arXiv:1309.4038v1 [math.FA] 16 Sep 2013 ml httessulna form sesquilinear the that imply es that sense oooyon topology h space the h ojgt ulof dual conjugate the ftemlilcto fdsrbtosb h ult metho duality of the study by distributions the of for multiplication background the basic of the severa provide coefficient in also singular tool and with fundamental forms) operators a (differential are Analysis them in on funct rigge test acting of of operators examples spaces relevant and The constitute 25]. them qu 16, over 14, to tributions 15, applications 12, their 13, for [8, Schr¨odinge and fields...) of 31] resonances eigenvalues, 28, (generalized pure 23, ries a [22, from view both literature of the point in considered been often have IRI ELMNE AVTR IBLA N AIL TRAPAN CAMILLO AND BELLA, DI SALVATORE BELLOMONTE, GIORGIA 2010 Date pcso iermp cigo a on acting maps linear of Spaces e od n phrases. and words Key eoegigfrh efi oenttosadbscdefinition basic and notations some Let fix we forth, going Before PRTR NRGE IBR PCS SOME SPACES: HILBERT RIGGED IN OPERATORS D uy3,2021. 30, July : ahmtc ujc Classification. Subject Mathematics n utvle eovn ucinaedrvdadsm exa some and derived are the discussed. of function and resolvent set multivalued resolvent ing the of properties Some interspaces. ibr space Hilbert Abstract. fitreit oal ovxsae iigbetween living spaces convex locally intermediate of eadnelna usaeo ibr space Hilbert of subspace linear dense a be D H × D a eietfidwt usaeof subspace a with identified be can nrta h oooyidcdb h ibr om Then norm. Hilbert the by induced topology the than finer , falcniuu ojgt ierfntoason functionals linear conjugate continuous all of oino eovn e o noeao cigi rigged a in acting operator an for set resolvent of notion A D ⊂ H ⊂ D PCRLPROPERTIES SPECTRAL D igdHletsae,oeaos spectrum. operators, spaces, Hilbert Rigged [ t ,i iervco pc and space vector linear a is ], 1. × D ⊂ H ⊂ D Introduction spooe.Ti e eed nafamily a on depends set This proposed. is B ( · igdHletspace Hilbert rigged 1 , 76,47L05. 47L60, · htputs that ) × D D H × D hs identifications These . and and prtr,quantum operators, r and contains RS o short) for (RHS, correspond- t 2,2,34]. 27, [24, d D osadtedis- the and ions D ibr spaces Hilbert d ,Fuirtrans- Fourier s, pe are mples oal convex locally a × × called , mathematical nu theo- antum ndaiyis duality in h problem the s. problems l H D nthe in , [ t ,i.e., ], I 2 GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI an extension of the inner product of ; i.e. B(ξ, η) = ξ η , for every D h | i ξ, η (to simplify notations we adopt the symbol for both of them). ∈ D h· |·i The space will always be considered as endowed with the strong dual D× topology t = β( , ). The is dense in [t ]. × D× D H D× × We get in this way a Gelfand triplet or rigged Hilbert space (RHS)

,[×t] ֒ ֒ ×[t] (1) D →H → D ,where ֒ denotes a continuous embedding with dense range. As it is usual → we will systematically read (1) as a chain of inclusions and we will write [t] [t ] or ( [t], , [t ]) for denoting a RHS. D ⊂H⊂D× × D H D× × Let L( , ) denote the vector space of all continuous linear maps from D D× [t] into [t ]. In L( , ) an involution X X can be introduced by D D× × D D× 7→ † the equality Xξ η = X η ξ , ξ, η . h | i h † | i ∀ ∈ D Hence L( , ) is a *-invariant vector space. As we shall see, L( , ) D D× D D× can be made into a partial *-algebra by selecting an appropriate family of intermediate spaces (interspaces) between and and, for this reason, D D× this paper is a continuation of the study on the spectral properties of locally convex quasi *-algebras or partial *-algebras on which several results of a certain interest have been recently obtained, see e.g. [3, 5, 6, 9, 10, 30].

The problem we want to face in this paper is that of giving a reason- able notion of spectrum of an operator X L( , ); where reasonable ∈ D D× means that it gives sufficient information on the behavior of the operator. Indeed, we propose a definition of resolvent set which is closely linked to the intermediate structure of interspaces that can be found between and D , as it happens in many concrete examples: a spectral analysis can be D× performed each time we fix one of these families. Actually, the definition of resolvent set we will give depends on the choice of a family F0 of inter- spaces. This is not a major problem if we take into account the problem that originated the spectral analysis of operators in Hilbert spaces. If, for instance, A ( ) (the C*-algebra of bounded operators in Hilbert space ∈ B H ), looking for the resolvent set of A simply means looking for the λ’s in C H for which the equation Aξ λξ = η − has a unique solution ξ , for every choice of η , with ξ depending ∈ H ∈ H continuously on η. OPERATORSINRHS:SPECTRALPROPERTIES 3

The same problem can be posed in the framework of rigged Hilbert spaces.

