Constant J-Unitary Factor and Operator-Valued Transfer Functions

PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ON DYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONS May 24 – 27, 2002, Wilmington, NC, USA pp. 48–56

CONSTANT J-UNITARY FACTOR AND OPERATOR-VALUED TRANSFER FUNCTIONS

Yury Arlinski˘ı Department of Mathematics East Ukrainian National University Lugansk, 91034, Ukraine

Eduard Tsekanovski˘ı Department of Mathematics POB 2044 Niagara University, NY, 14109, USA

Abstract. We study linear, conservative, stationary dynamical systems (l.c.s.d.s) with unbounded main operator and their transfer operator-valued functions. A new class of transfer operator-valued functions is introduced and shown that this class is invariant with respect to multiplication (from the left, from the right or both) by a constant J-unitary operator. The problem how to realize the product W (z)B as a transfer function of some l.c.s.d.s with an arbitrary J-unitary operator B and given realization of W (z) is considered.

1. Introduction. We keep the following notations: L(H1, H2) denotes the Ba- nach space of all continuous linear operators acting from the Hilbert space H1 into the Hilbert space H2, L(H) = L(H, H) and D(Q), R(Q), Ker Q, ρ(Q) denote the domain, the range, the null-space and the resolvent set of a linear operator Q, respectively. Let H be a separable Hilbert space with the inner product (·, ·) and let H+ ⊂ H ⊂ H− be a rigged Hilbert space [5]. If E is a Hilbert space and F is a bounded operator × acting from E into H+ (H−), then by F we will denote the adjoint operator with respect to inner product (·, ·) which is acting from H− (H+) into E. Let A be a × bounded operator from H+ into H−. The adjoint operator A we also consider as operator from H+ into H−. Consider a linear, conservative, stationary dynamical system Θ of the form ½ (A − zI)x = KJϕ− ∗ ϕ+ = ϕ− − 2iK x where ϕ− ∈ E is an input vector, ϕ+ ∈ E is an output vector, and x is a state space vector in H, K is a bounded operator from a Hilbert space E into H− and J is a signature operator in E (J = J ∗ = J −1) and 2iK×JK = A − A×. For the systems Θ we use also the following notation in the form of Brodskii-Livsic rigged operator colligations [1], [4] [6], [7], [9] [10] : µ ¶ A KJ Θ = (1) H+ ⊂ H ⊂ H− E

1991 Mathematics Subject Classiﬁcation. Primary: 47A63, 47B25; Secondary: 47B65.

48 CONSTANT J-UNITARY FACTOR AND OPERATOR-VALUED TRANSFER FUNCTIONS 49

The importance of such type of systems was pointed out in [7], [8]. In this paper we deal with the special class of systems with the main operator A which satisfies the condition A ⊃ T, A× ⊃ T ∗, (2) where T is some densely defined unbounded operator acting on Hilbert space H with nonempty resolvent set ρ(T ) and the additional property: the linear manifold D(A˚) := {f ∈ D(T ) ∩ D(T ∗), T f = T ∗f} is dense in H and the symmetric operator A˚ := T ¹ D(A˚) has finite and equal defect numbers hn, ni. ∗ The Hilbert space H+ in this case is the linear manifold D(A˚ ) equipped by the inner product ∗ ∗ (u, v)+ = (u, v) + (A˚ u, A˚ v), the Hilbert space E is finite-dimensional with dim E = n. and the operator A apart from (2) is chosen as satisfying the condition the linear manifold Ker (A − A×) is the domain (3) of some selfadjoint operator. The transfer operator-valued function of the system takes the form × −1 WΘ(z) = I − 2iK (A − zI) KJ, z ∈ ρ(T ), (J) is well defined and belongs to the following class ΛE [1], [2], [9]: 1. Wθ(z) is holomorphic function on ρ(T ); 2. the fractional linear transformation −1 VΘ(z) = i (WΘ(z) + I) (WΘ(z) − I) J initially defined on ρ(T ) admits analytical continuation to a Nevanlinna operator valued function which has the integral representation Z µ ¶ +∞ 1 t VΘ(z) = ∆ + − 2 dΣ(t), −∞ t − z t + 1 where ∆ is selfadjoint operator in E and Σ(t) is nondecreasing operator-valued function such that Z +∞ Z +∞ d(Σ(t)e, e)E 2 < ∞, d(Σ(t)e, e)E = +∞, e ∈ E \{0}. −∞ 1 + t −∞ It is known [9] that × −1 VΘ(z) = K (AR − zI) K, × where AR = (A + A )/2 is the real part of A. Recall that a bounded operator B acting on Hilbert space E is called J-unitary if BJB∗ = J and B∗JB = J.

