal operator 2.1. Boundary conditions Consider the operator A of the form (0.1). It is obvious that A is a closed simple symmetric operator√ with deficiency indices n±(A)=2. We denote by z the branch of the multifunction√ on the complex plane C with the cut along R−, singled out by the condition −1+i0=i. Theorem 2.1. (i) The adjoint operator A∗ has the form 2 ∗ d ∗ 2 2 2 A = −(sgn x) , dom(A )=W (R \{0}):=W (R−) ⊕ W (R+). (2.1) dx2 2 2 2 2 2 (ii) Let mappings Γ : W2 (R \{0}) → C , j = {0, 1}, be given by j f(+0) f (+0) Γ0f = , Γ1f = . (2.2) f (−0) −f(−0) 2 ∗ Then Π={C , Γ0, Γ1} is a boundary triplet for A . (iii) The corresponding Weyl function M(·) is √ − −λ 0 M(λ):= √ ,λ∈ C \ R. (2.3) 0 −1/ λ Differential Operators with Indefinite Weights and a Point Interaction 175 2 (iv) The corresponding γ-field γ(λ):C → Nλ is √ √ c+ − −λx c− λx γ(λ) := c+ · e χ+(x)+√ · e χ−(x),c± ∈ C. (2.4) c− λ Proof. The first statement is obvious. Moreover, we have (A∗f, g) − (f, A∗g)=f (+0)g(+0) + f (−0)g(−0)− − f(+0)g (+0) − f(−0)g (−0),f,g∈ dom(A∗). (2.5) Hence (ii) follows from Definition 1.3. Note that √ √ − −λx λx Nλ = {fλ(x):=c+ · e χ+(x)+c− · e χ−(x):c± ∈ C}. (2.6) Combining (2.2) and (2.6), one gets √ c√+ −c+ −λ Γ0fλ = , Γ1fλ = . (2.7) c− λ −c− By Definitions 1.6 and 1.7, we easily obtain (2.3) and (2.4). Let us introduce the following boundary conditions at zero a11f(−0) + a12f (−0) + a13f(+0) + a14f (+0) = 0 ,aij ∈ C. (2.8) a21f(−0) + a22f (−0) + a23f(+0) + a24f (+0) = 0 By Definition 1.2, a quasi-self-adjoint extension A˜ of the operator A has the form ˜ ∗| A = A(aij ) = A dom(A(aij )), { ∈ 2 R \{ } } dom(A(aij ))= f W2 ( 0 ): f satisfies conditions (2.8) , (2.9) with the matrix a11 a12 a13 a14 (aij ):= a21 a22 a23 a24 such that rank(aij )=n±(A)=2. (2.10) Consider three cases. 1) Let a11 a12 00 a13 a14 00 = or = . a21 a22 00 a23 a24 00 2 Then A˜ = A− ⊕ A+,whereA± is an operator in L (R±). By condition (2.10), we see that one of the operators A−, A+ is a symmetric with deficiency indices (1,1) and another one is an adjoint to a symmetric operator with deficiency indices (1,1). Hence C \ R ⊂ σp(A˜)andA˜ is not similar to a normal operator. 176 I.M. Karabash and A.S. Kostenko a11 a12 a13 a14 2) Suppose that and have a zero column. Since a21 a22 a23 a24 rank(aij ) = 2, it follows that A˜ = A− ⊕ A+,whereA+ and A− are self-adjoint 2 2 ∗ operators in L (R+)andL (R−), respectively. Thus A˜ = A˜ . 3) Suppose that there are three nonzero columns in (aij ). In this case one of the determinants a11 a14 a12 a13 ∆1 = , ∆2 = , a21 a24 a22 a23 (2.11) a12 a14 a11 a13 ∆3 = , ∆4 = a22 a24 a21 a23 does not vanish. Evidently, only the case (3) is of interest to us. 2.2. The case ∆1 =0 Let ∆1 = 0. (The cases ∆2 =0,∆ 3 = 0, and ∆4 = 0 will be considered in Section 4.) Then conditions (2.8) take the form f (+0) = b11f(+0) + b12f (−0) (2.12) −f(−0) = b21f(+0) + b22f (−0). ∗| − Hence, by Definition 1.5, A(aij ) = AB = A ker(Γ1 BΓ0). Here b11 b12 B = ∈ C2×2 b21 b22 2 and the boundary triplet Π = {C , Γ0, Γ1} is of the form (2.2). In what follows AB stands for the operator d2 A := − sgn x , dom(A )={f ∈ W 2(R \{0}):f satisfies (2.12)}. (2.13) B dx2 B 2 For B ∈ C2×2 and the Weyl function M(·) of the form (2.3) we define the function ϕB (·):C \ R → C by ϕB(λ) := det(B − M(λ)),λ∈ C \ R. (2.14) Lemma 2.1. Suppose AB is the operator of the form (2.13) and |b12| + |b21| =0 ; then: (i) σc(AB )=R; (ii) σr(AB )=∅; (iii) σp(AB)={λ ∈ C \ R : ϕB(λ)=0} = {λ ∈ C+ : ϕB(λ)=0}∪{λ ∈ C− : ϕB∗ (λ)=0}. Proof. Simple calculations show that there are no eigenvalues on the real axis and σc(AB )=R. The second and the third statements evidently follow from Proposition 1.2. Differential Operators with Indefinite Weights and a Point Interaction 177 Lemma 2.2. Let Π={H, Γ0, Γ1} be a boundary triplet of the form (2.2).Then ⎛ ⎞ +0∞ √ − −λt ⎜ f(t)e dt ⎟ ∗ ⎜ 0 ⎟ ∈ 2 R γ (λ)f = ⎝ 00 √ ⎠ ,fL ( ). (2.15) √1 f(t)e λtdt λ −∞ Proof. By Theorem 2.1.(iv), γ∗(·)isamapfromL2(R)toC2. To find γ∗(·)weuse the equation ∗ c1 2 2 (γ(λ)c, f)L2(R) =(c, γ (λ)f)C2 ,c= ∈ C ,f∈ L (R). (2.16) c2 Combining (2.4) and (2.16), one gets +∞ 0 √ √ − −λt c2 λt ∗ ∗ c1 · f(t)e dt+ √ · f(t)e dt = c1 ·(γ (λ)f)1 +c2 ·(γ (λ)f)2. (2.17) λ 0 −∞ Hence (2.15) immediately follows from (2.17). Let us denote +∞ 0 √ √ − −λt λt y+(f,λ):= f(t)e dt, y−(f,λ):= f(t)e dt, λ ∈ C \ R. 0 −∞ (2.18) ∗ Lemma 2.3. Let the operator AB be of the form (2.13) and A0 = A | ker Γ0.Then −1 −1 (AB − λ) f − (A0 − λ) f (x) √ − −λx e · χ+(x) 1 b12 = b22 + √ y+(f,λ) − √ · y−(f,λ) ϕB (λ) λ λ √ √ λx e · χ−(x) b11 + −λ + √ √ · y−(f,λ) − b21 · y+(f,λ) , λ · ϕB(λ) λ 2 f ∈ L (R),λ∈ ρ(AB). (2.19) Here ϕB(·) and y±(f,·) are given by (2.14) and (2.18), respectively. Proof. By (2.3), for λ ∈ ρ(AB ) (see Lemma 2.1) one obtains √ −1 b11 + −λb12 (B − M(λ))−1 = √ b21 b22 +1/ λ √ (2.20) 1 b22 +1/ λ −b12 = √ . ϕB(λ) −b21 b11 + −λ Combining (2.4), (2.15), (2.20) with formula (1.14), we get (2.19). 178 I.M. Karabash and A.S. Kostenko 3. Similarity to a normal operator 3.1. The main result ∈ C2×2 + C → C For each B let us define the function ϕB : + in the following way. We set + − ∈ C ϕB(λ):=ϕB (λ)=det(B M(λ)) for λ +, (3.1) ∈ R + C and for x by ϕB (x) we denote the boundary values of ϕB(λ)in +, ϕ+(x) := lim det(B − M(z)),x∈ R ∪{∞}. (3.2) B z→x z∈C+ + C C \{ } Note that the function ϕB is analytic on + and continuous on + 0 . The following similarity criterion is the main result of the paper. Theorem 3.1 (Main Theorem). Assume that ∆1 =0 and the operator AB is defined + + | | | | by (2.13).LetϕB and ϕB∗ be the functions defined in (3.1)–(3.2) and b12 + b21 = 0.ThenAB is similar to a normal operator if and only if the following conditions hold: + + R ∪{∞} (i) ϕB and ϕB∗ have no zeroes in the set ; + + C (ii) ϕB and ϕB∗ have no zeroes of the second order in +. Remark 3.1. Suppose that |b12| + |b21| =0. Then the operator AB has the form AB = A− ⊕ A+, 2 2 where the operators A± : L (R±) → L (R±) are given by 2 d 2 A± := ∓ , dom(A±)={f ∈ W (R±):f(±0) + b± · f (±0) = 0}. (3.3) dx2 2 Here b+ := −1/b11,b− := b22.OperatorsA± are well studied. + Remark 3.2. The function ϕB has a simple form. Indeed, by (2.14) and (2.3),we have √ + b11 ϕ (λ)=−ib22 λ +(b11b22 − b12b21) − i + √ ,λ∈ C+ \{0}. (3.4) B λ So the conditions of Theorem 3.1 can be easily checked (see Section 5).Letus + remark that the function ϕB has at most two zeroes (a zero of multiplicity k is counted as k zeroes). A criterion of similarity to a self-adjoint operator immediately follows from Theorem 3.1. Theorem 3.2. Let |b12|+|b21| =0 . Then the operator AB is similar to a self-adjoint + + C one iff the functions ϕB and ϕB∗ do not vanish in +. R + + Proof. By Lemma 2.1.(iii), σ(AB )= iff the functions ϕB and ϕB∗ have no zeroes in C+. Combining this fact with Theorem 3.1, we get Theorem 3.2. To prove the main theorem we recall the following Lemma. Differential Operators with Indefinite Weights and a Point Interaction 179 Lemma 3.1. If an operator T is similar to a normal one, then the inequality C (T − λI)−1 ≤ (3.5) dist(λ, σ(T )) holds with some constant C>0. 3.2. Some estimates We start with the following lemma. Lemma 3.2. Let |b12| + |b21| =0 . Suppose there exists λ0 ∈ R ∪{∞} such that + + ϕB(λ0)=0or ϕB∗ (λ0)=0. Then the operator AB of the form (2.13) is not similar to a normal operator. + Proof. Without loss of generality suppose that b21 =0andϕB(λ0)=0forsome λ0 ∈ R. It is obvious that for λ ∈ C \ R 2 √ | |2 − −λt sup y+(f,λ) =sup f(t)e dt f≤1 f≤1 R+ (3.6) √ − −λx 2 1 1 = e χ+(x) 2 = √ = √ . L |2Re −λ| |2Im λ| 2 Further, we set f+(·):=f(·)χ+(·),f∈ L (R). By (2.19), we have −1 −1 2 (AB − λI) f+ − (A0 − λI) f+ 2 L √ 2 − −λx · e χ+(x) √1 = + b22 + y+(f+,λ) (3.7) ϕ (λ) λ B L2 √ 2 λx e · χ−(x) + |b21|· √ · y+(f+,λ) ,λ∈ ρ(AB ) ∩ C+. λ · ϕ+(λ) B L2 Combining (3.6) and (3.7), one obtains −1 −1 2 (AB − λI) − (A0 − λI) 2 (3.8) L 2 2 1 1 b21 1 ≥ b22 + √ √ + √ · √ √ . + · · + | · | λ 2ϕB(λ) Im λ 4 λ ϕB(λ) Re λ Im λ Now if we recall Lemma 2.1, we obtain that dist(λ, σ(AB )) = | Im λ| in some neighborhood of λ0. Therefore, for sufficiently small ε and λ = λ0 + iε 2 −1 −1 2 dist(λ, σ(AB )) · (AB − λI) − (A0 − λI) 2 L 2 b21 · Im λ 1 1 ≥ √ · √ √ ≥ C1 · . (3.9) · + | · | | Im λ| 4 λ ϕB(λ) Re λ Im λ 180 I.M. Karabash and A.S. Kostenko Then the left part of inequality (3.9) is unbounded in the neighborhood of λ0.Note that A0 is a self-adjoint operator. Hence inequality (3.5) is valid for A0. Therefore, the function · − −1 dist(λ, σ(AB )) (AB λI) L2 (3.10) is unbounded in the neighborhood of λ0. By Lemma 3.1, the operator AB is not similar to a normal operator. √ + ∞ + ∈ C If ϕB( ) = 0, then formula (3.4) implies ϕB(λ)=b11/ λ for λ +. Hence for ε large enough 2 −1 −1 2 · − − 0 − dist(iε, σ(AB )) (AB iεI) (A iεI) L2 2 |b21| ε ≥ · √ √ = C2ε. (3.11) 4|b11| | Re iε · Im iε| Since the right part of (3.11) is unbounded in C+,weseethatAB is not similar to a normal operator. | | | | + Lemma 3.3. Let b12 + b21 =0. Suppose that the function ϕB has a zero of algebraic multiplicity 2 in C+.ThenAB is not similar to a normal operator. Proof. Let b21 = 0 (the case b12 = 0 can be considered in the same way). Suppose + λ0 ∈ C+ is a zero of multiplicity 2 of ϕ (·). By (3.8), we have for λ ∈ ρ(AB) ∩ C+ B −1 −1 √ b21 √ 1 √ (AB − λI) − (A0 − λI) 2 ≥ · . (3.12) L · + | · |1/2 4 λ ϕB (λ) Re λ Im λ Since λ0 ∈ ρ(A0), we see that (3.12) implies C (A − λI)−1 ≥ λ0 ,C= const > 0, (3.13) B L2 | + | λ0 ϕB (λ) in some neighborhood of λ0. Therefore λ0 is a pole of multiplicity 2 of the resolvent −1 (AB − λI) . Consequently, the operator AB is not similar to a normal one. We also need the following estimates. Lemma 3.4. Let λ = µ + iε, (ε>0).Lety±(f,λ) be of the form (2.18). Then the following inequalities +∞ √ 2 y±(f,λ) √ · − −λx ≤ · 2 ε e χ+(x) dµ 2π C1 f L2 , (3.14) λ 2 −∞ L +∞ √ 2 y±(f,λ) √ · λx ≤ · 2 ε e χ−(x) dµ 2π C2 f L2 (3.15) λ 2 −∞ L 2 are valid for all f ∈ L (R) with constants C1,C2 independent of ε and f. Differential Operators with Indefinite Weights and a Point Interaction 181 Proof. Let us prove the inequality (3.14) for y−(f,λ). 2 Put f−(·):=f(·)χ−(·),f∈ L (R). Denote by F (z) the Fourier transform of f−, +∞ 1 −izt F (z):=√ f−(t)e dt, z ∈ C+. (3.16) 2π −∞ 2 Note that F (·) ∈ H (C+)and F H2 = f− L2 ≤ f L2 . Further, we obtain +∞ √ 2 y−(f,λ) − −λx √ · e χ+(x) dµ λ 2 −∞ L +∞ √ √ 1 2 − −λx 2 =2π |F (i λ)| e 2 dµ |λ| L −∞ +∞ 1 √ 2 dµ =2π √ √ F (i λ) ) 2| λ Im λ| 2 |λ| −∞ iε+∞ 1 √ 2 √ ≤ 2π √ √ F (i λ) |d λ| 2| Im λ Re λ| iε−∞ iε+∞ 1 √ 2 √ =2π F (i λ) |d λ| Im λ iε−∞ iε+∞ 2π √ 2 √ = F (i λ) |d λ|. (3.17) ε iε−∞ Hence we find the estimate +∞ iε+∞ √ 2 √ 2 √ y−(f,λ) − −λx ε √ · e χ+(x) dµ ≤ 2π F (i λ) |d λ|. (3.18) λ L2 −∞ iε−∞ √ Finally, let us remark that |d λ| is the Carleson measure for all ε>0(see[17]).It is easy to see that the Carleson norms of these measures are uniformly bounded. Then, by the Carleson embedding theorem (see [17]), there exists C1 > 0such that for all ε>0 the inequality iε+∞ √ 2 √ 2 | |≤ 2 ≤ 2 F (i λ) d λ C1 F H2 = C1 f− L C1 f L (3.19) iε−∞ holds. Combining (3.18) and (3.19), one gets (3.14). Other inequalities can be obtained analogously. 182 I.M. Karabash and A.S. Kostenko 3.3. Proof of Theorem 3.1 3.3.1. Necessity. Follows immediately from Lemmas 3.2 and 3.3. 3.3.2. Sufficiency. a) Suppose that conditions (i)and(ii) of Theorem 3.1 hold. ∈ C + · + · Suppose also that λ1 + is a unique zero of ϕB ( )andϕB∗ ( )doesnotvanish in C+. Then, by Lemma 2.1, σ(AB )=R ∪{λ1}. Denote by B(λ1) a closed neighborhood of λ1 such that B(λ1) ⊂ C+.Letus consider the Riesz projection 1 −1 P1 := (AB − λ) dλ , (3.20) 2πi ∂B(λ1) where ∂B(λ1) is the boundary of B(λ1). 2 Then (see [25]) P1 ∈ [L (R)] and AB P1 = P1AB.SinceA0 is a self-adjoint operator, it follows that 1 −1 −1 1 −1 (AB − λ) − (A0 − λ) dλ = (AB − λ) dλ. (3.21) 2πi ∂B(λ1) 2πi ∂B(λ1) 2 It is not hard to show that P1 is a one-dimensional operator in L (R). Actually, λ−λ1 we set mλ1 := lim ϕ(λ) . Using (3.21) and (2.19), one gets λ→λ1 √ (P1f)(x) − −λ1x 1 b12 =e ·χ+(x)· b22 + √ y+(f,λ1)− √ ·y−(f,λ1) mλ1 λ1 λ1 √ √ λ1x e ·χ−(x) b11 + −λ1 2 + √ √ ·y−(f,λ1)−b21 ·y+(f,λ1) ,f∈L (R). (3.22) λ1 λ1 Let us write (3.22) in the following form √ √ λ1x · (P1f)(x) − −λ1x 1 e χ−(x) =y+(f,λ1)· e χ+(x)· b22 + √ −b21 √ mλ1 λ1 λ1 √ √ λ1x · ) y−(f,λ1) − −λ1x e χ−(x) + √ · e χ+(x)·(−b12)+ √ · b11 + −λ1 . (3.23) λ1 λ1 Note that 1 (b22 + √ ) −b12 λ1 √ det =det(B − M(λ1)) = ϕB (λ1)=0. (3.24) −b21 (b11 − i λ1) Hence P1 is a one-dimensional operator. b) By step (a), the space H = L2(R) can be decomposed as (see [25]) H = H0 H1, Hj := PjH,j∈{0, 1},P0 := I − P1. (3.25) Moreover, ABPj = PjAB ,j ∈{0, 1}, and the operator AB admits the following decomposition 0 1 j ∈{ } AB = AB AB,AB := Pj ABPj,j 0, 1 . (3.26) 1 { } 0 R We also have σ(AB )= λ1 and σ(AB )= . Differential Operators with Indefinite Weights and a Point Interaction 183 Let us show that the inequality +∞ 2 2 sup ε R 0 (µ + iε) f dµ ≤ C f ,f∈ H0, (3.27) AB ε>0 −∞ holds with some constant C>0. + · R ∪{∞} Since ϕB ( )doesnotvanishin , we see that Lemma 3.4 and (2.19) imply 2 ≤ 2 ∈ 2 R sup RAB (µ + iε)f dµ C1 f ,fL ( ) . (3.28) ε>0 µ∈R µ+iε∈B(λ1 ) Therefore 2 2 sup R 0 (µ + iε)f dµ ≤ C1 f ,f∈ H0. (3.29) AB ε>0 µ∈R µ+iε∈B(λ1) 0 Further, let us recall that λ1 ∈ ρ(A ). It means that the operator function R 0 (λ) B AB is bounded on B(λ1). If we combine this with (3.29), we get (3.27). 