For instance, between (R), the of rapidly decreasing C∞ S functions, and ×(R), the space of tempered distributions, live many clas- S sical families of spaces like Sobolev spaces, Besov spaces, Bessel potential spaces, etc. Let us call F0 one of these families. Then, finding solutions of the equation

Xξ λξ = η, − with X L( (R), ×(R)) should be intended in a more general sense: there ∈ S S exists a continuous extension of X to a space F where solutions of our E ∈ 0 equation do exist. The fact that ξ means that the solution satisfies ∈ E regularity conditions milder than those needed for ξ to belong to (R). S Rigged Hilbert spaces are a relevant example of partial inner product (Pip-) spaces [7]. A Pip-space V is characterized by the fact that the inner product is defined only for compatible pairs of elements of the space V . It contains a complete lattice of subspaces (the so called assaying subspaces) fully determined by the compatibility relation. An assaying subspace is nothing but an interspace, in the terminology adopted here. The point of view here is however different: we start from a RHS and look for convenient families of interspaces for which certain properties are satisfied. Neverthe- less, we believe that an analysis similar to that undertaken here could also be performed in the more general framework of Pip-spaces, but this problem will not be considered here. The paper is organized as follows. In Section 2 we collect some basic facts on rigged Hilbert spaces and operators on them. In Section 3 we introduce the resolvent and spectrum of an operator X L( , ). This ∈ D D× definition, as announced before, depends on the choice of a family F0 of interspaces living between and and the crucial assumption is that the D D× operator X extends continuously to some of them. Section 4 is devoted to elements of L( , ) that can be considered also as closable operators in D D× Hilbert space. In particular, we give an extension of Gelfand theorem on the existence of generalized eigenvectors of a symmetric operator X L( , ) ∈ D D× having a self-adjoint extension in the Hilbert space . We will prove, under H the assumptions that = (A), where A is a self-adjoint operator in D D∞ H having a Hilbert-Schmidt inverse, that the operator X has a complete set of generalized eigenvectors without requiring, as done in Gelfand theorem, that X leaves invariant. Moreover, it is shown that these generalized D 4 GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI eigenvectors all belong to a certain element of the chain of Hilbert spaces generated by A. Finally, in Section 5 we collect some examples.

2. Notations and preliminaries

For general aspects of the theory of partial *-algebras and of their repre- sentations, we refer to the monograph [4]. For reader’s convenience, however, we repeat here the essential definitions. A partial *-algebra A is a complex vector space with conjugate linear involution and a distributive partial multiplication , defined on a subset ∗ · Γ A A, satisfying the property that (x,y) Γ if, and only if, (y ,x ) Γ ⊂ × ∈ ∗ ∗ ∈ and (x y) = y x . From now on, we will write simply xy instead of x y · ∗ ∗ · ∗ · whenever (x,y) Γ. For every y A, the set of left (resp. right) multipliers ∈ ∈ of y is denoted by L(y) (resp. R(y)), i.e., L(y)= x A : (x,y) Γ , (resp. { ∈ ∈ } R(y) = x A : (y,x) Γ ). We denote by LA (resp. RA) the space of { ∈ ∈ } universal left (resp. right) multipliers of A. In general, a partial *-algebra is not associative. The unit of partial *-algebra A, if any, is an element e A such that ∈ e = e , e RA LA and xe = ex = x, for every x A. ∗ ∈ ∩ ∈ Let [t] [t ] be a RHS and L( , ) the vector space of all D ⊂H⊂D× × D D× continuous linear maps from [t] into [t ]. D D× × To every X L( , ) there corresponds a separately continuous sesquilin- ∈ D D× ear form θ on defined by X D × D (2) θ (ξ, η)= Xξ η , ξ,η . X h | i ∈ D The space of all jointly continuous sesquilinear forms on will be D × D denoted with B( , ). We denote by LB( , ) the subspace of all X D D D D× ∈ L( , ) such that θ B( , ). D D× X ∈ D D We denote by L ( ) the *-algebra consisting of all X L( , ) such † D ∈ D D× that X and X . D ⊆ D †D ⊆ D Let [t] ×[t×] be a rigged Hilbert space and [t ] a locally D ⊂H⊂D E E convex space such that