In this paper we prove that if an operator-valued function W (z) acting on ﬁnite- (J) dimensional Hilbert space E belongs to the class ΛE , then for an arbitrary J- unitary operator B the functions W (z)B and BW (z) belong again to the same (J) class ΛE and can be realized as transfer operator-valued functions of the same type of l.c.s.d.s as W (z) and with the same unbounded operator T and diﬀerent main operators A. 50 YURY ARLINSKI˘I AND EDUARD TSEKANOVSKI˘I

2. Construction of the main operator of a system. Let H be a separable Hilbert space and let T be a densely defined on D(T ) linear operator with nonempty resolvent set ρ(T ). Consider HT = D(T ) as a Hilbert space HT with the inner 0 product (u, v)T := (u, v) + (T u, T v), u, v ∈ HT . Let HT ⊂ H ⊂ HT be the rigged Hilbert space. Since T is a bounded operator from HT into H, the adjoint operator × 0 × T is bounded from H into HT and satisfies the condition (T u, f) = (u, T f) for ∗ × ∗ all u ∈ HT and all f ∈ H. Let T be the adjoint of T in H. Then T ⊃ T . The −1 resolvent Rξ(T ) = (T − ξI) of T is bounded operator from H onto D(T ) = HT . It ∗ ∗ ¯ −1 ∗ ∗ follows that the resolvent Rξ¯(T ) = (T − ξI) of T has the continuation Rξ¯(T ) 0 0 ∗ × ¯ −1 on HT and maps HT onto H. Moreover, Rξ¯(T ) = (T − ξI) . × We suppose that the imaginary part (T − T )/2i which is defined on HT is singular with respect to H. This means that Ker (T − T×) is dense linear manifold 0 in H or equivalently the subspace in HT defined by × 0 Φ := R(T − T ) (the closure in HT ) (4) satisfies the condition Φ ∩ H = {0}. Let us define the following operator in H × D(A˚) := Ker (T − T ) = {h ∈ HT :(h, ϕ) = 0 for all ϕ ∈ Φ} , (5) A˚ = T ¹ D(A˚) The operator A˚ is a densely defined, closed symmetric operator and T ⊃ A˚, T ∗ ⊃ A˚. ∗ Let Nz = Ker (A˚ − zI) be the defect subspaces of A˚. Then the following identity holds: × −1 ∗ (T − zI) Φ = Nz, z ∈ ρ(T ) (6) Our main assumptions on A˚ defined by (5): 1. A˚ has equal defect numbers, 2. there exists a selfadjoint extension Ae of A˚ in H such that D(T ) + D(Ae) = D(A˚∗). (7) ∗ Denote by H+ the Hilbert space D(A˚ ) equipped by the inner product ∗ ∗ (u, v)+ = (u, v) + (A˚ u, A˚ v). Then (+)-orthogonal decomposition holds:

H+ = D(A˚) ⊕ Ni ⊕ N−i. By the von Neumann formula [3] the domain of every selfadjoint extension Ae of A˚ can be represented in the form

D(Ae) = D(A˚) ⊕ (I + Ve)Ni, where Ve is an isometry from Ni onto N−i. Let Ae be a selfadjoint extension of A˚ satisfying the condition (7). Then the direct decomposition takes place