0 R + · C Since σ(AB )= and ϕB∗ ( ) have no zeroes in +, we can obtain in the same way the estimate +∞ 2 ∗ 2 ∗ sup R 0 ∗ (µ + iε)f dµ ≤ C f ,f∈ H0,C=const> 0. (AB) ε>0 −∞ (3.30) 0 Hence, by Theorem 1.2, the operator AB is similar to a self-adjoint operator. 1 Moreover, AB is a one-dimensional operator. Thus the operator AB is similar to anormalone. + · + · c) General case. Suppose that conditions (i) and (ii) hold, i.e., ϕB ( )andϕB∗ ( ) do not vanish on R ∪{∞}and have only simple zeros in C+. Let us denote by + + · C + + n(ϕB ) the number of zeroes of ϕB( )in +.Thenn(AB):=n(ϕB)+n(ϕB∗ )is the number of eigenvalues of AB , σp(AB)={λi : i =1,...,n(AB )}, (σp(AB)=∅ if n(AB )=0). + · C ≤ By (3.4), the function ϕB( ) has at most two zeroes in +. Hence n(AB) 4. It can be shown in the same way as in step (b) that exists a decomposition 2 R H H ··· H L ( )= 0 1 n(AB ), 0 1 ··· n(AB ) AB = AB AB AB . (3.31) 0 i Here AB is similar to a self-adjoint operator and AB are one-dimensional operators. Thus, AB is similar to a normal operator. This completes the proof. 184 I.M. Karabash and A.S. Kostenko 4. Other boundary conditions 4.1. The case ∆2 =0 The following theorem is an obvious corollary of Theorem 2.1. Theorem 4.1. 2 2 2 (i) The triplet Π2 = {C , Γ0, Γ1},whereΓj : W2 (R \{0}) → C ,j ∈{0, 1}, −f (+0) f(+0) Γ0f = , Γ1f = , (4.1) f(−0) f (−0) is a boundary triplet for A∗. (ii) The corresponding Weyl function M(·) is √ 1/ −λ √0 M(λ)=M2(λ):= ,λ∈ C \ R. (4.2) 0 λ 2 (iii) The corresponding γ-field γ(λ):C → Nλ is √ √ c+ 1 − −λx λx γ(λ) := c+ √ · e χ+(x)+c− · e χ−(x),c± ∈ C. (4.3) c− −λ If ∆2 = 0 then the boundary conditions (2.8) take the form f(+0) = −b11f (+0) + b12f(−0) . (4.4) f (−0) = −b21f (+0) + b22f(−0) Further, by Definition 1.5, we get ∗| − b11 b12 ∈ C2×2 A(aij ) = AB = A ker(Γ1 BΓ0),B= . (4.5) b21 b22 As before, denote by ϕB(·) the function as in (2.14) with M(·) as in (4.2). Combining Proposition 1.2 and Theorem 1.2, one obtains Lemma 4.1. If AB is the operator (4.5) and |b12| + |b21| =0 then: (i) σc(AB )=R, σr(AB )=∅; (ii) σp(AB)={λ ∈ C \ R : ϕB(λ)=0} = {λ ∈ C+ : ϕB(λ)=0}∪{λ ∈ C− : ϕB∗ (λ)=0}. (iii) The Krein formula has the form −1 −1 (AB − λ) f − (A0 − λ) f (x) √ & √ ' − −λx e · χ+(x) b22 − λ = √ √ · y+(f,λ) − b12 · y−(f,λ) ϕB(λ) · −λ −λ √ λx e · χ−(x) b21 1 + −√ · y+(f,λ)+ b11 − √ · y−(f,λ) , ϕB (λ) −λ −λ f ∈ L2(R), (4.6) ∗ where A0 = A | ker Γ0, λ ∈ ρ(AB)∩ρ(A0),andy±(f,λ) are defined by (2.18). Differential Operators with Indefinite Weights and a Point Interaction 185 4.2. The case ∆3 =0 Let ∆3 = 0. In this case we write the boundary conditions (2.8) in the following form f (+0) = b11f(+0) + b12f(−0) (4.7) f (−0) = b21f(+0) + b22f(−0). b11 b12 Let B = and b21 b22 ∗ 2 AB := A | dom(AB), dom(AB )={f ∈ W2 (R \{0}):f satisfies (4.7)}. (4.8) Theorem 4.2. 2 2 2 (i) The triplet Π3 = {C , Γ0, Γ1},whereΓj : W2 (R \{0}) → C ,j ∈{0, 1}, f(+0) f (+0) Γ0f = , Γ1f = , (4.9) f(−0) f (−0) is a boundary triplet for A∗. (ii) The corresponding Weyl function M(·) is √ − −λ √0 M(λ)=M3(λ):= ,λ∈ C \ R. (4.10) 0 λ 2 (iii) The corresponding γ-field γ(λ):C → Nλ is √ √ c+ − −λx λx γ(λ) := c+ · e χ+(x)+c− · e χ−(x),c± ∈ C. (4.11) c− Therefore, AB of the form (4.8) is an almost solvable extension of A and ∗ AB = A | ker(Γ1 − BΓ0), where Γi, i ∈{0, 1}, are defined by (4.9). Lemma 4.2. Let the function ϕB (·) be of the form (2.14) with M(·) defined by (4.10); let the operator AB is given by (4.8) and |b12| + |b21| =0 .Then: (i) σc(AB )=R, σr(AB )=∅; (ii) σp(AB)={λ ∈ C \ R : ϕB(λ)=0} = {λ ∈ C+ : ϕB(λ)=0}∪{λ ∈ C− : ϕB∗ (λ)=0}. (iii) The Krein formula has the form −1 −1 (AB − λ) f − (A0 − λ) f (x) √ − −λx √ e · χ+(x) = (b22 − λ) · y+(f,λ) − b12 · y−(f,λ) ϕB (λ) √ λx √ e · χ−(x) + −b21 · y+(f,λ)+(b11 + −λ) · y−(f,λ) , ϕB (λ) f ∈ L2(R), (4.12) ∗ where A0 = A | ker Γ0, λ ∈ ρ(AB ) ∩ ρ(A0),andy±(f,λ) are given by (2.18). 186 I.M. Karabash and A.S. Kostenko 4.3. The case ∆4 =0 If ∆4 = 0, then boundary conditions (2.8) take the form −f(+0) = b11f (+0) + b12f (−0) (4.13) −f(−0) = b21f (+0) + b22f (−0). b11 b12 For B = we set b21 b22 ∗ AB := A | dom(AB ), 2 dom(AB)={f ∈ W2 (R \{0}):f satisfies (4.13)}. (4.14) Theorem 4.3. 2 2 2 (i) The triplet Π4 = {C , Γ0, Γ1},whereΓj : W2 (R \{0}) → C ,j ∈{0, 1}, f (+0) −f(+0) Γ0f = , Γ1f = , (4.15) f (−0) −f(−0) is a boundary triplet for A∗. (ii) The corresponding Weyl function M(·) is √ 1/ −λ 0√ M(λ)=M4(λ):= ,λ∈ C \ R. (4.16) 0 −1/ λ 2 (iii) The corresponding γ-field γ(λ):C → Nλ is √ √ c+ c+ − −λx c− λx γ(λ) := −√ · e χ+(x)+√ · e χ−(x),c± ∈ C. (4.17) c− −λ λ Therefore, AB as in (4.14) is an almost solvable extension of A and AB = ∗ A | ker(Γ1 − BΓ0), where Γj, j ∈{0, 1} are given by (4.15). Lemma 4.3. Suppose the function ϕB(·) is given by (2.14) with M(·) as in (4.16). If the operator AB is given by (4.14) and |b12| + |b21| =0 then: (i) σc(AB )=R, σr(AB )=∅; (ii) σp(AB)={λ ∈ C \ R : ϕB(λ)=0} = {λ ∈ C+ : ϕB(λ)=0}∪{λ ∈ C− : ϕB∗ (λ)=0}. (iii) The Krein formula has the form −1 −1 (AB − λ) f − (A0 − λ) f (x) √ − −λx √ e · χ+(x) y+(f,λ) y−(f,λ) = √ (b22 +1/ λ) · √ + b12 · √ ϕB (λ) · −λ −λ λ √ λx e · χ−(x) y+(f,λ) √ y−(f,λ) + √ b21 · √ +(b11 − 1/ −λ) · √ , ϕB(λ) · λ −λ λ f ∈ L2(R), (4.18) ∗ where A0 = A | ker Γ0, λ ∈ ρ(AB ) ∩ ρ(A0),andy±(f,λ) are given by (2.18). Differential Operators with Indefinite Weights and a Point Interaction 187 4.4. The similarity criterion Arguing as above, we see that for the cases considered in Subsections 4.1–4.3 analogues of Theorems 3.1 and 3.2 hold. Theorem 4.4. Theorems 3.1 and 3.2 are valid for the extensions AB with boundary conditions (4.4), (4.7) or (4.13) if the function M(·) is replaced by (4.2), (4.10) or (4.16), respectively. x − d2 cδ x − d2 cδ 5. On similarity of (sgn ) dx2 + and (sgn ) dx2 + to normal operators 5.1. Let us illustrate the previous results by several examples. We start with the ˜ − d2 ˜ operator A = (sgn x) dx2 , see (0.3). It is obvious that A = AB,whereAB is an 01 almost solvable extension of the form (2.13) with B = . It follows from −10 (3.4) that in this case + + ≡ − ∈ C ϕB(λ)=ϕB∗ (λ) 1 i, λ +. Hence, by Theorem 3.2, one obtains the result of [7]. ˜ − d2 Theorem 5.1 ([7]). The operator A = (sgn x) dx2 is similar to a self-adjoint operator. 5.2. Let δ be the Dirac delta. Let us introduce the following differential expression d2 − + cδ, c ∈ C \{0}. (5.1) dx2 Here δ formally represents a contact interaction at zero if c ∈ R. 2 In L (R), expression (5.1) generates the differential operator Lcδ defined by 2 dom(Lcδ)={f ∈ W2 (R \{0}):f(+0) = f(−0),f (+0) − f (−0) = cf(−0)}, d2 L := − (5.2) cδ dx2 (see for example [2, 35]). We put Acδ := JLcδ,where(Jf)(x)=(sgnx)f(x), i.e., 2 the operator Acδ is defined in L (R) by the differential expression d2 (sgn x) − + cδ ,c∈ C \{0}. (5.3) dx2 It is clear that the operator Acδ is an extension of the operator A of the form (0.1). Moreover, Acδ = AB where AB is given by (2.13) and c 1 B = B := . (5.4) c −10 188 I.M. Karabash and A.S. Kostenko Theorem 5.2. Let c =0 . (i) The operator Acδ is similar to a normal one if and only if Re c = −| Im c|. (ii) Acδ is similar to a self-adjoint operator if and only if Re c>−| Im c|. Proof. Let B be given by (5.4). By (3.4), one gets + c + c ϕ (λ)=1− i + √ ,ϕ∗ (λ)=1− i + √ ,λ∈ C+ \{0}. (5.5) B λ B λ + · + · −| | Note that ϕB( )orϕB∗ ( ) have a real zero iff Re(c)= Im(c) .Furthermore, + · + · C −| | ϕB( )andϕB∗ ( )donotvanishin + iff Re(c) > Im c . Hence the statements of Theorem 5.2 obviously follow from Theorems 3.1 and 3.2. 5.3. Let us consider the extension AB of the form (2.13) with 01 B = B = ,c=0 . c −1 c This is a so-called “operator with δ -interaction” (see [2]). The formal differential expression corresponding to AB is d2 (sgn x) − + cδ ,c∈ C \{0}. (5.6) dx2 Therefore we will denote the operator AB by Acδ . Theorem 5.3. Let c =0 . (i) The operator Acδ issimilartoanormaloneifandonlyifRe c = −| Im c|. (ii) Acδ is similar to a self-adjoint operator if and only if Re c>−| Im c|. Proof. So, by (3.4), we have √ √ + − − + − − ∈ C ϕcδ (λ)=1 i ic λ, ϕcδ (λ)=1 i ic λ, λ +. (5.7) These functions have a real zero iff Re c = −| Im c|, have no zeros in C+ iff Re c> −| Im c|. Hence the statements of Theorem 5.3 follow from Theorems 3.1 and 3.2. Remark 5.1. Let L be a self-adjoint extension of the symmetric operator 2 (Lf)(x):=−f (x), dom(L)={f ∈ W2 (R):f(0) = f (0) = 0}. (5.8) Assume that the boundary conditions at 0 associated with the extension L are nonseparate, i.e., the operator L does not admit the following decomposition L+ ⊕ 2 L−,whereL± := L |(dom(L) ∩ L (R±)).ThenL can be considered as an operator with a singular interaction (see [27] for the details). Using the arguments of this section, one can describe the main spectral properties of the corresponding J-self- adjoint operator A := JL. Differential Operators with Indefinite Weights and a Point Interaction 189 Acknowledgment The authors are deeply grateful to professor M.M. Malamud for statement of the problem and his constant attention to this work. The authors thanks a referee for a careful reading of the manuscript and a number of remarks, which helped to improve it. 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Malamud, Similarity of a triangular operator to a diagonal operator. Zapiski Nauchnykh Seminarov POMI 270 (2000), 201–241 [in Russian]; English translation: J. Math. Sciences 115 (2000), 2199–2222. [31] S.N. Naboko, Conditions for similarity to unitary and selfadjoint operators. Funkt- sional. Anal. i Prilozhen. 18, no. 1, (1984), 16–27 [in Russian]. [32] S.G. Pyatkov, Properties of eigenfunctions of certain spectral problem with some ap- plications. Nekotorie prilozhenia funktsionalnogo analiza k zadacham matematich- eskoi fiziki, Novosibirsk: IM SO AN SSSR, 1986, 65–84 [in Russian]. Differential Operators with Indefinite Weights and a Point Interaction 191 [33] S.G. Pyatkov, Some properties of eigenfunctions of linear pencils. Sibirskii Mat. Zh. 30, no. 4 (1989), 111–124 [in Russian]; English translation: Siberian Math. J. 30 (1989), 587–597. [34] S.G. Pyatkov, Indefinite elliptic spectral problems. Sibirskii Mat. Zh. 39, no. 2 (1998), 409–426 [in Russian]; English translation: Siberian Math. J. 30 (1989), 358–372. [35] M. Reed, B. Simon, Methods of modern mathematical physics. II. Academic Press, New York-San Francisco-London, 1975. Ilia Karabash and Aleksey Kostenko Department of Mathematics Donetsk National University Universitetskaja 24 83055 Donetsk, Ukraine e-mail: [email protected]; [email protected] e-mail: [email protected]; [email protected] Operator Theory: Advances and Applications, Vol. 175, 193–209 c 2007 Birkh¨auser Verlag Basel/Switzerland Normal Matrices in Degenerate Indefinite Inner Product Spaces Christian Mehl and Carsten Trunk Abstract. Complex matrices that are structured with respect to a possibly degenerate indefinite inner product are studied. Based on the theory of linear relations, the notion of an adjoint is introduced: the adjoint of a matrix is defined as a linear relation which is a matrix if and only if the inner product is nondegenerate. This notion is then used to give alternative definitions of selfadjoint and unitary matrices in degenerate inner product spaces and it is shown that those coincide with the definitions that have been used in the literature before. Finally, a new definition for normal matrices is given which allows the generalization of an extension result for positive invariant subspaces from the case of nondegenerate inner products to the case of degenerate inner products. Mathematics Subject Classification (2000). Primary 15A57; Secondary 15A63. Keywords. Degenerate inner product space, adjoint, linear relations, H-self- adjoint, H-unitary, H-normal. 1. Introduction We consider the space Cn equipped with an indefinite inner product induced by a Hermitian matrix H ∈ Cn×n via ∗ [x, y]:=[x, y]H := Hx,y = y Hx, (1.1) where ·, · denotes the standard Euclidean scalar product on Cn. We will sup- press the subscript H when it is clear that H induces the indefinite inner product. If H is invertible, i.e., if the indefinite inner product is nondegenerate, then for a matrix M ∈ Cn×n there exists a unique matrix M [∗]H satisfying [x, My]=[M [∗]H x, y] for all x, y ∈ Cn. (1.2) (Again, we will suppress the subscript H if it is clear from the context which inner product is under consideration.) This matrix M [∗] is called the H-adjoint of M. 194 C. Mehl and C. Trunk From (1.2), one easily obtains the matrix identity M [∗] = H−1M ∗H. (1.3) AmatrixM ∈ Cn×n is called H-selfadjoint, H-skewadjoint,orH-unitary, respec- tively, if M [∗] = M, M [∗] = −M,orM [∗] = M −1, respectively. Using (1.3), we obtain the matrix identities M ∗H = HM, M∗H + HM =0, or M ∗HM = H (1.4) for H-selfadjoint, H-skewadjoint, or H-unitary matrices, respectively. These three types of matrices have been widely discussed in the literature, both in terms of theory and numerical analysis. Extensive lists of references can be found in [1, 9, 13, 21]. H-selfadjoint, H-skewadjoint, and H-unitary matrices are special cases of H- normal matrices. A matrix M ∈ Cn×n is called H-normal if M commutes with its H-adjoint, i.e., if MM[∗] = M [∗]M, or, in other words, if and only if MH−1M ∗H = H−1M ∗HM, or, HMH−1M ∗H = M ∗HM. (1.5) In recent years, there has been great interest in H-normal matrices, see [8, 9, 10, 17, 18] and the references therein. Spaces with degenerate inner products, that is, H is singular, are less familiar, although this case does appear in applications [14]. Some works here, primarily concerning infinite-dimensional degenerate Pontryagin spaces, include [23], [11] (and references there), [2], [12], and parts of the book [3]. The main problem in the context of degenerate inner products is that there is no straightforward definition of an H-adjoint. Indeed, if H is singular, then for a matrix M ∈ Cn×n an H-adjoint, i.e., a matrix N ∈ Cn×n satisfying [x, My]=[Nx,y] for all x, y ∈ Cn need not exist, and if it exists, it need not be unique. Example 1.1. Let a, b ∈ C and 10 10 11 10 H = ,M1 = ,M2 = ,N(a, b)= . 