.[×t] ֒ [t ] ֒ ×[t] (3) D →E E → D Let × be the conjugate dual of [t ] endowed with its own strong dual E E E topology t×. Then by duality, × is continuously embedded in × and the E E D embedding has dense range. Also is continuously embedded in , but in D E this case the image of is not necessarily dense in [4, Example 10.2.21], D E× unless is endowed with the Mackey topology τ( , ×) =: τ ; in which case E E E E OPERATORSINRHS:SPECTRALPROPERTIES 5 we say that is an interspace. If , are interspaces and , then τ E E F E ⊂ F F is coarser than τ . E Let , be interspaces. Let us define E F ( , ) := X L( , ×) : Y L( , ),Yξ = Xξ, ξ , C E F { ∈ D D ∃ ∈ E F ∀ ∈ D} where L( , ) denotes the vector space of all continuous linear maps from E F [τ ] into [τ ]. It is clear that X ( , ) if and only if it has a continuous E E F F ∈C E F extension X : [τ ] [τ ]. In particular, if X ( , ), then X E E E → F F ∈ C E F ∈ ( , ). The continuous extension of X from into clearly coincides C E D× E D× with X . Obviously, if X,Y ( , ×), then (X + Y ) = X + Y . E ∈C E D E E E If X ( , ) then X L( , ), hence there exists a Mackey continuous ∈C E F E ∈ E F linear map X‡ : × × such that E F →E

X ξ η = X‡ η ξ , ξ , η ×. h E | i E | ∀ ∈E ∀ ∈ F Since the sesquilinear form whichD putsE all pairs ( , ) in duality extends E E× the inner product of , it follows that X‡ is an extension of X† to × with D E F values in . Hence X ( , ). Interchanging the roles of X and X E× † ∈C F × E× † (and recalling that every interspace carries its Mackey topology) we obtain

(4) X ( , ) X† ( ×, ×). ∈C E F ⇔ ∈C F E Moreover, a simple comparison of topologies shows that if , are in- E F terspaces and Y L( , ), then there exists X L( , ) such that ∈ E F ∈ D D× X = Y ↾ . D Let now X,Y L( , ) and assume there exists an interspace such ∈ D D× E that Y L( , ) and X ( , ); it would then be natural to define ∈ D E ∈C E D× X Yξ = X (Yξ), ξ . · E ∈ D However, this product is not well-defined, because it may depend on the choice of the interspace . E Definition 2.1. A family F of interspaces in the rigged Hilbert space ( [t], , [t ]) is called a multiplication framework if D H D× × (i) F; D∈ (ii) F, its conjugate dual also belongs to F; ∀E ∈ E× (iii) , F, F. ∀E F ∈ E∩F∈ Definition 2.2. Let F be a multiplication framework in the rigged Hilbert space ( [t], , [t ]). The product X Y of two elements of L( , ) is D H D× × · D D× defined, with respect to F, if there exist three interspaces , , F such E F G ∈ 6 GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI that X ( , ) and Y ( , ). In this case, the multiplication X Y is ∈C F G ∈C E F · defined by X Y = (X Y ) ↾ · F E D or, equivalently, by X Yξ = X Yξ, ξ . · F ∈ D Actually, the product so defined does not depend on the particular choice of the interspaces , , F but it may depend (and it does!) on the choice E F G ∈ of F. For a more detailed analysis see [4, 19]. As shown in [4, Theorem 10.2.30], we have

Theorem 2.3. Let F be a multiplication framework in the rigged Hilbert space ( [t], , [t ]). Then L( , ), with the multiplication defined D H D× × D D× above, is a (non-associative) partial *-algebra.

3. Inverses and resolvents

Let us consider an element X L( , ). We denote by (X) the ∈ D D× R range of X. If X is injective and (X) = ×, then there exists a linear 1 R 1 D 1 1 map X− : × such that XX− = I × and X− X = I . If X− is D → D D D continuous, then its restriction to is a and it leaves D 1 D invariant. Conditions for the continuity of X− are given in [18, Sect. 38]. The next proposition shows that, even though it is natural, the notion of invertibility considered above is too restrictive.

Proposition 3.1. Let [t] ֒ ֒ [t ] be a rigged Hilbert space and D → H → D× × X L( , ) a linear bijection. Then there exists a triplet of Hilbert spaces ∈ D D× . × such that and × ֒ ֒ HX →H →HX D⊆HX D× ⊆HX Proof. Since X is a bijection, the linear operator X 1 : is well- − D× → D defined. Then we can introduce an inner product on by D× 1 1 (5) ζ ζ′ = X− ζ X− ζ′ , ζ,ζ′ ×. X ∈ D

By (5) it follows easily that X is an isometry of [ ] onto ×[ X ] so it D k · k D k · k extends to a of onto the Hilbert space completion × of H HX [ ]. Since the embedding of into × is continuous, the conjugate D× k · kX H HX dual ( × ) =: of × , with respect to the inner product of contains HX × HX HX H as dense subspace.  D OPERATORSINRHS:SPECTRALPROPERTIES 7