H+ = HT +(˙ I + Ve)Ni. (8) ∗ ∗ ∗ ∗ Let HT ∗ = D(T ), (u, v)T ∗ = (u, v) + (T u, T v), u, v ∈ D(T ) and let HT ∗ ⊂ H ⊂ 0 0 HT ∗ be the corresponding rigging. The operator T : H → HT ∗ which is deﬁned as the adjoint to T ∗: (T ∗v, f) = (v, Tf) 0 is the continuation of the operator T . Note that the subspace in HT ∗ ∗ 0 Φ∗ := R(T − T ) ( the closure is taken in HT ∗ ) (9) CONSTANT J-UNITARY FACTOR AND OPERATOR-VALUED TRANSFER FUNCTIONS 51 satisﬁes the condition Φ∗∩H = {0}. The Hilbert spaces HT and HT ∗ are subspaces of ∗ the Hilbert space H+ and one can easily show that if (7) holds then D(T )+D(Ae) = D(A˚∗). Therefore,

H+ = HT ∗ +(˙ I + Ve)Ni. (10)

We denote by Pe and Pe∗ the skew projections in H+ onto HT and HT ∗ with respect to the decompositions (8) and (10), respectively. Let

H+ ⊂ H ⊂ H− e× 0 e× 0 be the rigged space and let P : HT → H− and P∗ : HT ∗ → H− be the adjoint operators. These operators are deﬁned by the relations e e× 0 (Pu, ϕ) = (u, P ϕ), u ∈ H+, ϕ ∈ HT and e e× 0 (P∗v, ψ) = (v, P∗ ψ), u ∈ H+, ψ ∈ HT ∗ .

Observe that in view of (8) and (10) the operators Pe and Pe∗ are isomorphisms from HT ∗ onto HT and HT onto HT ∗ , respectively. Moreover the relations hold

PePe∗fT = fT , fT ∈ HT , (11) Pe∗PefT ∗ = fT ∗ , fT ∗ ∈ HT ∗ and e× e× e P Φ = P∗ Φ∗ = Ψ, where n o Ψe := ϕ ∈ H− :(ϕ, h) = 0 for all h ∈ D(Ae) .

Theorem 1. Let the operator APe : H+ → H− be deﬁned by the formula ˚∗ e× × ˚∗ e APe = A + P (T − A )(I − P). (12) Then its adjoint A× : H → H acts by the rule Pe + − A× = A˚∗ + Pe×(T − A˚∗)(I − Pe ). (13) Pe ∗ ∗ Moreover, T ⊂ A , T ∗ ⊂ A× and the following relations hold Pe Pe A = A˚∗ + Pe×(T − A˚∗), A× = A˚∗ + Pe×(T× − A˚∗), Pe ∗ Pe A − A× = Pe×(T − T×)Pe, A − A× = Pe×(T − T ∗)Pe , Pe Pe Pe Pe ∗ ∗ −1 e× −1 (APe − zI) P∗ ϕ = (T − zI) ϕ, ϕ ∈ Φ∗ (A× − zI¯ )−1Pe×ϕ = (T× − zI¯ )−1ϕ, ϕ ∈ Φ, Pe ¡ ¢ ³ ´ (A − zI)−1ψ, ψ = ψ, (A× − zI¯ )−1ψ , ψ ∈ Ψe, z ∈ ρ(T ). Pe Pe The resolvents (A − zI)−1, (A× − zI¯ )−1 map H+˙ Ψe onto H for all z ∈ ρ(T ). Pe Pe + The real part A = (A + A×)/2 satisﬁes the condition A ⊃ Ab, where Ab is a PeR Pe Pe PeR ˚ e −1 ˙ e selfadjoint extension of A transversal to A. The resolvent (APeR −λI) maps H+Ψ onto H+ for all λ, Im λ 6= 0. 52 YURY ARLINSKI˘I AND EDUARD TSEKANOVSKI˘I

Proof. Let APe be an operator deﬁned by (12). Then for fT ∈ D(T ) = HT we get APefT = T fT , i.e. T ⊂ APe. ∗ Since A˚ ¹ (I + Ve)Ni is symmetric operator, we have ∗ ∗ (A˚ (I − Pe)u, (I − Pe)v) = ((I − Pe)u, A˚ (I − Pe)v), u, v ∈ H+.