00 01 01 ab Then for all possible choices a, b ∈ C, N(a, b)satisfies[x, M1y]=[N(a, b)x, y] for all x, y ∈ Cn. On the other hand, there is no matrix N ∈ C2×2 such that n [x, M2y]=[Nx,y] for all x, y ∈ C . Despite the lack of the notion of an H-adjoint, one can define H-selfadjoint, H-skewadjoint and H-unitary for the case of singular H by the matrix identities (1.4) and this definition has been used in many sources, see, e.g., [20] and the ref- erences therein. The corresponding matrix identities (1.5) for H-normal matrices, however, require an inverse of H. A standard approach to circumvent this difficulty is the use of some generalized inverse, in particular the Moore-Penrose generalized inverse H†, instead. However, it seems that the use of the Moore-Penrose inverse leads to some inconsistencies in the theory of degenerate inner products. We illus- trate this by an example. Normal Matrices in Inner Product Spaces 195 Example 1.2. Let 10 10 01 H = ,A= ,N= . 00 11 10 Then we have H† = H.ObservethatA∗H = HA, but A = H†A∗H. So, should A be considered as an H-selfadjoint matrix or not? On the other hand, note that NH†N ∗H = H†N ∗HN, but HNH†N ∗H = N ∗HN. So, should N be considered as an H-normal matrix or not? In [16] H-normal matrices have been defined as matrices M satisfying HMH†M ∗H = M ∗HM (1.6) and this definition has later been taken up in [4, 17]. (This way minimizes the num- ber of times the Moore-Penrose generalized inverse of H appears in the defining equation of H-normal matrices.) In this paper, we will call matrices M satisfy- ing (1.6) Moore-Penrose H-normal matrices in order to highlight the occurrence of the Moore-Penrose generalized inverse in the definition. Although it is possible to prove some interesting results for Moore-Penrose H- normal matrices (e.g., concerning existence of invariant maximal semidefinite sub- spaces), there is an unpleasant mismatch between H-selfadjoint, H-skewadjoint, and H-unitary matrices on the one hand and Moore-Penrose H-normal matrices on the other hand. It is easy to check (see also [20]) that the kernel of H is always an invariant subspace for H-selfadjoint, H-skewadjoint, and H-unitary matrices (defined as in (1.4)). However, it has been shown in [17, Example 6.1] that there exist Moore-Penrose H-normal matrices A such that ker H is not A-invariant. It is exactly the fact that ker H need not be invariant which makes the in- vestigation of Moore-Penrose H-normal matrices challenging. For example, let us consider the problem of existence of semidefinite invariant subspaces. Recall that a subspace M⊆Cn is called H-nonnegative if [x, x] ≥ 0foreveryx ∈M, H- positive if [x, x] > 0 for every nonzero x ∈M,andH-neutral if [x, x]=0for every x ∈M.AnH-nonnegative subspace is said to be maximal H-nonnegative if it is not properly contained in any larger H-nonnegative subspace. It is easy to see that an H-nonnegative subspace is maximal if and only if its dimension is equal to ν+(H)+ν0(H), where ν+(H)andν0(H) denote the number (counted with multiplicities) of positive and zero eigenvalues of H, respectively. It has been shown in [17, Theorem 6.6] that any Moore-Penrose H-normal matrix has a max- imal H-nonnegative invariant subspace, but the problem which H-nonnegative, H-positive, or H-neutral invariant subspaces of Moore-Penrose H-normal matri- ces can be extended to a maximal H-nonnegative invariant subspaces has only been solved for the case of invertible H so far [18, 19]. It is the aim of this paper to propose a different definition of H-normal matrices in degenerate inner product spaces that is based on a generalization of the H-adjoint A[∗]H of a matrix A for singular H. This generalization is obtained by dropping the assumption that the H-adjoint of a matrix is a matrix itself. Instead, the H-adjoint A[∗]H is defined to be a linear relation in Cn, i.e., a linear 196 C. Mehl and C. Trunk subspace of C2n. For basic facts on linear relations and further references see, e.g., [5, 6, 7, 22]. Throughout this paper we identify a matrix A ∈ Cn×n with its graph x : x ∈ Cn ⊆ C2n Ax which is a linear relation, also denoted by A.IfH ∈ Cn×n is invertible, then, by (1.2), we obtain that A[∗]H coincides with the linear relation y A[∗]H = ∈ C2n :[y,Ax] =[z,x] for all x ∈ Cn . z H H This representation allows a direct generalization of the concept of adjoint to the case of degenerate inner products and even to the case of starting with a linear relation rather than a matrix A, see [22]. Definition 1.3. Let H ∈ Cn×n be Hermitian and let A be a linear relation in Cn . Then the linear relation y x A[∗]H = ∈ C2n :[y,w] =[z,x] for all ∈ A z H H w is called the H-adjoint of A. Again, if there is no risk of ambiguity, we will suppress the subscript H in the notation. In this setting, we obtain (see Proposition 2.6 below) that A[∗] = H−1A∗H, where H−1 is the inverse of H in the sense of linear relations (see Section 2). Observe that this coincides with (1.3) if H is invertible. Hence, the H-adjoint in degenerate inner product spaces is a natural generalization of the H-adjoint in nondegenerate inner product spaces. The remainder of the paper is organized as follows. In Section 2, we discuss basic properties of the H-adjoint. In Section 3 we use the H-adjoint to define H-symmetric and H-isometric linear relations and we show that in the case of matrices these definitions coincide with the definitions of H-selfadjoint and H- unitary matrices via the identities (1.4). In Section 4, we present a new definition for H-normal matrices. We show that the set of H-normal matrices is a proper subset of the set of Moore-Penrose H-normal matrices and that H-normal matri- ces share the property with H-selfadjoint and H-unitary matrices that the kernel of H is always an invariant subspace. The latter fact allows us to obtain sufficient conditions for an H-positive invariant subspace of an H-normal matrix to be con- tained in a maximal H-nonnegative invariant subspace. This generalizes a result obtained in [18]. 2. The adjoint in degenerate inner product spaces We study linear relations in Cn, i.e., linear subspaces of C2n. For the definitions of linear operations with relations and the inverse of relations we refer to [6]. We Normal Matrices in Inner Product Spaces 197 only mention the following. For linear relations A, B ⊆ C2n we define x dom A = x : ∈ A , the domain of A, y x ran A = y : ∈ A , the range of A, y 0 mul A = y : ∈ A , the multivalued part of A, y y x A−1 = : ∈ A , the inverse of A x y and the product of A and B, x y x AB = : there exists a y ∈ Cn with ∈ A, ∈ B . z z y In all cases, x, y, z are understood to be from Cn. For a subset M ⊆ Cn we define M [⊥] = {x :[x, y] = 0 for all y ∈ M} . The following lemma is needed for the proof of Proposition 2.2 below. It is contained in [15, proof of Lemma 2.2], but for the sake of completeness we give a separate proof. Lemma 2.1. Let M ⊆ Cn be a subspace. Then (M [⊥])[⊥] = M +kerH. Proof. Obviously, we have M +kerH ⊆ (M [⊥])[⊥].Ifu/∈ M +kerH then, as the quotient space (Cn/ker H, [., .]∼)with[x+kerH, y+kerH]∼ := [x, y], x, y ∈ Cn,is nondegenerate, there exists an v ∈ Cn such that [v, M]=[v+kerH, M +kerH]∼ = {0} and [v, u]=[v +kerH, u +kerH]∼ = 0. Therefore u/∈ (M [⊥])[⊥],which completes the proof. In the next proposition we collect some properties of the H-adjoint. Proposition 2.2. Let A, B ⊆ C2n be linear relations. Then we have (i) A ⊆ B =⇒ B[∗] ⊆ A[∗]. (ii) mul A[∗] =(domA)[⊥].IfA is a matrix, we have mul A[∗] =kerH. (iii) ker A[∗] =(ranA)[⊥]. (iv) (A[∗])[∗] = A +(kerH × ker H). Proof. Assertions (i), (ii), (iii) are easy consequences of Definition 1.3 and can be found, e.g., in [22]. In order to prove assertion (iv) we equip the space Cn × Cn with the inner product y x 0 −iH y x , := i [y,w] − [z,x] = , z w iH 0 z w 198 C. Mehl and C. Trunk where x, y, w, z ∈ Cn.Then A[∗] = A[[ ⊥]] and (A[∗])[∗] =(A[[ ⊥]] )[[ ⊥]] . (2.1) Lemma 2.1 remains true if we replace Cn by C2n,[⊥]by[[⊥]] a n d k e r H by (ker H×ker H). This and (2.1) shows assertion (iv) of Proposition 2.2. If A is a matrix, then, by Proposition 2.2 (iv), we have that y ∈ mul (A[∗])[∗] if and only if there exists f,g ∈ ker H such that 0 f −f 0 = + = . y Af g Af + g This proves the following lemma. Lemma 2.3. Let A ∈ Cn×n be a matrix. Then ker H ⊆ mul (A[∗])[∗]. If , in addition, ker H is A-invariant, then we have mul (A[∗])[∗] =kerH. The following formula for the H-adjoint of matrices will be used frequently. Lemma 2.4. Let A ∈ Cn×n be a matrix. Then y A[∗] = ∈ C2n : A∗Hy = Hz . (2.2) z In particular, A[∗] is a matrix if and only if H is invertible. Proof. Clearly, since A is a matrix, we have y ∈ A[∗] ⇐⇒ x∗A∗Hy = x∗Hz for all x ∈ Cn. z This implies (2.2). The remaining assertion of Lemma 2.4 follows from Proposi- tion 2.2 (ii). ByLemma2.4wehavethatdomA[∗] = Cn if and only if ran (A∗H) ⊆ ran H. Example 2.5. Using the notations of Example 1.1 we have [∗] [∗] M1 =graphI +({0}×{0}×{0}×C)andM2 = {0}×C ×{0}×C, where + denotes the sum of linear subspaces. We then obtain [∗] 2 [∗] dom M1 = C , mul M1 = {0}×C, [∗] [∗] dom M2 = {0}×C, mul M2 = {0}×C. Let A ⊆ C2n be a linear relation. Introducing a change of basis x → Px on Cn,whereP ∈ Cn×n is a nonsingular matrix, yields P −1x x P −1AP = : ∈ A . (2.3) P −1w w Normal Matrices in Inner Product Spaces 199 Therefore, for y x ∈ (P −1AP )[∗]P ∗HP and arbitrary ∈ A z w −1 −1 we have 0 = [y,P w]P ∗ HP − [z,P x]P ∗HP =[Py,w]H − [Pz,x]H , i.e., Py y ∈ A[∗]H or, equivalently, ∈ P −1A[∗]H P. Pz z This gives (P −1AP )[∗]P ∗HP = P −1A[∗]H P. (2.4) Moreover, with the help of (2.3), it is easily deduced that (P −1AP )−1 = P −1A−1P, (2.5) where the inverses are understood in the sense of linear relations. Thus, if A is a matrix, then changing the basis of Cn accordingly, we may always assume that H and A have the forms mn− m mn− m m H1 0 m A1 A2 H = and A = ∈ Cn×n, (2.6) n − m 00 n − m A3 A4 m×m where H1 ∈ C is nonsingular. When using these forms and identifying A with the linear relation A ⊆ C2n, then for the ease of simple notation we will usually omit the indication of dimensions of vectors if those are clear from the context. Thus, for example, we write ⎧⎛ ⎞ ⎫ ⎧⎛ ⎞⎫ ⎪ x1 ⎪ ⎪ x1 ⎪ ⎨⎜ ⎟ ⎬ ⎨⎜ ⎟⎬ ⎜ x2 ⎟ ∈ Cm ∈ Cn−m ⎜ x2 ⎟ A = ⎪⎝ ⎠ : x1 ,x2 ⎪ = ⎪⎝ ⎠⎪ . ⎩⎪ A1x1 + A2x2 ⎭⎪ ⎩⎪ A1x1 + A2x2 ⎭⎪ A3x1 + A4x2 A3x1 + A4x2 Proposition 2.6. Let A ∈ Cn×n be a matrix. Then A[∗] = H−1A∗H, (2.7) where H−1 is the inverse in the sense of linear relations. In particular, if H and A have the forms as in (2.6) then ⎧⎛ ⎞ ⎫ ⎪ y1 ⎪ ⎨⎜ ⎟ ⎬ [∗] ⎜ y2 ⎟ ∗ A = [∗] : A H1y1 =0 . (2.8) ⎪⎝ H1 ⎠ 2 ⎪ ⎩⎪ A1 y1 ⎭⎪ z2 [∗] n Moreover, we have dom A = C if and only if A2 =0. 200 C. Mehl and C. Trunk Proof. We have x z Hz H−1 = : ∈ H = z x z and, using Lemma 2.4, y y H−1A∗H = H−1 = H−1 : Hz = A∗Hy A∗Hy Hz y = : Hz = A∗Hy = A[∗] z and (2.7) is proved. Let H and A be in the forms (2.6). By Lemma 2.4, we have that ⎧⎛ ⎞ ⎫ ⎪ y1 ⎪ ⎨⎜ ⎟ ⎬ [∗] ⎜ y2 ⎟ ∗ y1 z1 A = ⎪⎝ ⎠ : A H = H ⎪ ⎩⎪ z1 y2 z2 ⎭⎪ z2 ⎧⎛ ⎞ ⎫ ⎪ y1 ⎪ ⎨⎜ ⎟ ⎬ ⎜ y2 ⎟ ∗ ∗ = ⎪⎝ ⎠ : A1H1y1 = H1z1 and A2H1y1 =0⎪ . ⎩⎪ z1 ⎭⎪ z2 −1 ∗ [∗]H1 Since H1 A1H1y1 = A1 y1, we obtain (2.8) and since H1 is invertible, we have [∗] n dom A = C if and only if A2 =0. 