Since the existence of global inverses of operators of L( , ) is a so D D× strong condition, one may try to exploit the intermediate structure between and for a more appropriate definition of the inversion procedure. D D× The fact that, once fixed a multiplication framework F, L( , ) becomes D D× a partial *-algebra [Theorem 2.3] suggests, say, an algebraic definition: Y ∈ L( , ) is the inverse of X L( , ) if D D× ∈ D D× (6) X Y and Y X are well-defined and X Yξ = Y Xξ = ξ, ξ . · · · · ∀ ∈ D This equality, however, does not define Y uniquely, because of possible lack of associativity. Let X L( , ) and F a family of interspaces. Assume that there ∈ D D× 0 exist , F0 such that X ( , ). If the extension X is bijective E F ∈ 1 ∈ C E F1 E from into , then X− exists. If X− is continuous from onto , then E F E E F E its restriction to is automatically continuous from [t] into ×[t×]; i.e. 1 D 1 D D X− ↾ L( , ×) and, moreover, X− ↾ ( , ). If this is the case, E D∈ D D E D∈ C F E 1 and if F0 is a multiplication framework, then (6) holds. So that X− ↾ E D is the algebraic inverse of X. The converse may fail to be true. For this reason there is no need, in what follows, to consider F0 as a multiplication framework.

Remark 3.2. By (4) we know that X ( , ) if, and only if, X ∈ C E F † ∈ ( , ); we denote by X† × the continuous extension of X to . As- C F × E× † F × sume in particular that , F are Hilbert spaces and X ( , ). Then, E F ∈ C E F if X has a continuous inverse, X† × also has a continuous inverse and E 1 1 F ((X )− )† = (X† × )− . E F Remark 3.3. Assume that there exists a second pair ′, ′, such that 1 E F X ( ′, ′) and X ′ has a continuous inverse X−′ from ′ onto ′. If F0 ∈C E F E E F E is a multiplication framework, then ′ F0 and is dense in ′ with E∩E ∈ D 1 E∩E1 the projective topology [4, Proposition 10.2.24]. But X− and X−′ need E E not have the same restriction to , unlessL:= X ( ′) is an interspace D E E∩E of F and X ( , L). 0 ∈C E∩E′

Definition 3.4. Let X L( , ×) and λ C. We say that λ is a gen- ∈ D D ∈ eralized eigenvalue of X if there exists an interspace such that X has a E continuous extension X from [t ] into ×[t×] and X λI is not in- E E E D E − E jective. Any nonzero vector ξ N(X λI ) is called a generalized ∈ E − E ⊂ E eigenvector. If = we say that λ is an eigenvalue of X and elements of E D N(X λI ) are called eigenvectors. D − D 8 GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI

1 Remark 3.5. If X λI has a global inverse (X λI )− : × and − D − D D → D X ( , ), for some , F, with $ , then there exists ζ 0 ∈ C E F E F ∈ D E ∈E\{ } such that (X λI )ζ = 0. Indeed, let ξ′ , then, (X λI )ξ′ = E − E ∈E\D E − E (X λI )ξ for a unique ξ . Hence (X λI )(ξ′ ξ) = 0, with ξ′ ξ = 0. − D ∈ D E − E − − 6 This implies that λ is (also) a generalized eigenvalue.

Example 3.6. Let := (R) be the Schwartz space of rapidly decreas- S S ing C∞ functions and × := ×(R) its conjugate dual (tempered distribu- S S tions) and consider the very familiar example of the operator p = id/dx − ∈ L( , ). It is clear that p has no eigenvalues in . But, since p L ( ), S S× S ∈ † S has a continuous extension p to the whole of ×. For every λ R, the iλx S ∈ function fλ(x)= e is an eigenvector of p corresponding to the eigenvalue λ. Thus, every real number λe is a generalized eigenvalue of p. This is just an example of a more generale situation: every symmetric oper- ator A on a nuclear rigged Hilbert space , having a self-adjoint D⊂H⊂D× extension to and invariant in possesses a complete set of generalized H D eigenvectors [17, Ch.IV, Sect.5, Theorem 1]. We will come back to this point in Section 4.

Definition 3.7. Let X L( , ×) and F0 be a family of interspaces. The ∈F0 D D F0-resolvent set of X, ̺ (X), consists of the set of complex numbers λ satisfying the following conditions: there exist , F , with , such that E F ∈ 0 E ⊆ F (i.1) X ( , ) and (X λI ) = ; E E ∈C E F 1 − E F (i.2) (X λI )− exists and it is continuous from [τ ] onto [τ ]. E − E F F E E F0 F0 The set σ (X) := C ̺ (X) will be called the F0-spectrum of X. \ Remark 3.8. The assumption of continuity in condition (i.2) can be omitted if we suppose that , are Banach spaces; in this case, in fact, the inverse E F 1 mapping theorem guarantees the continuity of (X λI )− . E − E Remark 3.9. If the topology of is equivalent to the initial Hilbert norm, D then = and, in this case, X L( , ) if and only if X ( ). The D× H ∈ D D× ∈B H unique possible interspace is itself. Hence the spectrum of X coincides H with the usual spectrum defined in ( ). B H A crucial point in the previous definition is that there could exist different couples of interspaces for which the requirements of our definition are true. This point requires a careful analysis, which will be performed later. OPERATORSINRHS:SPECTRALPROPERTIES 9