Therefore for all u, v ∈ H+: e ˚∗ e e e (APeu, v) − (u, APev) = (T Pu + A (I − P)u, Pv + (I − P)v)− − (Peu + (I − Pe)u, T Pev + A˚∗(I − Pe)v) + ((I − Pe)u, T Pev)− − (T Peu, (I − Pe)v) − (A˚∗(I − Pe)u, Pev) + (Peu, A˚∗(I − Pe)v) = = ((T − T×)Peu, Pev) = (Pe×(T − T×)Peu, v). Thus, A − A× = Pe×(T − T×)Pe. It follows that Pe Pe A× = A − Pe×(T − T×)Pe = Pe Pe = A˚∗ + Pe×(T× − A˚∗)(I − Pe) − Pe×(T − T×)Pe = = A˚∗ + Pe×(T× − A˚∗). Further we show that e× × ˚∗ e e× ˚∗ P (T − A )(I − P) = P∗ (T − A ). (14)

Actually (14) holds for f ∈ HT . Let f ∈ (I + Ve)Ni and g ∈ H+, then Peg − Pe∗g = −(I − Pe)g + (I − Pe∗)g ∈ (I + Ve)Ni. Therefore, ∗ (A˚ f, Peg − Pe∗g) = (Af,e Peg − Pe∗g) = (f, Ae(Peg − Pe∗g)) = ∗ ∗ (f, A˚ (Peg − Pe∗g)) = (f, T Peg − T Pe∗g). Now we get × ∗ ∗ ((T − A˚ )f, Peg) = ((T − A˚ )f, Pe∗g). Consequently e× × ˚∗ e× ˚∗ e P (T − A )f = P∗ (T − A )f, f ∈ (I + V )Ni. Now from (12) and (14) we obtain (13). ∗ Let z ∈ ρ(T ) and let ϕz ∈ Nz then (A× − zI)ϕ = Pe×(T× − A˚∗)ϕ = Pe×(T× − zI)ϕ . Pe z z z Now from (6) we obtain the relation (A× − zI)−1Pe×ϕ = (T× − zI)−1ϕ, ϕ ∈ Φ. Pe ∗ × If z ∈ ρ(T ) then HT ∗ +˙ Nz = H+. This decomposition and the relations (T − zI)Nz = Φ, Pe∗Nz = (I + Ve)Ni imply × ∗ × ∗ (T − A˚ )H+ = (T − A˚ )(I + Ve)Ni = Φ. Let ³ ´−1 × ∗ −1 × ∗ (T − A˚ ) = (T − A˚ )¹ (I + Ve)Ni . For the real part we have the relation µ ¶ 1 1 A = (A + A×) = A˚∗ + Pe× (T× − A˚∗)(I − Pe) + (T× − T )Pe . PeR 2 Pe Pe 2 CONSTANT J-UNITARY FACTOR AND OPERATOR-VALUED TRANSFER FUNCTIONS 53

Let © ª b b b D(A) = u ∈ H+ : APeRu ∈ H , Au = APeRu, u ∈ D(A). Then µ ¶ 1 D(Ab) = I − (T× − A˚∗)−1(T× − T ) H . 2 T

It follows that D(Ab)+(˙ I + Ve)Ni = H+. Since Ab is a symmetric extension of A˚ the last equality implies the selfadjointness of A.b

3. Systems (Operator colligations) and their transfer operator-valued functions. Deﬁnition 2. The collection µ ¶ TK0 J Θ0 = 0 (15) HT ⊂ H ⊂ HT E is called the standard operator colligation if E is a Hilbert space, J is a signature ∗ −1 0 operator in E, i.e. J = J = J , K0 is a bounded linear operator from E into HT × × such that Ker K0 = {0}, R(K0) = R(T − T ) and the identity holds: T − T = × K0JK0 .