3. H-symmetric and H-isometric matrices In the nondegenerate case, H-selfadjoint matrices are defined as matrices A satis- fying A = A[∗]. When generalizing this concept to the degenerate case, however, we have to take into account that for any A ∈ Cn×n,therelationA[∗] is never amatrixwhenH is singular. Thus, matrices satisfying A = A[∗] do not exist. Instead, it is natural to consider matrices that are H-symmetric relations in the following sense. Definition 3.1. A linear relation A in Cn is called H-symmetric if A ⊆ A[∗]. We mention that H-symmetric relations in degenerate inner product spaces have been introduced in [22]. Example 3.2. In Example 1.1 we have that M1 is H-symmetric. Clearly, A is H-symmetric if and only if P −1AP is P ∗HP-symmetric for any invertible P ∈ Cn×n, cf. (2.3). At first sight, Definition 3.1 in the case of a matrix A ∈ Cn×n may look a little bit weird, but the following proposition shows that this definition does make sense, because we will show that H-symmetry is equivalent to the condition A∗H = HA which has been used as the definition for H-selfadjoint matrices in degenerate inner products in various sources. Normal Matrices in Inner Product Spaces 201 Proposition 3.3. Let A ∈ Cn×n be a matrix. Then the following statements are equivalent. i) A is H-symmetric, i.e., A ⊆ A[∗]. ii) A∗H = HA. If one of the conditions is satisfied, then ker H is A-invariant. In particular, if H and A have the forms as in (2.6) then A is H-symmetric if and only if A1 is H1-selfadjoint and A2 =0. y Proof. Let ∈ A. Then, by Lemma 2.4, we have Ay y ∈ A[∗] ⇐⇒ A∗Hy = HAy. Ay This shows the equivalence of the two statements in Proposition 3.3. (For the implication i) ⇒ ii) observe that A ⊆ A[∗] implies that the domain of A[∗] is Cn.) For the remainder of the proof, let H and A be in the forms as in (2.6) and assume that A is H-symmetric. Then using that the domain of A[∗] is Cn,we [∗] H1 ∗ m obtain from (2.8) that A1y1 + A2y2 = A1 y1 and A2H1y1 =0forally1 ∈ C . [∗]H1 Since H1 is invertible, this implies A2 = 0 and we also obtain A1 = A1 , i.e., A1 is H1-selfadjoint. Thus, a matrix is H-symmetric (in the sense of linear relations) if and only if it is H-selfadjoint (in the sense of matrices). At first sight, it may look a little bit disappointing that only one inclusion of the usual concept of “selfadjointness” (in the sense of linear relations) is fulfilled. However, this changes if we adjoin the inclusion A ⊆ A[∗] once more, as the following proposition shows. Proposition 3.4. Let A ∈ Cn×n be a matrix. Then A is H-symmetric if and only if A[∗] =(A[∗])[∗]. Proof. If A[∗] =(A[∗])[∗] then, by Proposition 2.2 (iv), A is H-symmetric. For the converse assume A ⊆ A[∗]. Then Proposition 2.2 (i) implies (A[∗])[∗] ⊆ A[∗].For the other inclusion observe that A ⊆ A[∗] and Proposition 2.2 (iv) give dom A[∗] =dom(A[∗])[∗] = Cn. Moreover, Proposition 2.2 (ii) and Lemma 2.3 together with Proposition 3.3 imply mul A[∗] =mul(A[∗])[∗] =kerH. Hence, A[∗] =(A[∗])[∗]. Proposition 2.2 (iv) together with Proposition 3.4 imply the following Corollary 3.5. Let A ∈ Cn×n be a matrix. Then A is H-symmetric if and only if A[∗] = A +(kerH × ker H). 202 C. Mehl and C. Trunk Similar to H-symmetric matrices, H-isometric matrices can be defined by passing to the concept of linear relations. (We mention that H-isometric relations in degenerate inner product spaces have been introduced in [22].) Definition 3.6. A linear relation U in Cn is called H-isometric if U −1 ⊆ U [∗]. We note that in the definition above U −1 is the inverse in the sense of linear relations. E.g., if H = 0, then every matrix is H-isometric. It follows from (2.4) and (2.5) that U is H-isometric if and only if P −1UP is P ∗HP-isometric for any invertible P ∈ Cn×n. Proposition 3.7. Let U ∈ Cn×n be a matrix. Then the following statements are equivalent. i) U is H-isometric, i.e., U −1 ⊆ U [∗]. ii) U ∗HU = H. If one of the conditions is satisfied, then ker H is U-invariant. In particular, if H and U have the forms as in (2.6), i.e., U1 U2 H1 0 U = ,H= , (3.1) U3 U4 00 where H1 is invertible, then U is H-isometric if and only if U1 is H1-unitary and U2 =0. Moreover, we have (ran U)[⊥] =kerH. (3.2) Proof. By Lemma 2.4, we have Uy U −1 ⊆ U [∗] ⇐⇒ ∈ U [∗] for all y ∈ Cn ⇐⇒ U ∗HUy = Hy for all y ∈ Cn. y This shows the equivalence of the two statements in Proposition 3.7. For the remainder of the proof, let H and U be in the forms as in (3.1) and assume that U is H-isometric. Then we obtain from the identity U ∗HU = H that ∗ ∗ U1 H1U1 U1 H1U2 H1 0 ∗ ∗ = U2 H1U1 U2 H1U2 00 ∗ This immediately implies U1 H1U1 = H1, i.e., U1 is H1-unitary. In particular, with H1 also U1 must be invertible. This finally yields U2 = 0 and (3.2). If we adjoin the inclusion U −1 ⊆ U [∗] once more we get, similar to Proposi- tion 3.4, the following characterization of H-isometric matrices. Proposition 3.8. Let U ∈ Cn×n be a matrix. Then U is H-isometric if and only if (U −1)[∗] =(U [∗])[∗] Proof. Let U be H-isometric, i.e., U −1 ⊆ U [∗]. Then, by Proposition 2.2 (i), we have (U [∗])[∗] ⊆ (U −1)[∗]. For the other inclusion, observe that, by Proposition 2.2 (iv), we have dom (U [∗])[∗] = Cn. Normal Matrices in Inner Product Spaces 203 Thus, using (U [∗])[∗] ⊆ (U −1)[∗], we obtain that dom (U [∗])[∗] =dom(U −1)[∗] = Cn. Proposition 2.2 (ii) and (3.2) imply mul (U −1)[∗] =(domU −1)[⊥] =(ranU)[⊥] =kerH and, with Lemma 2.3 and the U-invariance of ker H (Proposition 3.7), we conclude mul (U [∗])[∗] =kerH =mul(U −1)[∗]. Hence (U −1)[∗] =(U [∗])[∗]. The contrary follows from Proposition 2.2 and U −1 ⊂ ((U −1)[∗])[∗] =((U [∗])[∗])[∗] ⊂ U [∗]. 4. H-normal matrices Recall that in the case of invertible H,amatrixA is called H-normal if and only if AA[∗] = A[∗]A. For the case that H is singular and that H and A are given in the forms as in (2.6), a straightforward computation reveals ⎧⎛ ⎞ ⎫ ⎪ y1 ⎪ ⎨⎜ ⎟ ⎬ [∗] ⎜ y2 ⎟ ∗ AA = ⎝ [∗] ⎠ : A2H1y1 =0 (4.1) ⎪ A1A1 y1 + A2z2 ⎪ ⎩ [∗] ⎭ A3A1 y1 + A4z2 and ⎧⎛ ⎞ ⎫ ⎪ y1 ⎪ ⎨⎜ ⎟ ⎬ [∗] ⎜ y2 ⎟ ∗ ∗ A A = ⎪⎝ [∗] [∗] ⎠ : A2H1A1y1 + A2H1A2y2 =0⎪ . (4.2) ⎩⎪ A1 A1y1 + A1 A2y2 ⎭⎪ z2 However, even in the case that A is H-symmetric (i.e., in the identities (4.1) [∗] [∗] 2 and (4.2) we have A1A1 = A1 A1 = A1 and A2 = 0), we only obtain the inclusion [∗] [∗] [∗] [∗] AA ⊆ A A, while the other inclusion A A ⊆ AA is only satisfied if A4 is invertible. This motivates the following definition. Definition 4.1. ArelationA in Cn is called H-normal if AA[∗] ⊆ A[∗]A. As for the case of H-isometric and H-symmetric matrices, we obtain that the kernel of H is always an invariant subspace for H-normal matrices. Proposition 4.2. Let A ∈ Cn×n be an H-normal matrix. Then ker H is A-invariant. In particular, if A and H are in the forms as in (2.6),thenA is H-normal if and only if A1 is H1-normal and A2 =0. Proof. Without loss of generality, we may assume that A and H are in the forms as in (2.6). Clearly, if A1 is H1-normal and A2 = 0 then it follows directly from (4.1) and (4.2) that AA[∗] ⊆ A[∗]A, i.e., A is H-normal. For the converse, assume that 204 C. Mehl and C. Trunk A is H-normal. Then AA[∗] ⊆ A[∗]A implies in particular that AA[∗] ∩ ({0}×Cn) ⊆ A[∗]A ∩ ({0}×Cn) Comparing the third block components of (4.1) and (4.2) this reduces to A2z =0 n−m [∗] [∗] for all z2 ∈ C and this is only possible if A2 =0.ButthenAA ⊆ A A [∗] [∗] m implies A1A1 y1 = A1 A1y1 for all y1 ∈ C and we obtain that A1 is H1-normal. Clearly, ker H is A-invariant, because of A2 = 0. This concludes the proof. With Propositions 3.3 and 3.7 we immediately obtain the following corollary. Corollary 4.3. H-symmetric and H-isometric matrices are H-normal. The question arises, if we obtain a different characterization of H-normality in the style of Propositions 3.4 and 3.8 by the identity A[∗](A[∗])[∗] =(A[∗])[∗]A[∗]. (4.3) Let us investigate this question in detail. Without loss of generality assume that the matrix A ∈ Cn×n and H are given in the forms (2.6). Then using Proposition 2.2 (iv), we obtain ⎧⎛ ⎞ ⎛ ⎞⎫ ⎧⎛ ⎞⎫ ⎪ x1 0 ⎪ ⎪ x1 ⎪ ⎨⎜ ⎟ ⎜ ⎟⎬ ⎨⎜ ⎟⎬ [∗] [∗] ⎜ x2 ⎟ ⎜ w2 ⎟ ⎜ w2 ⎟ (A ) = ⎪⎝ ⎠ + ⎝ ⎠⎪ = ⎪⎝ ⎠⎪ . ⎩⎪ A1x1 + A2x2 0 ⎭⎪ ⎩⎪ A1x1 + A2x2 ⎭⎪ A3x1 + A4x2 z2 z2 Together with (2.8) this implies ⎧⎛ ⎞ ⎫ ⎪ x1 ⎪ ⎨⎜ ⎟ ⎬ [∗] [∗] [∗] ⎜ w2 ⎟ ∗ A (A ) = ⎪⎝ [∗] [∗] ⎠ : A2H1(A1x1 + A2x2)=0⎪ (4.4) ⎩⎪ A1 A1x1 + A1 A2x2 ⎭⎪ z2 and ⎧⎛ ⎞ ⎫ ⎪ y1 ⎪ ⎨⎜ ⎟ ⎬ [∗] [∗] [∗] ⎜ y2 ⎟ ∗ (A ) A = ⎪⎝ [∗] ⎠ : A2H1y1 =0⎪ . (4.5) ⎩⎪ A1A1 y1 + A2x2 ⎭⎪ z2 With the help of these formulas, we obtain that (4.3) may be satisfied even if the matrix A ∈ Cn×n is not H-normal, see Example 4.4 below. Example 4.4. Let A and H be given as ⎛ ⎞ ⎛ ⎞ 101 010 A1 A2 H1 0 A = := ⎝ 010 ⎠ ,H= := ⎝ 100 ⎠ . A3 A4 0 0 000 000 Then A is not H-normal, because ker H is not A-invariant. Normal Matrices in Inner Product Spaces 205 However, from (4.4) and (4.5), we immediately obtain ⎧⎛ ⎞ ⎫ ⎪ x1 ⎪ ⎨⎜ ⎟ ⎬ [∗] [∗] [∗] ⎜ w2 ⎟ ∗ [∗] [∗] [∗] A (A ) = ⎪⎝ ⎠ : A2H1x1 =0⎪ =(A ) A . ⎩⎪ x1 + A2x2 ⎭⎪ z2 The following result shows that the set of matrices satisfying (4.3) contains the set of H-normal matrices. Proposition 4.5. Let A ∈ Cn×n be a matrix. If A is H-normal, then A[∗](A[∗])[∗] =(A[∗])[∗]A[∗]. Proof. Without loss of generality let A and H be in the forms (2.6). Then we obtain by Proposition 4.2 that A2 = 0. The identities (4.4) and (4.5) imply ⎧⎛ ⎞⎫ ⎧⎛ ⎞⎫ ⎪ y1 ⎪ ⎪ x1 ⎪ ⎨⎜ ⎟⎬ ⎨⎜ ⎟⎬ [∗] [∗] [∗] ⎜ y2 ⎟ [∗] [∗] [∗] ⎜ x2 ⎟ (A ) A = ⎪⎝ [∗] ⎠⎪ and A (A ) = ⎪⎝ [∗] ⎠⎪ . ⎩⎪ A1A1 y1 ⎭⎪ ⎩⎪ A1 A1x1 ⎭⎪ w2 z2 Indeed, these two sets are equal because of the H1-normality of A1 which is guar- anteed by Proposition 4.2. Next, let us compare H-normal matrices with Moore-Penrose H-normal ma- trices. We obtain the following result. Proposition 4.6. Let A ∈ Cn×n be a matrix. Then the following statements are equivalent. i) A is H-normal, i.e., AA[∗] ⊆ A[∗]A. ii) A is Moore-Penrose H-normal and A[∗](A[∗])[∗] =(A[∗])[∗]A[∗]. Proof. Without loss of generality, let A and H have the forms (2.6). Then the Moore-Penrose generalized inverse of H is given by H−1 0 H† = 1 00 and the matrix A is Moore-Penrose H-normal if and only if ∗ ∗ −1 ∗ A1H1A1 A1H1A2 H1A1H1 A1H1 0 ∗ ∗ = . (4.6) A2H1A1 A2H1A2 00 “i) ⇒ ii)”: If A is H-normal, then by Proposition 4.2 we have that A1 is H1- normal and that A2 = 0. Then (4.6) is satisfied and the remainder follows from Proposition 4.5. “ii) ⇒ i)”: Let A be Moore-Penrose H-normal and let A[∗](A[∗])[∗] =(A[∗])[∗]A[∗]. ∗ ∗ Then by (4.6) we have A2H1A1 =0andA2H1A2 = 0. Comparing this with (4.4), we find that dom A[∗](A[∗])[∗] = Cn.Butthen,wemusthavedom(A[∗])[∗]A[∗] = Cn ∗ m as well which, using (4.5), implies A2H1y1 =0forally1 ∈ C .Fromthis,weobtain [∗] [∗] [∗] [∗] [∗] [∗] [∗] [∗] A2 =0.ButthenA (A ) =(A ) A reduces to A1 A1x1 = A1A1 x1 for 206 C. Mehl and C. Trunk m all x1 ∈ C which implies H1-normality of A1. From this and Proposition 4.2, we finally obtain that A is H-normal. As a consequence, we obtain that the set of H-normal matrices is a strict subset of the set of Moore-Penrose H-normal matrices, because it has been shown in [17, Example 6.1] that there exist Moore-Penrose H-normal matrices A such that ker H is not A-invariant. The fact that the kernel of H is invariant for H-normal matrices allows the generalization of extension results for H-semidefinite invariant subspaces of normal matrices for invertible H to the case of singular H. For example, if A is H-normal and H is invertible, then any H-nonnegative subspace that is invariant for both A and A[∗] can be extended to an A-invariant maximal H-nonnegative subspace, see [15]. This result now easily generalizes to the case of singular H. Here, an invariant subspace U⊆Cn of a linear relation A in Cn is defined by the implication x x ∈U and ∈ A =⇒ y ∈U. y n×n Theorem 4.7. Let A ∈ C be H-normal, and let M0 be an H-nonnegative A-in- variant subspace that is also invariant for A[∗]. Then there exists an A-invariant [∗] maximal H-nonnegative subspace M containing M0 that is also invariant for A . Proof. Without loss of generality assume that A and H are in the forms (2.6) and that M0 can be written as a direct sum M0 = M1+˙ M2,whereM2 ⊆ ker H and x1 ( M1 = : x1 ∈ M1 0 ( m [∗] for some subspace M1 ⊆ C .ItiseasytoverifythattheA-andA -invariance ( [∗] of M0 imply that M1 is A1-invariant as well as A1 -invariant. Thus, by [15] there exists an A1-invariant maximal H-nonnegative subspace Mmax that is also [∗] invariant for A1 . Setting M = Mmax+ker˙ H, we obtain that M contains M0 and is A-invariant and maximal H-nonnegative. It is easy to check that M is also A[∗]-invariant. [∗] If we drop the assumption that M0 is A -invariant, then extension results are not as immediate. Indeed, it has been shown in [18] that there exist H-normal matrices (in the case of invertible H)thathaveaninvariantH-nonnegative sub- space that cannot be extended to an invariant maximal H-nonnegative subspace. (There still exist such counterexamples if one restricts the subspace to be H- positive rather than H-nonnegative.) Thus, stronger conditions have to be imposed on an H-normal matrix such that extension of semidefinite invariant subspaces can be guaranteed, see [18, 19]. We conclude the paper by generalizing a result concerning the extension of H-positive invariant subspaces of H-normal matrices obtained in [18] to the case of singular H.ThefactthatkerH is always an invariant subspace for H-normal matrices plays a key role in this proof. Normal Matrices in Inner Product Spaces 207 n×n Theorem 4.8. Let A ∈ C be H-normal, and let M0 be an H-positive A-in- [⊥] variant subspace. Let Mcom be a direct complement of ker H in M0 , that is, [⊥] M0 = Mcom+ker˙ H. Define A22 := PA|Mcom : Mcom →Mcom, where P is the projection onto Mcom along M0+ker˙ H.ThenMcom is nonde- generate. Equip Mcom with the indefinite inner product induced by H. Assume that [∗] [∗] σ(A22 + A22 ) ⊆ R or σ(A22 − A22 ) ⊆ iR. (4.7) Then there exists an A-invariant maximal H-nonnegative subspace M that con- [∗] tains M0 and that is also A -invariant. The condition (4.7) is independent of the particular choice of a direct com- [⊥] plement Mcom of ker H in M0 . [⊥] Proof. Since M0 is H-positive, we have that M0 is a direct complement of M0. Moreover, it is clear that a complement of ker H is nondegenerate. (By de- fault, i.e., nonexistence of H-neutral vectors, {0} is a nondegenerate subspace.) Thus, without loss of generality, we may assume that M0 =span(e1,...,ek)and [⊥] Mcom =span(ek+1,...,ek+l), l ≥ 0, and M0 =span(ek+1,...,en), where ej denotes the jth unit vector. Then A and H have the corresponding block forms ⎛ ⎞ ⎛ ⎞ A11 A12 A13 Ik 00 ⎝ ⎠ ⎝ ⎠ A = 0 A22 A23 ,H= 0 H22 0 , 0 A32 A33 000 where H22 is invertible. In this representation of A,wehave A11 A12 Ik 0 A1 = , and H1 = , 0 A22 0 H22 where A1 and H1 are defined in analogy to the decomposition (2.6). By Proposi- tion 4.2 we have that A1 is H1-normal and that ker H is A-invariant which implies A13 =0andA23 = 0. Setting ( Ik k+l M0 =ran ⊆ C 0 which is an A1-invariant H1-positive subspace, we obtain, given the condition on A22, that by [18, Theorem 4.4] there exists an A1-invariant maximal H1- . ( nonnegative subspace M0 of dimension ν+(H)=ν+(H1) containing M0.We . choose appropriate matrices M11, M12 and write M0 in the following way . Ik M11 M0 =ran . 