In order to study the F -resolvent of an operator X L( , ) it is 0 ∈ D D× useful to introduce the notion of regular point of X, in analogy with what is usually done for operators in Hilbert spaces. We will do this by supposing that F0 is a family of interspaces whose elements are Hilbert spaces. In this case, we will prefer the notation ( , ) to L( , ) as a remainder of the B E F E F fact that its elements are bounded operators from into . The norm of E F the ( , ) will be denoted by , . B E F k · kE F Let now , F and X ( , ). The inverse of X, when it exists, is, E F ∈ 0 ∈B E F clearly, the unique Y ( , ) such that ∈B F E XY = I , Y X = I . F E The set of all invertible elements of ( , ) will be denoted by G( ( , )). B E F B E F

Definition 3.10. A complex number λ is called a F0-regular point for + X L( , ×) if there exist , F0, with , and cλ, dλ R such ∈ D D E F ∈ E ⊆ F ∈ that:

(7) cλ ξ (X λI)ξ dλ ξ , ξ . k kE ≤ k − kF ≤ k kE ∀ ∈ D The set of regular points of X will be denoted by π(X).

Clearly, λ π(X) if and only if there exist , F , such that (X λI) ∈ E F ∈ 0 − ∈ ( , ) and both (X λI) and its extension (X λI ) to are injective. E E C E F − , − E If X ( , ), we denote by πE F (X) the set of λ C for which (7) ∈ C E F ∈ holds, with , fixed. It is clear that λ π , (X) if, and only if, the E F ∈ E F extension X λI of X λI to satisfies E − E − E (8) cλ ξ (X λI )ξ dλ ξ , ξ . k kE ≤ k E − E kF ≤ k kE ∀ ∈E , , Definition 3.11. Let X L( , ) and λ π (X). We call d E F (X) := ∈ D D× ∈ E F λ dim (X λI )⊥ (the orthogonal being taken in ) the defect number of R E − E F X at λ w. r. to , . E F

Let X ( , ). Then, when ′ runs over F0, we have: ∈C E F F ′ ′ , , if dλE F (X) dλE F (X); • F ⊂ F ⇒ ≤ ′ ′ , , if and (X λI ) ′ dE F (X) dE F (X); • F ⊂ F R E − E ⊂ F ⇒ λ ≥ λ if and are not comparable, then there is no a priori relation • F F ′ between the corresponding defect numbers.

On the other hand, if , are fixed and ′ runs over F0, E F ′ E , , if dE F (X) dE F (X); • E′ ⊂E⇒ λ ≤ λ 10 GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI

′ ′ , , if and (X ′ λI ′ ) dE F (X) dE F (X); • E⊂E R E − E ⊂F⇒ λ ≥ λ if and are not comparable, then there is no a priori relation • E E′ between the corresponding defect numbers.

Proposition 3.12. Let , F0, X ( , ) and λ C. , E F ∈ ∈C E F ∈ 1 (i) λ πE F (X) if and only if (X λI ) has a bounded inverse (X λI )− ∈ E − E E − E defined on (X λI ) . R E − E ⊆ F (ii) (X λI )= (X λI), for each λ π , (X). R E − E R − ∈ E F (iii) If λ π , (X), then (X λI) is closed in . ∈ E F R − F Proof. (i) follows easily from (8). (ii) Let η be in the closure of (X λI) in . Then there is a sequence R − F (ξn)n N of vectors ξn such that ηn = (X λI)ξn η in . We have ∈ ∈ D − → F 1 ξn ξm c− (X λI)(ξn ξm) = ηn ηm . k − kE ≤ λ k − − kF k − kF Hence (ξ ) is a Cauchy sequence in and so it is convergent. Let ξ = n E limn ξn. Then Xξn = ηn + λξn η + λξ. By the definition of X it →∞ → E follows that X ξ = η + λξ, so that η = (X λI )ξ (X λI ). Thus E E − E ∈ R E − E (X λI) (X λI ). Using once more the definition of X , it follows R − ⊆ R E − E E easily that (X λI ) (X λI) and, thus, the equality holds. R E − E ⊆ R − (iii): It follows from (ii). 

Proposition 3.13. The set of the regular points of X L( , ) is an ∈ D D× open subset of C.