Assume that operator T satisﬁes conditions from section 2 and let Θ0 be a standard operator colligation. Suppose that Ae is a selfadjoint extension of A˚ satisfying the condition (7). If Pe is corresponding projection then µ ¶ e× APe P K0,J ΘPe = H+ ⊂ H ⊂ H+ E is the l.c.s.d.s (operator colligation) with the main operator APe since by Theorem 1 we have A − A× = Pe×(T − T×)Pe = Pe×K JK×Pe. Pe Pe 0 0 The operator-valued function W (z) = I − 2iK×Pe(A − zI)−1Pe×K J ΘPe 0 Pe 0 is a transfer function of the system ΘPe. Observe that ∗ × × −1 × W (z) = I + 2iJK Pe(A − zI¯ ) Pe K0. ΘPe 0 Pe By Theorem 1 it follows

∗ × × −1 W (z) = I + 2iJK Pe(T − zI¯ ) K0. (16) ΘPe 0 Our goal is to prove the following theorem.

Theorem 3. Let Θ0 be a standard operator colligation and let Ae1 and Ae2 be two selfadjoint extensions of A˚ satisfying the condition of transversality (7). If Pe1 and Pe are two corresponding projections and Θ , Θ are two systems (operator col- 2 Pe1 Pe2 ligations) with the main operators A and A , then for all z ∈ ρ(T ) the identity Pe1 Pe2 holds

WΘ e (z) = WΘ e (z)B, P2 P1 where B is a J-unitary operator acting in Hilbert space E. 54 YURY ARLINSKI˘I AND EDUARD TSEKANOVSKI˘I

Proof. Let MT = HT ª D(A˚) be the orthogonal complement of D(A˚) in the Hilbert × ×−1 ¡ × ¢−1 space HT . Then K0 f 6= 0 for all f ∈ MT \{0}. Let K0 = K0 ¹ MT . For all f ∈ MT we have from (16) ³ ´ ∗ × × e × −1 × WΘ (z)JK0 f = JK0 f + Pm(T − zI¯ ) (T − T )f = Pem × e ∗ −1 = JK0 Pm(T − zI¯ ) (T − zI¯ )f, m = 1, 2. By (11) we obtain 2i(T ∗ − zI¯ )−1(T − zI¯ )f = Pe K×−1JW ∗ (z)JK×f ∗1 0 Θ e 0 P1 and W ∗ (z)JK×f = JK×Pe Pe K×−1JW ∗ (z)JK×f. Θ e 0 0 2 ∗1 0 Θ e 0 P2 P1 Set × e e ×−1 B = JK0 P2P∗1K0 J. × The operator B is defined on the linear manifold D(B) = JR(K0 ) which is dense in E. Let us show that B is a bounded operator in E. Since B(W ∗ (z) − I) = 2iJK×Pe Pe Pe (T× − zI¯ )−1K , Θ e 0 2 ∗1 1 0 P1 we get that B(W ∗ (z) − I) is the bounded operator in E. For e ∈ D(B) we have Θ e P1 already obtained W ∗ (z)e = BW ∗ (z)e. Θ e Θ e P2 P1 Therefore, Be = BW ∗ (z)e − B(W ∗ (z) − I)e, e ∈ D(B). Θ e Θ e P1 P1 It follows that the operator B is bounded and preserve the notation B for its continuation on E. If again e ∈ D(B), then using Theorem 1 we get × ×−1 × ×−1 (JBe, Be)E = (K Pe2Pe∗1K Je, JK Pe2Pe∗1K Je)E = ³ 0 0 0 ´ 0 × e e ×−1 e e ×−1 = (T − T )P2P∗1K0 Je, P2P∗1K0 Je = ³ ´ × e ×−1 e ×−1 = (A e − A )P∗1K0 Je, P∗1K0 Je = P2 Pe2 ×−1 ∗ ×−1 ∗ ×−1 ×−1 = (Pe∗1K Je, T Pe∗1K Je) − (T Pe∗1K Je, Pe∗1K Je) = ³ 0 0 ´ 0 0 ∗ e ×−1 e ×−1 = (T − T )P∗1K0 Je, P∗1K0 Je = ³ ´ × ×−1 ×−1 × ×−1 ×−1 = (A e − A )K0 Je, K0 Je = (K0JK0 K0 Je, K0 Je) = P1 Pe1 = (Je, e)E. This implies the identity B∗JB = J. Since R(B) is dense in E, from the identity (BJB∗ − J)JBe = 0 we get BJB∗ = J. Therefore, B is J-unitary operator in E. In the case of finite-dimensional E we will establish an important inverse theorem. Theorem 4. Let E be a finite-dimensional Hilbert space and let J be a signature (J) operator in E. If the operator valued function W (z) in E belongs to the class ΛE and B is a J-unitary operator in E, then the function WB(z) = W (z)B also belongs (J) to the class ΛE . CONSTANT J-UNITARY FACTOR AND OPERATOR-VALUED TRANSFER FUNCTIONS 55