0 M12 Let ⎛ ⎞ Ik M11 0 ⎝ ⎠ M := ran 0 M12 0 . 00In−k−l 208 C. Mehl and C. Trunk Then it is straightforward to check that M is an A-invariant, H-nonnegative sub- space of dimension ν+(H)+ν0(H) containing M0. Clearly, M is also invariant for A[∗]. It remains to show that the condition (4.7) is independent of the particular choice of the direct complement Mcom. But choosing a different direct complement [⊥] Mnew of ker H in M0 = span(ek+1,...,en) amounts to a change of basis given by a matrix of the form ⎛ ⎞ Ik 00 ⎝ ⎠ S = 0 S22 0 0 S32 In−k−l n with S22 invertible. With respect to the decomposition C = M0+˙ Mnew+ker˙ H and the new basis, A and H take the forms ⎛ ⎞ ⎛ ⎞ A11 ∗ 0 Ik 00 −1 ⎝ −1 ⎠ ∗ ⎝ ∗ ⎠ A˜ = S AS = 0 S22 A22S22 0 , H˜ = S HS = 0 S22H22S22 0 . 0 ∗ A33 000 −1 ∗ The compressions of A˜ and H˜ to Mnew are S22 A22S22 and S22H22S22. Clearly, condition (4.7) is satisfied for these compressions if and only if (4.7) is satisfied for A22 and H22. Acknowledgment We thank Heinz Langer for asking a question that initiated this research and Peter Jonas for fruitful comments on an earlier version of this paper. References [1] G. Ammar, C. Mehl, and V. Mehrmann, Schur-like Forms for Matrix Lie Groups, Lie Algebras and Jordan Algebras. Linear Algebra Appl. 287 (1999), 11–39. [2] T.Ya. Azizov, Completely Continuous Operators that are Selfadjoint with Respect to a Degenerate Indefinite Metric (Russian). Mat. Issled. 7 (1972), 237–240, 259. [3] T.Ya. Azizov and I.S. Iohvidov, Linear Operators in Spaces with an Indefinite Metric. John Wiley and Sons, Ltd., Chichester, 1989. (Translated from Russian.) [4] V. Bolotnikov, C.K. Li, P. Meade, C. Mehl, and L. Rodman, Shells of Matrices in Indefinite Inner Product Spaces. Electron. J. Linear Algebra 9 (2002), 67–92. [5] R. Cross, Multivalued Linear Operators. Marcel Dekker Inc., 1998. [6] A. Dijksma and H.S.V. de Snoo, Symmetric and Selfadjoint Relations in Krein Spaces I. Oper. Theory Adv. Appl. 24 (1987), 145–166. [7] A. Dijksma and H.S.V. de Snoo, Symmetric and Selfadjoint Relations in Krein Spaces II. Ann. Acad. Sci. Fenn. Math. 12 (1987), 199–216. [8] I. Gohberg, P. Lancaster, and L. Rodman, Matrices and Indefinite Scalar Products. Birkh¨auser, 1983. [9] I. Gohberg, P. Lancaster, and L. Rodman, Indefinite Linear Algebra. Birkh¨auser, 2005. Normal Matrices in Inner Product Spaces 209 [10] I. Gohberg and B. Reichstein, On Classification of Normal Matrices in an Indefinite Scalar Product. Integral Equations Operator Theory 13 (1990), 364–394. [11] M. Kaltenb¨ack and H. Woracek, Selfadjoint Extensions of Symmetric operators in Degenerated Inner Product Spaces, Integral Equations Operator Theory 28 (1997), 289–320. [12] P. Lancaster, A.S. Markus, and P. Zizler, The Order of Neutrality for Linear Oper- ators on Inner Product Spaces. Linear Algebra Appl. 259 (1997), 25–29. [13] P. Lancaster and L. Rodman, Algebraic Riccati Equations. Clarendon Press, 1995. [14] H. Langer, R. Mennicken, and C. Tretter, A Self-Adjoint Linear Pencil Q − λP of Ordinary Differential Operators. Methods Funct. Anal. Topology 2 (1996), 38–54. [15] H. Langer, Invariante Teilr¨aume definisierbarer J-selbstadjungierter Operatoren. Ann. Acad. Sci. Fenn. Ser. A.I. Math. 475 (1971), 1–23 [16] C.K. Li, N.K. Tsing, and F. Uhlig, Numerical Ranges of an Operator on an Indefinite Inner Product Space. Electron. J. Linear Algebra 1 (1996), 1–17. [17] C. Mehl, A. Ran, and L. Rodman, Semidefinite Invariant Subspaces: Degenerate Inner Products. Oper. Theory Adv. Appl. 149 (2004), 467–486. [18] C. Mehl, A. Ran, and L. Rodman, Hyponormal Matrices and Semidefinite Invariant Subspaces in Indefinite Inner Products. Electron. J. Linear Algebra 11 (2004), 192– 204. [19] C. Mehl, A. Ran, and L. Rodman, Extension to Maximal Semidefinite Invariant Subspaces for Hyponormal Matrices in Indefinite Inner Products. Linear Algebra Appl. 421 (2007), 110–116. [20] C. Mehl and L. Rodman, Symmetric Matrices with Respect to Sesquilinear Forms. Linear Algebra Appl. 349 (2002), 55–75. [21] V. Mehrmann, Existence, Uniqueness, and Stability of Solutions to Singular Linear Quadratic Optimal Control Problems. Linear Algebra Appl. 121 (1989), 291–331. [22] B. C. Ritsner, The Theory of Linear Relations. (Russian) Voronezh, Dep. VINITI, No. 846-82, 1982. [23] H. Woracek, Resolvent Matrices in Degenerated Inner Product Spaces. Math. Nachr. 213 (2000), 155–175. Christian Mehl Institut f¨ur Mathematik, MA 4-5 Technische Universit¨at Berlin Str. des 17. Juni 136 D-10623 Berlin, Germany e-mail: [email protected] Carsten Trunk Institut f¨ur Mathematik, MA 6-3 Technische Universit¨at Berlin Str. des 17. Juni 136 D-10623 Berlin, Germany e-mail: [email protected] Operator Theory: Advances and Applications, Vol. 175, 211–224 c 2007 Birkh¨auser Verlag Basel/Switzerland Symmetric Hermite-Biehler Polynomials with Defect Vyacheslav Pivovarchik Abstract. The polynomial ω = P (λ)+iQ(λ)withrealP (λ)andQ(λ)which belongs to Hermite-Biehler class (all its zeros lie in the open upper half-plane) and is symmetric (ω(−λ)=ω(λ)) is modified as follows 2 ˜ 2 ωc(λ)=P˜(λ + c)+iλQˆ(λ + c),c>0. 2 ˜ 2 −1 Here P˜(λ )=P (λ), Qˆ(λ )=λ Q(λ)and ω(λ)+ω(−λ) ω(λ) − ω(−λ) P (λ)= ,Q(λ)= . 2 2i The conditions are obtained necessary and sufficient for a set of complex numbers to be the zeros of a polynomial of the form ωc(λ). Mathematics Subject Classification (2000). Primary 12D10; Secondary 26C10, 30C15. Keywords. Hermite-Biehler polynomial, Hurwitz polynomials, zeros in the lower half-plane. 1. Introduction In [1] the following so-called generalized Regge boundary problem was considered: y − q(x)y = λ2y, x ∈ (0,a), (1.1) y(0) = y (a)+(iλα + β)y(a)=0,α>0,β∈ R. It was shown that the spectrum of this problem coincides with the set of zeros of the characteristic function ϕ(λ)=s (λ, a)+(iαλ + β)s(λ, a), (1.2) where s(λ, x) is the solution of (1.1) which satisfies the conditions s(λ, 0) = s (λ, 0) − 1 = 0. The function ϕ(λ) is symmetric with respect to the imaginary axis, i.e., ϕ(−λ)=ϕ(λ),λ∈ C. 212 V. Pivovarchik This function is of exponential type. Moreover, if the corresponding operator A acting in L2(0,a) defined by 2 D(A)={f : f ∈ W2 (0,a),f(0) = f (a)+βf(a)=0}, Af = −f + q(x)f, is strictly positive, then the function ϕ(λ) belongs to the Hermite-Biehler class. The function ω(λ) of exponential type is said to belong to SHB (symmetric Hermite-Biehler) class if 1) ω(−λ)=ω(λ),λ∈ C (1.3) 2) all its zeros are located in the open upper half-plane and ω(λ) 3) < 1forImλ>0. ω(λ) But in general the operator A is not strictly positive but bounded below only and ϕ(λ) in (1.2) does not belong to SHB. However, it belongs to a more general class defined as follows. Due to the symmetry of the problem ϕ(λ)+ϕ(−λ) P (λ)= , (1.4) 2 ϕ(λ) − ϕ(−λ) Q(λ)= (1.5) 2i and the functions P (λ)andQˆ(λ)=λ−1Q(λ) are even and, therefore 2 ˜ 2 ϕ(λ)=P˜(λ )+iλQˆ(λ ), (1.6) 2 def ˜ 2 def where P˜(λ ) = P (λ)andQˆ(λ ) = Qˆ(λ) are entire functions. The functions of the class which occurred in [1] can be obtained from the symmetric Hermite-Biehler functions (1.6) by shifting: 2 ˜ 2 ϕ(λ, c)=P˜(λ + c)+iλQˆ(λ + c), (1.7) where c is a real constant. Of course, if c<0 the function ω(λ, c) is still symmetric Hermite-Biehler function. But if c>0 is large sufficiently, then ϕ(λ, c)doesnot belong to Hermite-Biehler class. Also it was discovered in [1] that ϕ(λ, c)canhave only a finite number of zeros in the closed lower half-plane. These zeros are purely imaginary and simple. To prove this the authors used the theory of operator poly- nomials (polynomial operator pencils). A certain order was found also in location of the purely imaginary zeros lying in the upper half-plane. It was shown also that these properties of purely imaginary zeros together with the corresponding asymptotics are sufficient for a sequence to be the set of zeros of an entire function of the form (1.2), where ϕ(λ) is symmetric Hermite-Biehler function. The men- tioned order in location of purely imaginary zeros is a common feature of many entire functions occurring in applications (see [2], [3]). A more complicated order is present in location of pure imaginary eigenvalues of boundary problems gen- erated by the fourth order differential equation (see [4]). Here we show that this Symmetric Hermite-Biehler Polynomials with Defect 213 order is valid for pure imaginary zeros of polynomials of the corresponding class. Moreover, we show that the symmetry with respect to the imaginary axis and the mentioned order in location of the zeros (see below) are the characteristic proper- ties of the zeros of a polynomial of the form (1.7). Thus, we obtain a generalization of the Hermite-Biehler theorem (see Theorem 2.3 below) to the case of polyno- mials possessing certain symmetry with respect to the imaginary axis and having pure imaginary zeros in the lower half-plane. It is well known that problem (1.1) is connected with the problem of small transversal vibrations of a nonhomogeneous smooth string with one end fixed and the other end free to move with damping in the direction orthogonal to the equilibrium position of the string. If we assume that the string consists of weightless intervals (threads) joining a finite number of point masses, then the characteristic function will be a polynomial whose degree depends on the number of point masses. 2. Direct problem We are going to obtain a generalization of the well-known Hermite-Biehler theorem [5]. Let us recall it. Definition 2.1. A polynomial is said to be real if it takes on real values on the real axis. Definition 2.2. A polynomial is said to be Hermite-Biehler (HB) if all its zeros lie in the open upper half-plane. It should be mentioned that the transformation λ → iλ transforms a HB polynomial into a so-called Hurwitz polynomial [6]. Theorem 2.3. (Hermite-Biehler theorem, see [5]). In order that the polynomial ω (λ)=P (λ)+iQ(λ) where P (λ) and Q(λ) are real polynomials, not have any zeros in the closed lower half-plane Im λ ≤ 0, i.e., belong to HB, it is necessary and sufficient that the following conditions be satisfied: (1) the polynomials P (λ) and Q(λ) have only simple real zeros, while these ze- ros separate one another, i.e., between two successive zeros of one of these polynomials there lies exactly one zero of the other; (2) at some point λ0 of the real axis Q (λ0)P (λ0) − Q(λ0)P (λ0) > 0. (2.1) The fact that the two polynomials satisfy condition (1) will be expressed by saying that “the zeros of the polynomials P (λ)andQ(λ) are interlaced”. Definition 2.4. The polynomial ω(λ) is said to be symmetric if (1.3) is true. The polynomial ω(λ) is said to belong to the class SHB if it is Hermite-Biehler and symmetric. 214 V. Pivovarchik For a symmetric polynomial ω(λ) the following is valid 2 ˜ 2 ω(λ)=P (λ)+iQ(λ)=P (λ)+iλQˆ(λ)=P˜(λ )+iλQˆ(λ ), (2.2) where P (λ)andQˆ(λ) are real even function. Here P˜(λ2)=P (λ), (2.3) ˜ 2 Qˆ(λ )=Qˆ(λ). (2.4) Lemma 2.5. Let the polynomial ω(λ) of the form (2.2) belong to SHB. Then the polynomial ˜ P˜(λ)+iQˆ(λ), ˜ where P˜ and Qˆ are given by (2.3) and (2.4) belongs to HB. Proof. Due to Theorem 2.3 n n 2 − 2 − n+n 2 − 2 P (λ)= (λ ak)andQ(λ)=( 1) λ (λ bk), 1 1 def def where n = n or n = n − 1, a−k = −ak, b−k = −bk, and the zeros ak and bk interlace as follows: ··· n n ˜ − n 2 ˆ˜ − n+2n 2 P (0) = ( 1) ak, Q(0) = ( 1) bk, 1 1 -n n -n n ˜ − n−1 2 ˆ˜ − n+2n −1 2 P (0) = ( 1) ak, Q (0) = ( 1) bk. k =1 k=1,k=k k =1 k=1,k=k Symmetric Hermite-Biehler Polynomials with Defect 215 Using the above expressions we obtain ⎛ ⎞ -n n n -n n n ˆ˜ ˜ − ˆ˜ ˆ˜ − ⎝ 2 2 − 2 2⎠ Q (0)P (0) Q(0)P (0) = bk ak ak bk k =1 k=1,k=k k=1 k =1 k=1,k=k k=1 ⎛ ⎞ n n -n -n 2 2 ⎝ −2 − −2⎠ = bk ak ak bk . (2.6) k=1 k=1 k=1 k=1 Taking into account (2.5) and the inequality n ≤ n we have -n -n −2 − −2 ak bk > 0. (2.7) k=1 k=1 Now (2.6) and (2.7) imply ˜ ˜ ˜ Qˆ (0)P˜(0) − Qˆ(0)Pˆ (0) > 0. Using again Theorem 2.3 we finish the proof. Corollary 2.6. Let the polynomial ω(λ) of the form (2.2) belong to SHB. Then the polynomial ˜ P˜(λ + c)+iQˆ(λ + c) belongs to HB for every real c. Definition 2.7. Two real polynomials P (λ) and Q(λ) are said to compose a real pair, if they have no common zeros and if any linear combination µP (λ)+νQ(λ) of them with real coefficients µ and ν has no complex zeros. (We refer to not lying on the real axis as complex zeros). By Theorem 4 ([7], Chap. 7, Sec. 2, p. 315) we have the following statement. ˜ Corollary 2.8. The polynomials P˜(λ) and Qˆ(λ) form a real pair and ∂ ˜ ˜ ∂ P˜(λ + c) Qˆ(λ + c) − Qˆ(λ + c) P˜(λ + c) > 0, forλ ∈ R, c ∈ R. (2.8) ∂λ ∂λ To describe the sets of zeros of polynomials of the form (2.2) let us make use of the following definitions introduced in [1]. Definition 2.9. A finite or infinite sequence of complex numbers {ζk}k∈Z or {ζk}0=k∈Z is said to be properly enumerated if 1) Re ζk ≥ Re ζp for all k>p. 2) ζ−k = −ζk for all ζk not purely imaginary. 3) A certain complex number appears in the sequence at most finitely many times. 