Proof. Let λ0 π(X). Then there exist , F0, , such that , ∈ E F ∈ E ⊆ F λ0 πE F (X). Let λ C and suppose that λ λ0 < cλ0 . Then for ξ , ∈ ∈ | − | ∈ D we have, on one hand,

(X λI)ξ (X λ0I)ξ λ λ0 ξ (cλ0 λ λ0 ) ξ . k − kF ≥ k − kF − | − |k kE ≥ − | − | k kE On the other hand

(X λI)ξ (X λ0I)ξ + λ λ0 ξ k − kF ≤ k − kF | − |k kF (X λ0I)ξ + cλ0 ξ (dλ0 + cλ0 ) ξ . ≤ k − kF k kE ≤ k kE Hence λ π , (X). This in turn implies that π(X) is open.  ∈ E F With minor modifications of a classical argument (see, e.g. [29, Proposi- tion 2.4]) one obtains the following

, Proposition 3.14. Let X ( , ). Then the defect number dλE F (X) is ∈ C E F , constant on each connected component of the open set πE F (X). OPERATORSINRHS:SPECTRALPROPERTIES 11

Since the map X ( , ) X ( , ) is a topological isomorphism, ∈C E F → E ∈B E F it can be more convenient to study certain properties in ( , ) rather than B E F in ( , ). We begin with the notion of resolvent in ( , ). C E F B E F Definition 3.15. Let , F with . For A ( , ) we define E F ∈ 0 E ⊆ F ∈B E F 1 ̺ , (A) := λ C : (A λI )− exists in ( , ) . E F { ∈ − E B F E } , 1 If λ ̺ , (A) we put RE F (A) := (A λI )− . ∈ E F λ − E

Let X L( , ×) and suppose that X ( , ), with , F0 with ∈ D D ∈ C E F, E F ∈ 1 . Then we put ̺ , (X) := ̺ , (X ) and RE F (X) := (X λI )− , E F E F E λ E E E ⊆ F − , if λ ̺ , (X). It is clear that λ ̺ , (X) if and only if λ πE F (X) ∈ E F ∈ E F ∈ and (X λI ) is surjective in . For consistency of notations we put E − E F ̺ , (X)= if X ( , ). With this convention, one has E F ∅ 6∈ C E F F0 (9) ̺ (X)= ̺ , (X). E F , F0 E F∈[ Remark 3.16. If F is stable under duality (i.e., F if and only if 0 E ∈ 0 × F0), then by Remark 3.2 it follows that λ ̺ , (X) if and only if E ∈ ∈ E F F0 F0 λ ̺ ×, × (X†). This, in turn, implies that ̺ (X)= ̺ (X†). ∈ F E Theorem 3.17. Let , F . The set G( ( , )) of all invertible ele- E F ∈ 0 B E F ments of ( , ) is open. B E F Proof. Let A G( ( , )) and B ( , ). We can write A + B = 1 ∈ B E F ∈ B E F 1 A(I + A− B) and if we choose B such that A− B , < 1, then I + E k kE E E A 1B G( ( , )), since ( , ) is a , and then A + B − ∈ B E E B E E ∈ G( ( , )).  B E F Theorem 3.18. Let , F . The map A G( ( , )) A 1 ( , ) E F ∈ 0 ∈ B E F → − ∈B F E is continuous.

Proof. Let A, B G( ( , )), then: ∈ B E F 1 1 1 1 A− B− , = B− (B A)A− , k − kF E k − kF E 1 1 1 1 1 = (B− A− )(B A)A− + A− (B A)A− , k − − − kF E 1 1 1 B− A− , B A , A− , ≤ k − kF E k − kE F k kF E 1 2 + B A , A− , . k − kE F k kF E 1 1 If we take B A , such that (1 B A , A− , ) , then E F E F F E 2 1 1k − k − k − k k k ≥ A− B− , can be made arbitrarily small.  k − kF E 12 GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI

Remark 3.19. Let A, B ( , ), with . Let us define A0 := A ↾ . ∈B F E E ⊆ F E Then A ( , ). The product A B is defined by A Bη = A (Bη), for 0 ∈ B E E · · 0 every η and ∈ F A0B , A0 , B , . k kF E k ≤ k kE E k kF E In particular, if A ( , ), with , the n-th power A(n) of A is ∈ B F E E ⊆ F defined by (2) (n) (n 1) A := A0A, and A = A0A − and one has (n) n 1 (10) A , A0 ,− A , . k kF E ≤ k kE E k kF E Lemma 3.20. Let A, B ( , ), with . The following resolvent ∈ B E F E ⊆ F identities hold: , , , , RE F (A) RE F (B)= Rλ(A)E F (A B)RE F (B), λ ̺ , (A) ̺ , (B); λ − λ − λ ∀ ∈ E F ∩ E F , , , , RE F (A) RE F (A) = (λ λ0)RE F (A)RE F (A), λ, λ0 ̺ , (A). λ − λ0 − λ λ0 ∀ ∈ E F Theorem 3.21. Let , F and A ( , ). The following statements E F ∈ 0 ∈B E F hold:

(i) ̺ , (A) is open. E F 1 (ii) the function λ ̺ , (A) (A λI )− ( , ) is analytic on ∈ E F → − E ∈ B F E every connected component of ̺ , (A). E F 1 Proof. (i): Put h(λ)= A λI . Then ̺ , (A)= h− (G( ( , )) and so it − E E F B F E is open, since h is clearly continuous. (ii): 1 1 If λ µ, (A λI )− (A µI )− , due to the continuity of the inver- → − E → − E sion. Since and I : is continuous, we have B↾ , E ⊂ F E E→E⊂F 1 k1 E kE E ≤ B , , for every B ( , ). Thus, (A λI )↾− (A µI )↾− , 0 k kF E ∈B F E k − E E − − E E kE E → as λ µ. Hence, → 1 1 1 1 (A λI )↾− (A µI )− (A µI )↾− (A µI )− , k − E E − E − − E E − E kF E 1 1 1 (A λI )↾− (A µI )↾− , (A µI )− , 0, ≤ k − E E − − E E kE E k − E kF E → as λ µ. Finally, we get → 1 1 (A λI )− (A µI )− 1 1 lim − E − − E = lim ((A λI )↾− (A µI )− ) λ µ λ µ λ µ − E − E → − → E 1 1 = (A µI )↾− (A µI )− , − E E − E in the norm of ( , ).  B F E By (i) of Theorem 3.21 and by (9), we get OPERATORSINRHS:SPECTRALPROPERTIES 13

Proposition 3.22. Let X L( , ) and F a family of interspaces. ∈ D D× 0 Then, ̺F0 (X) is open.

Adopting the notations of Remark 3.19 we have

Proposition 3.23. Let X L( , ×), and λ0 ̺ , (X). Then ∈ D D E ⊂ F ∈ E F there exists δ > 0 such that, for every λ C with λ λ0 < δ, λ ̺ , (X) ∈ | − | ∈ E F and + , ∞ n , (n+1) RE F (X)= (λ λ ) RE F (X) , λ − 0 λ0 n=0 X where the series converges in the operator norm of ( , ). B F E Proof. We already know that ̺ , (X) is an open set; thus there exists E F r > 0 such that the disk λ λ0 < r is contained in ̺ , (X). Let 1| − | E F , − δ = min r, RλE0F (X) , hence λ ̺ , (X) and 0 , ∈ E F    E E 

k+p k+p (n) n , (n+1) n , , (λ λ0) RλE0F (X) = (λ λ0) RλE0F (X) RλE0F (X) − − 0 n=k+1 , n=k+1   , X F E X F E k+p (n ) n , , (λ λ0) RλE0F (X) RλE0F (X) ≤ − 0 , n=k+1   , F E X E E , Since (λ λ0) RλE0F (X) < 1, the series − 0 ,   E E + ∞ n , (n+1) (λ λ ) RE F (X) − 0 λ0 nX=0 converges in the operator norm of ( , ). B F E Using the second identity in Lemma 3.20, we finally obtain, in standard way, + , ∞ n , (n+1) RE F (X)= (λ λ ) RE F (X) . λ − 0 λ0 n=0 X 

Lemma 3.24. Let λ ̺ , (X), , F0, . Then ∈ E F E F ∈ E ⊆ F (i) λ σ , ′ (X), for every ′ F0, ′ % ; ∈ E F F ∈ F F (ii) if & ′ and X ( ′, ), then λ is an eigenvalue of X ′ and, E E ⊆ F ∈C E F E hence, λ σ ′, (X). ∈ E F Proof. (i) is straightforward. As for (ii) the proof is similar to that given in Remark 3.5.  14 GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI

Recalling that, if B ( , ), then B ↾ L( , ×), it is natural to give ∈B E F D∈ D D the following

Definition 3.25. Let , , , F and B ( , ), C ( , ). We E E′ F F ′ ∈ 0 ∈B E F ∈B E′ F ′ say that B and C are equivalent, and write B C, if B ↾ = C ↾ . ≡ D D For fixed , F ,with , X ( , ) the function E F ∈ 0 E ⊆ F ∈C E F 1 λ ̺ , (X ) (X λI )− ( , ) ∈ E F E → E − E ∈B F E 1 1 is analytic on ̺ , (X ). Thus, if we define (X λI)− := (X λI )− ↾ , E F E ( , ) E E D 1 − E F − for every λ ̺ , (X ), (X λI)− L( , ×). ∈ E F E − ( , ) ∈ D D Moreover, the function E F