Proof. According to [1] (see also [2], [4], [9]) there exists a system (operator colligation) µ ¶ A KJ H+ ⊂ H ⊂ H− E which transfer operator-valued function coincides with W (z). This colligation can be obtained from the standard operator colligation µ ¶ TK0 J Θ0 = 0 HT ⊂ H ⊂ HT E

This means that HT = D(T ) = {h ∈ H+ : Ah ∈ H}, T = A¹ HT , A = APe, × K = Pe K0 with some projector Pe. Observe that ¡ × ¢ HT +˙ Ker (A − A ) ª D(A˚) = H+ and Pe is the projector on HT with respect to this decomposition. Let B be a J-unitary operator in E and let ³ ´ e ×−1 ∗−1 × e ∗ e ˚∗ e D(A1) = I − K0 B K0 P D(T ), A1 = A ¹ D(A1), (17) ×−1 e where K0 is the same as in Theorem 3. Let show that A1 is symmetric. For ×−1 ∗−1 × e ∗ u = g − K0 B K0 Pg, g ∈ D(T ) we have ˚∗ ˚∗ ∗ ×−1 ∗−1 × e ×−1 ∗−1 × e (A u, u) − (u, A u) = (T g − TK0 B K0 Pg, g − K0 B K0 Pg)− ¡ ¢ − (g − K×−1B∗−1K×Peg, T ∗g − TK×−1B∗−1K×Peg) = (A× − A)g, g + ³ 0 0 0 0 ´ × ×−1 ∗−1 × e ×−1 ∗−1 × e e× × e + (A − A )K0 B K0 Pg, K0 B K0 Pg = −2i(P K0JK0 Pg, g)+ e× × ×−1 ∗−1 × e ×−1 ∗−1 × e + 2i(P K0JK0 K0 B K0 Pg, K0 B K0 Pg) = e× × e e× −1 ∗−1 × e = −2i(P K0JK0 Pg, g) + 2i(P K0B JB K0 Pg, g) = 0.

Thus, Ae1 is a symmetric operator. Let dim E = n, Then dim R(K) = dim R(K0) = × ˚ dim ³R(A − A ) = n´. It follows that the defect numbers of A are equal to hn, ni and dim D(Ae1)/D(A˚) = n. Therefore, Ae1 is a selfadjoint extension of A˚. Moreover, ∗ ∗ ∗ D(Ae1) ∩ D(T ) = D(A˚). Therefore D(T ) + D(Ae1) = D(T ) + D(Ae1) = D(A˚ )H+. This means that Ae1 is transversal to T . Let Pe1 and Pe∗1 be the corresponding projections onto D(T ) and D(T ∗), respectively. From (17) we obtain the equality × e ∗−1 × e ∗ K0 P1fT ∗ = B K0 PfT ∗ for all fT ∗ ∈ D(T ). Therefore, using (11): × e e ∗−1 × K0 P1P∗fT = B K0 fT for all fT ∈ D(T ). Deﬁne the operator A by means of Pe and by formula (12). If K = Pe×K Pe1 1 1 1 0 then the system µ ¶ A K J Pe1 1 H+ ⊂ H ⊂ H− E has the transfer function W (z) = I − 2iK×(A − zI)−1K J and 1 1 Pe1 1 ∗ × e × −1 W1 (z) = I + 2iJK0 P1(T − zI) K0.

Using the proof of Theorem 3 we get the equality W1(z) = W (z)B. The same result as in Theorem 4 takes place for BW (z). 56 YURY ARLINSKI˘I AND EDUARD TSEKANOVSKI˘I

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