216 V. Pivovarchik Definition 2.10. Let κ be a nonnegative integer. Then the properly enumerated { }n { }n + − sequence ζk k=−n ( ζk k=−n, k=0) is said to have the SHBκ (resp., SHBκ ) property if 1) All but κ terms of the sequence lie in the open upper half-plane. 2) All terms in the closed lower half-plane are purely imaginary and occur only once. If κ ≥ 1, we denote them as ζ−j = −i|ζ−j| (j =1,...,κ). We assume that |ζ−j| < |ζ−(j+1)| (j =1,...,κ− 1). 3) If κ ≥ 1,thenumbersi|ζ−j| (j =1,...,κ)(with the exception of ζ−1 if it equals zero ) are not terms of the sequence. 4) If κ ≥ 2, then the number of terms in the intervals (i|ζ−j |,i|ζ−(j+1)|)(j = 1,...,κ− 1) is odd. 5) If |ζ−1| > 0, then the interval (0,i|ζ−1|) contains no terms at all or an even number of terms. 6) If κ ≥ 1, then the interval (i|ζ−κ|,i∞) contains a nonzero even (in the case + − of SHBκ ) or odd (inthecaseofSHBκ ) number of terms. + 7) If κ =0, then the sequence has an odd (in the case of SHBκ ) or even or − zero (inthecaseofSHBκ ) number of positive imaginary terms. Now we introduce the class of polynomials to which this paper is devoted. Definition 2.11. Let the polynomial ω(λ) ∈ SHB. Then the polynomial 2 ˜ 2 ω(λ, c)=P˜(λ + c)+iλQˆ(λ + c), ˜ where P˜ and Qˆ are given by (2.3)–(2.4) is said to belong to the class of shifted sym- metric Hermite-Biehler polynomials or in other words to the symmetric Hermite- Biehler class with a defect (SHBc). Lemma 2.12. If the polynomial ω(λ) ∈ SHBc, then it has no real zeros except of possible zero at λ =0. Proof. If 2 ˜ 2 P˜(λ + c)+iλQˆ(λ + c)=0 for real λ =0,then 2 ˜ 2 P˜(λ + c)=Qˆ(λ + c)=0, what contradicts Corollary 2.6. Lemma 2.13. If −iτ (τ ≥ 0)isazeroofω(λ, c) ∈ SHBc at c>0 fixed, then this zero is simple. Proof. If −iτ (τ>0) is a multiple zero of ω(λ, c),then 2 ˜ 2 P˜(−τ + c)+τQˆ(−τ + c)=0, (2.9) ˜ 2 2 ˜ −2iτP˜ (λ) + iQˆ(−τ + c) − 2iτ Qˆ (λ) =0. (2.10) λ=−τ 2+c λ=−τ 2+c Symmetric Hermite-Biehler Polynomials with Defect 217 Combining (2.9) with (2.10) we obtain 2 ˜ 2 2 ˜ 2P˜(−τ + c) P˜ (λ) Qˆ(−τ + c) − P˜(−τ + c) Qˆ (λ) λ=−τ 2+c λ=−τ 2+c ˜3 2 +Qˆ (−τ + c) = 0 (2.11) ˜ 2 ˜ 2 2 ˜ Qˆ(−τ + c) −2τ P˜ (λ) Qˆ(−τ + c) − P˜(−τ + c) Qˆ (λ) . =− 2+ =− 2+ λ τ c λ τ c ˜2 2 +Qˆ (−τ + c) = 0 (2.12) ˜ ˜ 2 As the zeros of P˜(λ)andQˆ(λ) interlace we obtain from (2.9) that Qˆ(−τ + c) =0 for τ>0. Then (2.12) implies ˜ 2 2 ˜ ˜2 2 −2τ P˜ (λ) Qˆ(−τ +c)−P˜(−τ +c) Qˆ (λ) +Qˆ (−τ +c)=0, λ=−τ 2+c λ=−τ 2+c what is impossible for τ>0 due to Corollary 2.8. Let now τ = 0. Then from (2.9) we obtain P˜(c)=0. (2.13) Suppose λ = 0 is a multiple zero. Then (2.11) implies ˜ Qˆ(c)=0 what together with (2.13) contradicts the interlacing conditions. Now we are going to prove that the set of zeros of any polynomial from SHBc + − belongs to SHBκ or SHBκ . First we consider the zeros in the lower half-plane. Lemma 2.14. Let ω(λ, c) ∈ SHBc. Then all the zeros of ω(λ, c) in the closed lower half-plane are purely imaginary. Proof. Let λ0 = −iτ,(τ>0) be a zero of ω(λ, c0), where c0 > 0. The polynomial ω(λ, c)isanalyticinλ for c fixed and analytic in c for λ fixed. Therefore (see for example [8]), the zeros λj (c)ofω are continuous and piece-wise analytic functions of c, i.e., in some neighborhood of c0 the following is true ∞ - k − 1/r λj(c)=λ0 + βk (c c0)j , (j = 1,r), (2.14) k=1 − 1/r where (c c0)j (j = 1,r) means the set of all branches of the root. Here due to the symmetry of the function each of the βj is either real or purely imaginary. When c changes from 0 to c1 > 0 the zeros of ω(λ, c) can come into the lower half-plane from infinity or crossing the real axis. The order of the polynomial ω(λ, c)does not depend on c. Thus, the zeros cannot come to the lower half-plane from infinity. Due to Lemma 2.12 they can cross the real axis only at the origin. Moving along the negative imaginary half-axis the zeros do not collide because at the collision a 218 V. Pivovarchik multiple zero should appear what would contradict Lemma 2.13. The zeros remain purely imaginary due to the symmetry of the problem. Now we are ready to state the main theorem of this section. Theorem 2.15. Let the polynomial ω(λ, c) belong to SHBc. Then the set of its + − zeros belongs to SHBκ (if the order of ω(λ, c) is odd) or to SHBκ (if the√ order of ω(λ, c) is even) with κ equal to the number of zeros of P (λ) lying on (0, c] if c>0 and κ =0if c ≤ 0. Proof. Let us consider the following polynomial of two variables λ and α (c is fixed): 2 ˜ 2 Ω(λ, α)=P˜(λ + c)+iαλQˆ(λ + c). It is clear that Ω(λ, 0) = P˜(λ2 + c)andΩ(λ, 1) = ω(λ, c). To continue the proof of Theorem 2.15 we need the following four lemmas. Lemma 2.16. If Ω(λ, 1) ∈ SHBc,thenΩ(λ, α) ∈ SHBc for all α>0. Proof. It is clear that if P (λ)andQ(λ) satisfy conditions (1) and (2) of Theorem 2.3, then P (λ)andαQ(λ) also satisfy (1) and (2) for all α>0. The assertion of Lemma 2.16 follows. Lemma 2.17. 1) If Ω(0, 0) =0 then Ω(0,α) =0 for all α ∈ R. 2) If Ω(0, 0) = 0 then λ =0is a double zero of Ω(λ, 0) and a simple zero of Ω(λ, α) for all α>0. Proof. The first statement follows from the identity Ω(0,α)=P˜(c). If Ω(0, 0) = 0 then P˜(c) = 0. This zero is double because the function P˜(λ2 + c) is even. The simplicity of λ = 0 as the zero of Ω(λ, α) for all α>0 follows from Lemma 2.13. Lemma 2.18. The number of zeros of Ω(λ, α) in the closed lower half-plane (all the zeros are simple and purely imaginary) does not depend on α for α ∈ (0, ∞). Proof. Let λk(α) be a purely imaginary zero of Ω(λ, α). Then differentiating ˜ 2 ˆ˜ 2 P (λk(α)+c)+iαλk(α)Q(λk(α)+c) = 0 (2.15) with respect to α we obtain − ˆ˜ 2 iλk(α)Q(λk(α)+c) λk(α)= . ˜ 2 ˆ˜ 2 2 ˆ˜ 2 2λk(α)P (λk(α)+c)+iαQ(λk(α)+c)+2iαλk(α)Q (λk(α)+c) (2.16) From (2.15) we express ˜ 2 P(λk(α)+c) λk(α)=i (2.17) ˆ˜ 2 αQ(λk(α)+c) Symmetric Hermite-Biehler Polynomials with Defect 219 and substitute it into (2.16) and obtain ˜2 2 λ (α)=(−iλ (α)Qˆ (λ (α)+c)) k k k ˆ˜ 2 ˜ 2 − ˜ 2 ˆ˜ 2 2λk(α)(Q(λk (α)+c)P (λk(α)+c) P (λk(α)+c)Q (λk(α)+c)) −1 ˆ˜2 2 +iαQ (λk(α)+c) . (2.18) Due to Corollary 2.8 ˆ˜ 2 ˜ 2 − ˜ 2 ˆ˜ 2 Q(λk(α)+c)P (λk(α)+c) P (λk(α)+c)Q (λk(α)+c) < 0.. (2.19) Using (2.19) we obtain from (2.18) that if Reλk(α)=0,Imλk(α) < 0 (we keep α>0), then Re λk(α)=0andImλk(α) > 0. Combining this result with Lemmas 2.13, 2.16 and 2.17 we finish the proof of Lemma 2.18. Let us continue the proof of Theorem 2.15. If λk(0) = 0 then from (2.18) we obtain − ˆ˜2 2 iQ (λk(0) + c) λk(0) = . (2.20) ˆ˜ 2 ˜ 2 − ˜ 2 ˆ˜ 2 2(Q(λk(0) + c)P (λk(0) + c) P (λk(0) + c)Q (λk(0) + c) Due to (2.19) equation (2.20) implies Re λk(0) = 0 and Im λk(0) > 0. Remark 2.19. One can prove Lemmas 2.13, 2.14 and 2.18 using the results of [9] on quadratic operator pencils. Remark 2.20. If α>0 then the denominator in (2.18) is equal to zero if and only if the corresponding purely imaginary zero λk(α) is multiple (see the proof of Lemma 2.13). All the other (simple) purely imaginary zeros can be classified according to the sign of the mentioned denominator. An analogue of such a classification for eigenvalues of quadratic operator pencils can be found in [10]. Definition 2.21. The purely imaginary zero λk of a symmetric polynomial is said to be of the first (second) kind if − ˆ˜ 2 ˜ 2 − ˜ 2 ˆ˜ 2 ˆ˜2 2 2iλk(Q(λk + c)P (λk + c) P (λk + c)Q (λk + c)) + Q (λk + c) > 0(< 0). Remark 2.22. The purely imaginary zeros in the open lower half-plane are all of the first kind and according to (2.18) and (2.19) they all move upwards when α>0 grows. The purely imaginary zeros of the first (second) kind in the open upper half-plane move downwards (upwards). Lemma 2.23. If −iτ (τ>0) isazeroofω(λ, c) ∈ SHBc then iτ is not a zero of it. Proof. Suppose the both −iτ and iτ to be zeros of ω(λ, c). Then 2 ˜ 2 2 ˜ 2 P˜(−τ + c)+τQˆ(−τ + c)=0 and P˜(−τ + c) − τQˆ(−τ + c)=0 2 ˜ 2 and, consequently, P˜(−τ + c)=Qˆ(−τ + c) = 0, what is impossible. 220 V. Pivovarchik Let us continue proving Theorem 2.15. Taking into account the symmetry of the problem on reflection with respect to the imaginary axis, we have λ−k(α)=−λk(α) for all not purely imaginary λ−k(α)withα ≥ 0, and hence new zeros can appear on the imaginary axis in pairs, which implies the statements of Theorem 2.15. 3. Inverse problem Let us consider the inverse theorem, i.e., we are going to prove that the condition of Theorem 2.15 is not only sufficient but also necessary. { }n { }n − + Theorem 3.1. Let the set λk k=−n, k=0 ( λk k=−n) belong to SHBκ (SHBκ ) with κ ≥ 0 Then the polynomial & ' n n ϕ(λ)= (λ − λk) ϕ(λ)= (λ − λk) (3.1) k=−n, k=0 k=−n belongs to SHBc,withsomec>0. { }n ∈ − Proof. Let λk k=−n, k=0 SHBκ . Consider the auxiliary polynomial n − (0) − (0) ϕ0(λ)= (λ λk )(λ λ−k), k=1 (0) ∈ where λk = k + i , >0. It is clear that ϕ0 SHB. Put ϕ0(λ)+ϕ0(−λ) P0(λ)= , (3.2) 2 ϕ0(λ) − ϕ0(−λ) Qˆ0(λ)= . (3.3) 2iλ Then P0(λ)andQˆ0(λ) are both even polynomials. Let us introduce the following real polynomials √ P˜0(λ)=P0( λ), (3.4) √ ˜ Qˆ0(λ)=Qˆ0( λ). (3.5) { (0)}n { (0)}n ˆ Since the zeros ak k=−n, k=0 of P0(λ) and the zeros bk k=−n, k=0 of Q0(λ) interlace, we have (0) (0) (0) (0) (0) (0) (0) (0) ··· Proof. If there are not purely imaginary λk then there exists ∈ N ∪{0} such that Re λs > 0fors ≥ +1 andReλs < 0fors ≤−( +1).Forsuchλs we define (0) − (0) λs(t)=λs +(λs λs )t. So it remains to construct the piecewise analytic functions involving the purely imaginary eigenvalues λs (0 < |s|≤). Now let λ−j be a (purely imaginary) term in the closed lower half-plane. Consider the interval (i|λ−j|,i|λ−(j+1)|). Then the number of terms on this interval is odd. Let us choose one of them and denote it by λ+j, thus resulting in the positive imaginary, so-called principal, eigenvalues λ+1,...,λ+κ. The remaining positive imaginary eigenvalues, which are even in number when multiplicities are taken into account, are grouped into pairs {λ+s,λ−s} (s = κ +1,...,) such that each of the intervals (i|λ−j|,i|λ−(j+1)|) contains an integer number of such pairs. (0) Let us move the points λs (0 < |s|≤) in various steps. First, we define for 0 ≤ t ≤ (1/(κ +2)) ⎧ ⎪ (0) | | | | ⎨⎪ λj , 0 < j < κ , (0) λ−κ+λ+κ (0) λj(t)= λ± +(κ +2)t − λ± ,j= ±κ, ⎪ κ 2 κ ⎩ (0) | | | |≤ λj , κ < j . − ∈ 1+s 2+s Next, for s =0,...,κ 2andt [ κ+2 , κ+2 ] we define ⎧ λj +λ−j − − λj −λ−j ± − ⎪ 2 +((κ +2)t s 1) 2 ,j = (κ s), ⎨⎪ (0) + (0) − − λj λ−j − ± − − λj +((κ +2)t s 1) 2 λj ,j= (κ s 1), λj(t)= ⎪ (0) | | − − ⎩⎪ λj , 0 < j <κ s 1, λj,κ− s<|j|≤. ∈ κ κ+1 Next, for t [ κ+2 , κ+2 ] we define ⎧ ⎪ λ1+λ−1 λ±1−λ∓1 ⎨ 2 +((κ +2)t − κ) 2 ,j= ±1, λ , 1 < |j|≤κ, λj(t)=⎪ j ⎩ (0) − λj +λ−j − (0) | |≤ λj +((κ +2)t κ) 2 λj ,κ Continuing the proof of Theorem 3.1, let us construct the function n ϕ(λ, t)= (λ − λk(t)). (3.7) k=−n, k=0 222 V. Pivovarchik Then n − (0) ϕ(λ, 0) = (λ λk ), k=−n, k=0 (3.8) n ϕ(λ):=ϕ(λ, 1) = (λ − λk). k=−n, k=0 Put ϕ(λ, t)+ϕ(−λ, t) P (λ, t)= , (3.9) 2 ϕ(λ, t) − ϕ(−λ, t) Qˆ(λ, t)= , (3.10) 2iλ and then define P (λ)=P (λ, 1) and Qˆ(λ)=Qˆ(λ, 1). Next, put √ P˜(λ, t)=P ( λ, t), √ ˜ Qˆ(λ, t)=Qˆ( λ, t), ˜ ˜ and then define P˜(λ)=P˜(λ, 1) and Qˆ(λ)=Qˆ(λ, 1). { }n { }n Denote by ak(t) k=−n, k=0 the set of zeros of P (λ, t)andby bk(t) k=−n, k=0 ˆ { 2}n ˜ { 2}n the set of zeros of Q(λ, t). Then ak(t) k=1 are the zeros of P (λ, t)and bk(t) k=1 ˜ are the zeros of Qˆ(λ, t). ∈ { 2}n Proposition 3.3. For any fixed t [0, 1], the sets of zeros ak(t) k=1 and { 2}n bk(t) k=1 interlace, i.e., 2 2 2 2 2 −∞ ϕ(bk(t1),t1)=ϕ(−bk(t1),t1)=0, where bk(t1) is real or purely imaginary. Since Lemma 2.12 implies that ϕ(λ, t) can have a real zero only at the origin for any t ∈ [0, 1], we conclude that bk(t1) is purely imaginary or zero. Since P (λ, t)andQˆ(λ, t) are even functions of λ,we have P (bk(t1),t1)=P (−bk(t1),t1)=0 and Qˆ(bk(t1),t1)=Qˆ(−bk(t1),t1)=0, Symmetric Hermite-Biehler Polynomials with Defect 223 and consequently ϕ(bk(t1),t1)=ϕ(−bk(t1),t1) = 0, which contradicts Proposition 3.2 if bk(t1) = 0. On the other hand, if bk(t1) = 0, then it is a double zero, because ˜ it is a zero of the even functions P˜(λ, t1)andQˆ(λ, t1) and consequently a zero of ω(λ, t1), which contradicts Proposition 3.2. Let us continue proving Theorem 3.1. Let us choose c such that 2 a1(1) >c. (3.12) Let us consider the polynomial 2 ˜ 2 ω(λ, c)=P˜(λ + c, 1) + iλQˆ(λ + c, 1). ) ˜ 2 ˆ˜ 2 { 2 − }n The) zeros of P (λ + c, 1) and of Q(λ + c, 1) are ak(1) c k=−n, k=0 and { 2 − }n ak(1) c k=−n,k=0, respectively. They interlace as follows − 2 − − 2 − − 2 − ··· − 2 − b−n(1) c< a−n(1) c< b−n+1(1) c< < a−1(1) c ) 2 − ··· 2 − < 0 < a1(1) c< < bn(1) c. (3.13) Thus, ω(λ, c) fulfils condition (1) of Theorem 2.3. To show the validity of condition (2) for ω(λ, c)wenoticethat ∂ ˜ 2 ˜ λQˆ(λ + c, 1) = Qˆ(c, 1), ∂λ λ=0 ˜ 2 λQˆ(λ + c, 1) =0. λ=0 Using these equations we obtain 2 ∂ ˜ 2 ˜ 2 ∂ 2 P˜(λ + c, 1) λQˆ(λ + c, 1) − λQˆ(λ + c, 1) P˜(λ + c, 1) ∂λ ∂λ λ=0 ˜ = Qˆ(c, 1)P˜(c, 1). Using (3.12) and (3.13) we obtain ˜ Qˆ(c, 1)P˜(c, 1) > 0 and therefore 2 ∂ ˜ 2 ˜ 2 ∂ 2 P˜(λ + c, 1) λQˆ(λ + c, 1) − λQˆ(λ + c, 1) P˜(λ + c, 1) > 0. ∂λ ∂λ λ=0 That means condition (2) of Theorem 2.3 is satisfied too. The first statement of Theorem 3.1 follows. In the same way one can prove the second statement, i.e., { }n ∈ + that if λk −n SHBκ ,then n − ∈ + ϕ(λ)= (λ λk) SHBc . k=−n, k=0 224 V. Pivovarchik Acknowledgment This work is partially supported by Grants UM1-2567-OD-03 and UKM2-2811- OD-06 of Civil Research and Development Foundation. References [1] V. Pivovarchik, C. van der Mee, The inverse generalized Regge problem. Inverse Problems, 17 (2000), 1831–1845. [2] C. van der Mee, V. Pivovarchik, A Sturm-Liouville Spectral Problem with Bound- ary Conditions Depending on the Spectral Parameter. Functional Analysis and Its Applications, 36, No. 4 (2002), 315–317. [3] C. van der Mee, V. Pivovarchik, The Sturm-Liouville inverse spectral problem with boundary conditions depending on the spectral parameter. To appear in Opuscula Mathematica. [4] M. M¨oller, V. Pivovarchik, Spectral properties of a forth order differential equation. To appear in Zeitschrift f¨ur Analysis und ihre Anwendungen. [5] C. Hermite, Extract d’une lettre de M. Ch. Hermite de Paris `a Mr. Borchardt de Berlin sur le nombre des racines d’une ´equation alg´ebrique comprises entre des lim- ites donn´ees. J. Reine Angew. Math. 52 (1856), 39–51; reprinted in his Oeuvres, Vol. 1, Gauthier-Villars, Paris, 1905, 397–414. [6] F.R. Gantmaher, Theory of Matrices (in Russian). Nauka, Moscow, 1988. [7] B.Ja. Levin, Distribution of Zeros of Entire Functions, Trans. Math. Monographs. 5, Amer. Math. Soc., Providence, RI, 1980. [8] A.I. Markushevich, Theory of Analytic Functions, (in Russian), 1, Nauka, Moscow, 1968. [9] V. Pivovarchik. On Positive Spectra of One Class of Polynomial Operator Pencils. Integral Equations and Operator Theory, 19 (1994), 314–326. [10] A.G. Kostyuchenko, M.B. Orazov, The problem of oscillations of an elastic half- cylinder and related self-adjoint quadratic pencil (in Russian). Trudy Sem. Petrovsk., N6 (1981), 97–147. Vyacheslav Pivovarchik Preobrazhenskaya str. 59/61, a.17 65045 Odessa, Ukraine e-mail: [email protected] Operator Theory: Advances and Applications, Vol. 175, 225–229 c 2007 Birkh¨auser Verlag Basel/Switzerland A Note on Indefinite Douglas’ Lemma Leiba Rodman Abstract. The Douglas lemma on majorization and factorization of Hilbert space operators is extended to the setting of Krein space operators. Mathematics Subject Classification (2000). 47A68, 47A63, 46C20. Keywords. Majorization, factorization, Hilbert space operators, Krein spaces. 1. Introduction and main results We let E, F, G be (complex) Hilbert spaces, with the inner products (x, y)E , (x, y)F ,and(x, y)G , respectively, and let L(E, F) denote the Banach space of all linear bounded operators acting from E into F. The celebrated Douglas lemma, or more precisely one of its several versions, states that if H ∈L(E, F)andG ∈L(E, G)aresuchthatH∗H ≥ G∗G,in other words, H∗H − G∗G is positive semidefinite, then there exists an operator F ∈L(F, G) satisfying Gx = FHx for every x ∈F. Although generally speaking the operator F is not unique, it can be always chosen so that F ≤1. The original reference for the Douglas lemma is [2]. It has been extensively used in operator theory, in particular in studies of division and quotients of operators (see [8, 4]), operator range inclusions [3], and operator inequalities [5]; see, e.g., [6] for connections with lifting and Leech’s theorem. A survey of results around the Douglas lemma in the context of Banach spaces is given in a recent paper [1]. In this note we extend the Douglas lemma to the setting of Krein space operators. Yu. L. Smul’janˇ [7, 9] was the first who obtained some interesting results in this direction; in particular, he pointed out that generally the Douglas lemma does not hold verbatim in Krein spaces, even the finite-dimensional ones. In what follows, for a bounded selfadjoint operator X,wedenotebyi−(X)the dimension (finite or infinite) of the spectral X-invariant subspace corresponding to the negative part of σ(X). Let JE ∈L(E, E), JF ∈L(F, F), JG ∈L(G, G) be operators that are simulta- neously selfadjoint and unitary, and such that i−(JG ) < ∞. The operators JE , JF , 226 L. Rodman and JG introduce the Krein space structure in E, F,andG, using the indefinite inner products ∈E ∈F [x, y]JE =(JE x, y)E ,x,y ;[x, y]JF =(JF x, y)F ,x,y ; ∈G [x, y]JG =(JG x, y)G,x,y ; ∈LE E ≥ respectively. An operator Q ( , ) is called JE -nonnegative if [Qx, x]JE 0for all x ∈E.The(JF ,JE )-adjoint of an operator H ∈L(E, F) is by definition the ∗ unique operator H JF ,JE ∈L(F, E) with the property that ∗ JF ,JE ∀ ∈E ∈F [Hx,y]JF =[x, H y]JE , x ,y . We now state the extension of the Douglas lemma to the Krein space operators. Theorem 1.1. Let H ∈L(E, F), G ∈L(E, G),andV ∈L(E, E) be such that the operator ∗ ∗ H JF ,JE H − G JG ,JE G + V (1.1) is JE -nonnegative and V has finite rank. Then there exists an operator F ∈L(F, G) such that F ≤1 and Gx = FHx for every x ∈N, (1.2) where N is a subspace of E of codimension at most i−(JG )+rankV. It is of interest to find out when (1.2) holds on the whole of E, not only on a certain subspace N . A criterion for this to happen is given in the next theorem. Theorem 1.2. Assume the hypotheses of Theorem 1.1. Then there exists an operator F ∈L(F, G) such that Gx = FHx for every x ∈Eif and only if the following { }∞ E → condition is satisfied: If xm m=1 is a sequence of vectors in such that Hxm Hf and Gxm → q for some vectors f and q, then necessarily q = Gf. In the next section we develop preliminary results for the proofs of Theorems 1.1 and 1.2. 2. Preliminary results Lemma 2.1. Let there be given operators H ∈L(E, F) and G ∈L(E, G), such that ∗ ∗ κ := i−(H H − G G) < ∞. (2.3) Then there exists an operator F ∈L(F, G) such that Gx = FHx for every x ∈N, (2.4) where N is a (closed) subspace of E of codimension at most κ. Moreover, F can be chosen with the additional property that F ≤1. Conversely, if (2.4) holds for some F ∈L(F, G), F ≤1, and some subspace N⊆Eof codimension κ<∞,then ∗ ∗ i−(H H − G G) ≤ κ. (2.5) Indefinite Douglas’ Lemma 227 Proof. We prove first the converse statement. If Gx = FHxfor every x ∈N,where F ≤1 and codim N = κ<∞,then H∗H − G∗G = H∗(I − F ∗F )H ≥ 0 (2.6) on N . If (2.5) were false, then we would find a (κ + 1)-dimensional subspace N0 ∗ ∗ such that ((H H − G G)y,y)E > 0 for every y ∈N0 \{0}.SinceN0 ∩N = {0}, we obtain a contradiction with (2.6). The direct statement follows upon applying the Douglas lemma to the oper- ators G and H restricted to the kernel of the (finite rank) orthogonal projection on the spectral subspace of H∗H − G∗G corresponding to the negative part of σ (H∗H − G∗G). The Douglas lemma shows that without additional hypotheses, one cannot expect that the operator F of Lemma 2.1 be defined on the whole of E. Indeed, such a H exists if and only if λH ∗H − G∗G ≥ 0 for some positive λ.Othertype of conditions are specified in the next result. Lemma 2.2. Under the hypotheses of Lemma 2.1, there exists an operator F ∈ L(F, G) such that Gx = FHx for every x ∈E if and only if the following property (A) holds true: { }∞ E → → (A): If xm m=1 is a sequence of vectors in such that Hxm Hf and Gxm q for some vectors f and q, then necessarily q = Gf. Proof. It is easy to see that the hypothesis (A) is necessary for existence of an operator F ∈L(F, G) such that Gx = FHx for every x ∈E. Assume now that (A) holds true. By Lemma 2.1, let N be the subspace of codimension at most κ such that Gx = F0Hx for every x ∈N, F0 ∈L(F, G). If for some f ∈N⊥ we have Hf ∈ closure of {Hx| x ∈N}, in other words, Hf = lim (Hxm),xm ∈N, m→∞ then F0Hf = F0( lim (Hxm)) = lim (F0Hxm) = lim Gxm = Gf, m→∞ m→∞ m→∞ where the latter equality follows in view of (A). Thus, Gx = F0Hx holds also for x ∈N0 := {x ∈E|Hx ∈ closure of {Hy| y ∈N}}. N N E Clearly, 0 is closed. If 0 = , we are done. Otherwise, let ξ1,...,ξp be a basis E N p ∈ C for modulo 0, and for every linear combination j=1 αjξj, αj ,let -p -p F (H( αjξj)) = G( αjξj). (2.7) j=1 j=1 228 L. Rodman Now define F = F0 on the closure of {Hy| y ∈N}, and define F as in (2.7) on {Hy| y ∈ Span {ξ1,...,ξp}}. The definition is correct, because, as one checks easily, {Hy| y ∈N}∩{Hy| y ∈ Span {ξ1,...,ξp}} = {0}. It follows that we have G = FH on E. 3. Proofs of Theorems 1.1 and 1.2. We start with Theorem 1.1. Using the easily verifiable equalities ∗J ,J −1 ∗ ∗J ,J −1 ∗ H F E = JE H JF ,GG E = JE G JG , the condition that the operator Q given by the left-hand side of (1.1) is JE - nonnegative takes the form ∗ ∗ H JF H − G JGG + JE V ≥ 0. Rewrite this inequality: ∗ ∗ ∗ ∗ H H + G G ≥−JE V + H (I − JF )H − G (I − JG)G. ∗ Note that H (I − JF )H ≥ 0and ∗ ∗ i−(−JE V − G (I − JG )G) ≤ rank V + i−(G (JG − I)G) ≤ rank V + i−(JG ). Thus, ∗ ∗ i−(H H + G G) ≤ rank V + i−(JG ), and it remains to apply Lemma 2.1. The proof of Theorem 1.2 follows the same line of argument, using Lemma 2.2. Acknowledgments The author is grateful to V. Bolotnikov for constructive discussions concerning the subject matter of this note, and to T. Azizov for useful suggestions and for pointing out the references [7, 9]. The research is partially supported by NSF Grant DMS-0456625. References [1] A.B. Barnes. Majorization, range inclusion, and factorization for bounded linear operators. Proc.Amer.Math.Soc.133 (2005), 155–162. [2] R.G. Douglas. On majorization, factorization, and range inclusion of operators on Hilbert space. Proc.Amer.Math.Soc.17 (1966), 413–415. [3] P.A. Fillmore and J.P. Williams. On operator ranges. Advances in Math. 7 (1971), 254–281. [4] S. Izumino. Quotients of bounded operators. Proc.Amer.Math.Soc.106 (1989), 427–435. [5] C.-S. Lin. On Douglas’s majorization and factorization theorem with applications. Int. J. Math. Sci. 3 (2004), 1–11. Indefinite Douglas’ Lemma 229 [6] M. Rosenblum and J. Rovnyak. Hardy classes and operator theory.OxfordUniversity Press, New York, 1985. [7] Ju.L. Smul’jan.ˇ Non-expanding operators in a finite-dimensional space with an in- definite metric. Uspehi Mat. Nauk, 18 (1963), 225–230. (Russian.) [8] Ju.L. Smul’jan.ˇ Two-sided division in the ring of operators. Mat. Zametki, 1 (1967), 605–610. (Russian). [9] Ju.L. Smul’jan.ˇ Division in the class of J-expansive operators. Matem. Sbornik (N.S.), 74 (116) (1967), 516–525. (Russian.) Leiba Rodman Department of Mathematics The College of William and Mary Williamsburg VA 23187-8795, USA e-mail: [email protected] Operator Theory: Advances and Applications, Vol. 175, 231–240 c 2007 Birkh¨auser Verlag Basel/Switzerland Some Basic Properties of Polynomials in a Linear Relation in Linear Spaces Adrian Sandovici Abstract. The behavior of the domain, the range, the kernel and the multi- valued part of a polynomial in a linear relation is analyzed, respectively. Mathematics Subject Classification (2000). Primary 47A05; Secondary 47A06. Keywords. Polynomial, linear space, linear relation, domain, range, kernel, multivalued part. 1. Introduction Let S be a linear relation in a linear space H over the field K of real or complex numbers, let n and mi,1≤ i ≤ n be some positive integers, and let λi ∈ K, 1 ≤ i ≤ n be some distinct constants. Then, the polynomial p in S given by n mi p(S)= (S − λi) (1.1) i=1 is a linear relation in H too. Certain properties of polynomials in a linear relation have been obtained long time ago (see for instance [1, 4, 5]). However, it seems that the behavior of the domain, the range, the kernel and the multivalued part of p(S) has not yet been described, cf. [3]. It is the goal of this technical note to cover this gap. For instance, if α and β are two constants and p and q are two positive integers, then the results in this note show that dom (S − α)p(S − β)q =domSp+q, (1.2) and mul (S − α)p(S − β)q =mulSp+q. (1.3) If additionally, α = β then also ran (S − α)p(S − β)q =ran(S − α)p ∩ ran (S − β)q, (1.4) 232 A. Sandovici and ker (S − α)p(S − β)q =ker(S − α)p + ker (S − β)q. (1.5) The results in this note generalize and complete some algebraic ones in [2]. The notion of standard symmetric linear relation has been introduced in [2], and in connection with this notion, a von Neumann like formula has been proved for the case of standard symmetric linear operators. The main ingredients of the proof of that formula are some descriptions of the range and the kernel of certain polyno- mial with simple squares in a linear operator. The results proved in this note will be used in [6] in order to develop an extension theory of standard symmetric linear relations in Pontryagin spaces. A generalization of the von Neumann like formula in [2] will be also stated. Next the contents of this note are shortly presented. For the convenience of the reader Section 2 contains some basic notions concerning linear relations in linear spaces. Also, certain elementary results about the commutation of linear relations are presented. The main results are proved in Section 3. 2. Linear relations in linear spaces A linear relation A in a linear space H is a linear subspace of the space H × H,the Cartesian product of H and itself. All linear spaces in this paper are assumed to be over the filed K of real or complex numbers. 2.1. Basic definitions and properties of linear relations The notations dom A and ran A denote the domain and the range of A, defined by dom A = { x : {x, y}∈A }, ran A = { y : {x, y}∈A }. Furthermore, ker A and mul A denote the kernel and the multivalued part of A, defined by ker A = { x : {x, 0}∈A }, mul A = { y : {0,y}∈A }. A linear relation A is the graph of an operator if and only if mul A = {0} and the inverse A−1 is given by {{y,x} : {x, y}∈A }. The following identities express the duality between A and its inverse A−1: dom A−1 =ranA, ran A−1 =domA, ker A−1 =mulA, mul A−1 =kerA. For linear relations A and B in a linear space H the operator-like sum A + B is the linear relation in H defined by A + B = {{x, y + z} : {x, y}∈A, {x, z}∈B }. For λ ∈ K the linear relation λA in H is defined by λA = {{x, λy} : {x, y}∈A }, while A − λ stands for A − λI,whereI is the identity operator on H.From A − λ = {{x, y − λx} : {x, y}∈A} it follows that ker (A − λ)={ x : {x, λx}∈A }. Polynomials in a Linear Relation 233 Furthermore, the following equality holds (A − λ)+λI = A for all λ ∈ K. (2.1) For linear relations A and B in a linear space H the product AB is defined as the relation AB = {{x, y} : {x, z}∈B,{z,y}∈A for some z ∈ H }. Clearly, the following inclusions hold dom AB ⊂ dom B, ran AB ⊂ ran A, (2.2) and ker B ⊂ ker AB, mul A ⊂ mul AB. (2.3) For λ ∈ K the notation λA agrees in this sense with (λI)A. The product of relations is clearly associative. Hence An, n ∈ Z,isdefinedasusualwithA0 = I and A1 = A. Then for all n ∈ N ∪{0} dom An+1 ⊂ dom An, ran An+1 ⊂ ran An, (2.4) ker An+1 ⊃ ker An, mul An+1 ⊃ mul An, (2.5) and for all p, k ∈ N ∪{0} ker Ap ⊂ dom Ak, mul Ap ⊂ ran Ak. (2.6) 2.2. Some commutation properties of linear relations Let A and B be two linear relations in a linear space H. Assume that A and B commute, i.e., AB = BA in the sense of the product of linear relations. It is easy to see that (A − α)B = B(A − α) (2.7) holds true for all α ∈ K, and that the following equality AnBm = BmAn (2.8) holds for all n, m ∈ N ∪{0}. Applying (2.7) with B − β instead of B, it follows that (A − α)(B − β)=(B − β)(A − α). Using (2.8) with A − α and B − β instead of A and B it follows that (A − α)n(B − β)m =(B − β)m(A − α)n (2.9) for all α, β ∈ K,andforalln, m ∈ N ∪{0}. Furthermore, the next result is straightforward; it is a generalization of [3, Proposition VI.5.1]. Corollary 2.1. Let S be a linear relation in a linear space H,letn be a positive number, let λi ∈ C, 1 ≤ i ≤ n,andletmi ∈ N, 1 ≤ i ≤ n. Then the identity n n mi mσ(i) (S − λi) = (S − λσ(i)) , (2.10) i=1 i=1 holds for every permutation σ of the set {1,...,n}. 234 A. Sandovici 3. Main results This section contains the main results of this note. They represent the generaliza- tion of (1.2)–(1.5) to the case of a polynomial in a linear relation S in a linear space H. Lemma 3.1. Let S be a linear relation in a linear space H and let λ ∈ K.Then dom (S − λ)n =domSn for all n ∈ N. (3.1) Proof. If λ = 0 then (3.1) is obvious. Assume that λ = 0 and proceed by induction. n The case n = 1 is clear. Assume now that n>1andletx0 ∈ dom (S − λ) ,so { }∈ − ∈ H ≤ ≤ − that xi,xi+1 S λ for some xi ,0 i n 1. Define the vectors yp, ≤ ≤ p p ≤ ≤ 0 p n by yp = j=1 j λj xp−j,0p n,sothat -p p p−j {y ,y +1} = λ {x ,x +1 + λx }∈S, 0 ≤ p ≤ n, p p j j j j j=0 n n which implies that {y0,yn}∈S . Hence, x0 = y0 ∈ dom S , which shows that dom (S − λ)n ⊂ dom Sn. The converse inclusion follows using similar arguments. Theorem 3.2. Let S be a linear relation in a linear space H,letn ∈ N, λj ∈ K, mj ∈ N, 1 ≤ j ≤ n. Assume that λj, 1 ≤ j ≤ n are distinct. Then n dom p(S)=domS j=1 mj . (3.2) Proof. Proceed by induction. The case n = 1 follows from Lemma 3.1. Assume ≥ k ∈ that (3.2) holds true for some n = k 1 and denote Mk = j=1 mj.Letx0 / +1 / k − mj { }∈ − mk+1 { }∈ k − dom j=1 (S λj) ,sothat x0,y0 (S λk+1) and y0,z0 j=1(S mj Mk λj) for some y0, z0 ∈ H. By induction hypothesis it follows that y0 ∈ dom S , so that {yj,yj+1}∈S,0≤ j ≤ Mk − 1forsomeyi ∈ H,1≤ j ≤ Mk. It follows mk+1 from {x0,y0}∈(S − λk+1) that {xp,xp+1}∈S − λk+1, 0 ≤ p ≤ mk+1 − 1 ∈ H ≤ ≤ for some xp ,1 p mk+1, with xmk+1 = y0.Thisimpliesthat {xp,xp+1 + λk+1xp}∈S, 0 ≤ p ≤ mk+1 − 1. Furthermore, define the vectors wp ∈ H,0≤ p ≤ mk+1 by -p p j w = λ x − , 0 ≤ p ≤ m +1, p j k+1 p j k j=0 so that -p p p−j {w ,w +1} = λ {x ,x +1 + λ +1x }∈S, 0 ≤ p ≤ m +1 − 1. p p j k+1 j j k j k j=0 Polynomials in a Linear Relation 235 ∈ H ≤ ≤ Define also the vectors wmk+1+q ,1 q Mk by −1 mk-+1 -q − − mk+1 + q mk+1+q−j mk+1 1+q i q−i wmk+1+q = λk+1 xj + λk+1yi, j m +1 − 1 j=0 i=0 k so that mk-+1−1 mk+1 + q mk+1+q−j {w + ,w + +1} = λ {x ,x +1 + λ +1x } mk+1 q mk+1 q j k+1 j j k j j=0 -q − − mk+1 1+q i q−i { }∈ + λk+1 yi,yi+1 S m +1 − 1 i=0 k ≤ ≤ − { }∈ mk+1+Mk for all 0 q Mk 1. Hence, w0,wmk+1+Mk S ,whichleadsto k+1 mk x0 = w0 ∈ dom S j=1 . This shows that k+1 k+1 mj j=1 mj dom (S − λj) ⊂ dom S . j=1 Thus, the induction is complete. The converse inclusion will be next proved. Denote M = n m and let µ ,1≤ j ≤ M be the λ ’s counted with their multiplicities, j/=1 j j / i qp M − n − mj ≤ ≤ so that j=1(S µj)= j=1(S λj) . Define the constants ηp ,0 p M, 0 0 ≤ q ≤ p by η0 =1forp =0,and p - p p−qp − qp ··· ≤ ≤ ηp =1,ηp =( 1) µi1 µiqp , 1 qp p 1≤i1<··· Theorem 3.3. Let S be a linear relation in a linear space H,letn ∈ N, λj ∈ K, mj ∈ K, 1 ≤ j ≤ n. Assume that λj, 1 ≤ j ≤ n are distinct. Then n mi ran p(S)= ran (S − λi) . (3.3) i=1 Proof. In order to prove that n n mi mi ran (S − λi) ⊂ ran (S − λi) , (3.4) = =1 / i i i ∈ n − mi let y ran i=i(S λi) ,andletj be arbitrarily chosen/ between 1 and n. ∈ − mj n − mi Applying Corollary 2.1 it follows that y ran (S λj) i=1,i=j(S λi) .By mj (2.2) it follows that y ∈ ran (S − λj ) for all 1 ≤ j ≤ n. Thus, (3.4) holds true. 236 A. Sandovici The converse inclusion in (3.3) will be next proved by induction on n.The case n = 1 is obvious. Assume that the inclusion n n mi mi ran (S − λi) ⊂ ran (S − λi) (3.5) i=1 i=i holds for some n = k and it will be proved for n = k + 1. For simplicity denote +1 ∈ k − mi λ = λk+1 and m = mk+1. Assume that y i=1 ran (S λi) ,sothat k k mi m mi m y ∈ ran (S − λi) ran (S − λ) ⊂ ran (S − λi) ran (S − λ) . i=1 i=i Since y ∈ ran (S − λ)m it follows that {zi+1,zi}∈S − λ, 0 ≤ i ≤ m − 1 / ∈ H ≤ ≤ ∈ k − mi for some vectors zi ,0 i m with z0 = y.Also,y ran i=1(S λi) implies that mi {xi,xi+1}∈(S − λi) , 1 ≤ i ≤ k, (3.6) for some vectors xi ∈ H,1≤ i ≤ k +1,withxk+1 = y.Furthermore,itfollows from (3.6) that { }∈ − ≤ ≤ ≤ ≤ xp,qp ,xp,qp−1 S λp, 1 p k, 1 qp mp (3.7) ∈ H ≤ ≤ for some vectors xp,qp ,withxk,0 = y, xp,mp = xp,1 p k,andxp−1,0 = ≤ ≤ p,qp ≤ ≤ ≤ ≤ ≤ ≤ xp,mp ,2 p k. Define the constants ξt ,1 p k,1 qp mp,1 t m by − − 1,q1 t m1 q1 + t 1 −(m1−q1+t) ξ =(−1) (λ − λ1) , t t − 1 for p =1,and p−1 −1 -t m − q + l ξp,qp =(−1)t (λ − λ )−mj (λ − λ )−(mp+1−qp) p p ψp (λ − λ )−l t j p l t,l p j=1 l=0 for p>1, where p−1 - m − 1+r p j j − −rj ≤ ≤ − ψt,l = (λ λj) , 0 l t 1. r p−1 j=1 j h=1 rh=t−1−l s ≤ ≤ ≤ Define also the constants ηt ,1 s t m by k - k m − 1+r s − t+s − −mj j j − −rj ηt =( 1) (λ λj) (λ λj) , r j=1 k j=1 j h=1 rh=t−s ∈ H ≤ ≤ and define the vectors wt ,0 t m by w0 = x1,m1 ,and -k -mp -t p,qp s ≤ ≤ wt = ξt xp,qp + ηt zs, 1 t m. p=1 qp=1 s=1 Polynomials in a Linear Relation 237 By a straightforward computation it follows that -k -mp -t { } p,qp { − − } s{ }∈ − wt,wt−1 = ξt xp,qp ,xp,qp−1 (λ λp)xp,qp + ηt zs,zs−1 S λ p=1 qp=1 s=1 m for all t between 1 and m. Therefore, {w ,w0}∈(S − λ) ,sothat{w ,x1 }∈ / m m ,m1 − m { }∈ k − mi (S λ) .Since x1,m1 ,y i=1(S λi) it follows that & ' k mi m {wm,y}∈ (S − λi) (S − λ) . i=1 / +1 ∈ k − mi Thus, y ran i=1 (S λi) and the proof is now complete. Theorem 3.4. Let S be a linear relation in a linear space H, let n ∈ N, λj ∈ K, mj ∈ K, 1 ≤ j ≤ n. Assume that λj, 1 ≤ j ≤ n are distinct. Then -n mi ker p(S)= ker (S − λi) . (3.8) i=i ⊃ ∈ n − mi Proof. To prove the inclusion “ ” in (3.8) let x i=i ker (S λi) ,sothat n ≤ ≤ ∈ − mi x = i=1 xi,1 i n for some xi ker (S λi) . It follows using (2.3) that n n mi mj mj xi ∈ ker (S − λi) (S − λj) =ker (S − λj) . j=1,j=i j=1 / n ∈ n − mj Hence x = i=1 xi ker j=1(S λj) . The inclusion “⊂” in (3.8) will be next proved by induction on n.Thecase n = 1 is obvious. Assume that the inclusion holds true for some n = k ≥ 1andlet / +1 ∈ k − mi ≤ x0 ker i=1 (S λi) . Without lose of generality assume that mi mk+1 for all 1 ≤ i ≤ k. For simplicity denote m = mk+1 and λ = λk+1.Since & ' k +1 k mi mi m {x0, 0}∈ (S − λi) = (S − λi) (S − λ) , i=1 i=1 / { }∈ − m { }∈ k − mi it follows that x0,y0 (S λ) ,and y0, 0 i=1(S λi) for some vector ∈ H { }∈ − ≤ ≤ − y0 , which further implies that xi,xi+1 S λ,0/ i m 1forsome k mi vectors x ∈ H,1≤ i ≤ m,withx = y0.Sincey0 ∈ ker (S −λ ) it follows i m i=1 i k ≤ ≤ ∈ − mi ≤ ≤ that y0 = i=1 yi0,1 i k for some vectors yi0 ker (S λi) ,1 i k. mi m Since mi ≤ m, it follows that ker (S − λi) ⊂ ker (S − λi) , which implies that { }∈ − ≤ ≤ ≤ ≤ − yi,ji ,yi,ji+1 S λi, 1 i k, 0 ji m 1 ∈ H for some vectors yi,ji with yi,m =0.Thus { − − }∈ − ≤ ≤ ≤ ≤ − yi,ji ,yi,ji+1 (λ λi)yi,ji S λ, 1 i k, 0 ji m 1. 238 A. Sandovici Define now the vectors wp ∈ H,0≤ p ≤ m − 1by −1 −1 m−p−1 m- -k m- - j + g w = x + (−1)q (λ − λ )−(j+q+1)y , p j q i ij j=p i=1 j=0 q=0 and define wm = 0. By a straightforward computation it follows that m-−1 {wp,wp+1} = {xj,xj+1} j=p -k m-−1 m-−p−1 q j + g −(j+q+1) + (−1) (λ − λ ) {y ,y +1 − (λ − λ )y }, q i ij i,ji i i,ji i=1 j=0 q=0 so that {wp,wp+1}∈S − λ for all 0 ≤ p ≤ m − 1. Since wm = 0 it follows ∈ − m−p ⊂ − m ≤ ≤ − that wp ker (S λ) ker (S λ) for all 0 p m 1, which implies ∈ − m−p k − mi inductively that xp ker (S λ) + i=1 ker (S λi) . Therefore, -k -k+1 m mi mi x0 ∈ ker (S − λ) + ker (S − λi) = ker (S − λi) , i=1 i=1 which shows that the induction’s step holds true. This completes the proof. Lemma 3.5. Let S be a linear relation in a linear space H,andletλ ∈ K.Then mul (S − λ)n =mulSn,n∈ N. (3.9) Proof. The case λ = 0 is trivial so assume that λ = 0. The proof is done by induction. If n =1letx ∈ mul (S − λ), so that {0,x}∈S − λ, which implies that {0,x}∈S and then mul (S − λ) ⊂ mul S. The converse inclusion, i.e., mul S ⊂ mul (S − λ) follows using a similar argument, so that the case n = 1 is clear. Assume next that (3.9) holds for some n = k ≥ 1andletx ∈ mul (S − λ)k+1, so that {0,x}∈(S − λ)k+1.Then{0,y}∈(S − λ)k and {y,x}∈S − λ for some y ∈ H. By induction hypothesis it follows that y ∈ mul Sk. Furthermore, it follows from {y,x}∈S − λ that {y,x + λy}∈S, which together with the fact that y ∈ mul Sk implies that x + λy ∈ mul Sk+1. Hence, x =(x + λy) − λy ∈ mul Sk+1 +mulSk =mulSk+1. Therefore, mul (S − λ)k+1 ⊂ mul Sk+1. Similar arguments show that the inclusion mul Sk+1 ⊂ mul (S − λ)k+1 holds also true. This completes the proof. Theorem 3.6. Let S be a linear relation in a linear space H,letn ∈ N, λj ∈ K, mj ∈ N, 1 ≤ j ≤ n. Assume that λj, 1 ≤ j ≤ n are distinct. Then n mul p(S)=mulS j=1 mj . (3.10) Polynomials in a Linear Relation 239 Proof. The proof is done by induction. The case n = 1 follows from Lemma/ 3.5. ≥ ∈ k+1 − Assume now that (3.10) holds for some n = k 1 and let x0 mul j=1 (S mj λj) ,sothat k mj mk+1 {0,y0}∈ (S − λj ) , {y0,x0}∈(S − λk+1) (3.11) j=1 ∈ H ∈ Mk for somey0 . By induction hypothesis it follows that y0 mul S where k Mk = j=1 mj. Furthermore, the second relation in (3.11) implies that there ∈ H ≤ ≤ exist the vectors yj ,1 j mk+1, with ymk+1 = x0, such that {yj,yj+1}∈S − λk+1, 0 ≤ j ≤ mk+1 − 1, which implies that {yj,yj+1 + λk+1yj}∈S, 0 ≤ j ≤ mk+1 − 1. (3.12) Define the vectors zp ∈ H,0≤ p ≤ mk+1 by -p p j z = λ y − , 0 ≤ p ≤ m +1. (3.13) p j k+1 p j k j=0 It follows from (3.12) that z0 = y0 and also that -p p p−j {z ,z +1} = λ {y ,y +1 + λ +1y }∈S, 0 ≤ p ≤ m +1 − 1, p p j k+1 j j k j k j=0 Mk +p Mk+p so that zp ∈ mul S for all 0 ≤ p ≤ mk+1, which implies that yp ∈ mul S k+1 ≤ ≤ ∈ j=i mj for all 0 p mk+1.Inparticular,x0 = ymk+1 mul S . Therefore, k+1 k+1 mj j=1 mj mul (S − λj) ⊂ mul S . j=1 / +1 ∈ k mj Conversely, let x0 mul j=1 S ,sothat k mj mk+1 {0,u0}∈ S , {u0,x0}∈S , (3.14) j=1 / ∈ H ∈ k − mj for some u0 . By induction hypothesis it follows that u0 mul j=1(S λj) . Furthermore, the second relation in (3.14) implies that there exist the vectors ∈ H ≤ ≤ uj ,1 j mk+1, with umk+1 = x0, such that {uj,uj+1}∈S, 0 ≤ j ≤ mk+1 − 1, which implies that {uj,uj+1 − λk+1yj}∈S − λk+1, 0 ≤ j ≤ mk+1 − 1. (3.15) 240 A. Sandovici Define the vectors wp ∈ H,0≤ p ≤ mk+1 by -p p j w = (−λ +1) y − , 0 ≤ p ≤ m +1. (3.16) p j k p j k j=0 It follows from (3.15) that w0 = u0 and also that -p p p−j {w ,w +1} = (−λ +1) {u ,u +1 − λ +1u }∈S − λ +1 p p j k j j k j k j=0 / p k for all 0 ≤ p ≤ m +1 − 1, so that w ∈ mul (S − λ +1) (S − λ ), for all k p k j=1 / j ≤ ≤ ∈ − p k − 0 p mk+1, which implies by (2.3) that up mul (S λk+1) j=1(S λj), / +1 ≤ ≤ ∈ k − mj for all 0 p mk+1.Inparticular,x0 = umk+1 mul j=1 (S λj) ,sothat k+1 k+1 j=1 mj mj mul S ⊂ mul (S − λj ) . j=1 This completes the proof. Remark 3.7. Let S be a linear relation in a linear space H,letλ ∈ K and let p, k ∈ N ∪{0}. A combination of (2.6), (3.1) and (3.9) leads to ker (A − λ)p ⊂ dom Ak, mul Ap ⊂ ran (A − λ)k. References [1] R. Arens, Operational calculus of linear relations. Pacific J. Math. 11 (1961), 9–23. [2] T. Azizov, B. Curgus and A. Dijksma, Standard symmetric operators in Pontryagin spaces: a generalized von Neumann formula and minimality of boundary coefficients. J. Funct. Anal. 198 (2003), 361–412. [3] R. Cross, Multivalued linear operators. Marcel Dekker, New York, 1998. [4] M.J. Kascic, Polynomials in linear relations. Pacific J. Math. 24 (1968), 291–295. [5] M.J. Kascic, Polynomials in linear relations. II. Pacific J. Math. 29 (1969), 593–607. [6] A. Sandovici, Standard symmetric linear relations in Pontryagin spaces (in prepara- tion). Adrian Sandovici Department of Mathematics University of Groningen P.O. Box 800 9700 AV Groningen, The Netherland e-mail: [email protected]