1 L λ ̺ , (X ) (X λI)(− , ) ( , ×) ∈ E F E → − E F ∈ D D is analytic if L( , ) is endowed with the inductive topology τ defined D D× in by the family of Banach spaces ( , ); , F . {B E F E F ∈ 0} Let us fix F0. If λ ̺ , (X), for some F0, , then this E ∈ ∈ E F F ∈ E ⊆ F F is unique. Hence, the function f defined by E 1 (11) f (λ) = (X λI)(− , ), if λ ̺ , (X) E − E F ∈ E F is a single valued function, analytic on every connected component of every open set ̺ , (X). E F More in general, when , run over F we get a whole family of resolvent E F 0 functions and several different situations are possible. F0 If λ0 ̺ (X), then λ0 ̺ , (X ) for some , F0, . Then ∈ ∈ E F E E F ∈ E ⊆ F there exists an open disk Dr = λ : λ λ0 < r ̺ , (X ) and the E F E 1 { | − | } ⊂ function λ Dr (X λI )− is analytic on Dr. But it may happen that ∈ → E − E λ0 ̺ ′, ′ (X ′ ) for another couple ′, ′ F0, ′ ′. The corresponding E F E ∈ E1 F ∈ E ⊆ F resolvent function λ (X ′ λI ′ )− is analytic in some open disk Dr′ , but E E 1 → − 1 (X λI )− and (X ′ λI ′ )− need not be equivalent on Dr Dr′ . E − E E − E ∩

Lemma 3.26. Let λ0 ̺ , (X ) ̺ ′, ′ (X ′ ) for some , ′, , ′ F0. ∈ E F E ∩ E F E E E F F ∈ Let us assume that . The corresponding resolvent functions are equiv- E⊂E′ alent on some open neighborhood of λ0 and they are direct analytic contin- uations of each other.

1 Proof. If ′, then ′ and (X ′ λI ′ )− is an extension of (X E E E 1 E⊂E F ⊂ F − − λI )− . Indeed, since X λI X ′ λI ′ , their inverses coincide on E E E E E − ⊆ − 1 1 (X λI ) = . This implies that (X ′ λI ′ )− (X λI )− in the E − E E F E − E ≡ E − E OPERATORSINRHS:SPECTRALPROPERTIES 15 sense of Definition 3.25 and the corresponding resolvent functions are direct analytic continuation one of the other. 

Proposition 3.27. Let λ0 ̺ , (X ) ̺ ′, ′ (X ′ ) for some , ′, , ′ ∈ E F E ∩ E F E E E F F ∈ F0. If there exist and ′ in F0 such that ′ and λ0 ̺ , ′ (X), G G G G 1 G⊆E∩E 1 ∈ then the functions λ (X λI )− and λ (X ′ λI ′ )− are analytic → E − E → E − E continuations of each other in some open connected set containing λ0.

1 Proof. One can apply the reasoning of Lemma 3.26 to (X λ0I )− and E E 1 1 1 − (X λ0I )− or to (X ′ λ0I ′ )− and (X λ0I )− . Then, the functions G G E E G G − 1 − 1 − λ (X λI )− and λ (X λI )− are equivalent in some open disk Dr → E − E → G − G and they are direct analytic continuation one of the other. The same happens 1 1 of course for the functions λ (X ′ λI ′ )− and λ (X λI )− . Thus → E − E → G − G the functions indexed by and are analytic continuations of each other E E′ (but they need not be direct analytic continuation one of the other). 

Remark 3.28. Let us finally examine the general situation. Let λ 0 ∈ ̺ , (X ) ̺ ′, ′ (X ′ ) for some , ′, , ′ F0. Since is dense in E F E ∩ E F E E E F F ∈ D E ∩ ′, for the Mackey topology, then (X λ0I ) ↾( ′) ( ′) = (X ′ E E − E E∩E E∩E E − λ0I ′ ) ↾( ′) ( ′) =L ′. The equalityL= ′ does not hold E E∩E E∩E 1 ⊆F∩F F ∩ F in general. (X λ0I )− is well-defined in and then also inL. The same E E − 1 F holds for (X ′ λ0I ′ )− but these operators need not be equivalent. E − E If λ ̺F0 (X), we denote by (X λ I) 1 the collection of all resolvent 0 ∈ − 0 − operators corresponding to λ and by λ ̺F0 (X) (X λI) 1 the mul- 0 ∈ → − − tivalued resolvent function described above. For each fixed F , the E ∈ 0 function f , defined in (11), can be viewed as a single valued branch of E λ (X λI) 1. In Example 4.2 we shall see a concrete realization of this → − − situation.

4. Hilbert space operators

Let X L( , ) and assume that both X and X map into . Then ∈ D D× † D H X can be viewed as a closable operator in . Let X be its closure and H ̺ (X) its usual resolvent set. We denote by X the Hilbert space obtained H H by endowing D(X) with the graph norm. If X , F0 then X ( X , ) F0 H H∈ ∈C H H and ̺ X , = ̺ (X); so that σ (X) σ (X) (this implies, in particular H H H ⊆ H that, if X is bounded, σF0 (X) is compact). As we will see below, however, σ (X) and σF0 (X) need not coincide. H 16 GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI

4.1. Rigged Hilbert spaces generated by symmetric operators. Let A be a self-adjoint operator in Hilbert space . The space = (A), H D D∞ endowed with its natural topology tA, defined by the seminorms pn(ξ) = Anξ , n N, generates in canonical way a RHS, with a Fr´echet space. k k ∈ D For every n N we denote by n the Hilbert space obtained by endowing n ∈ H 2n 1/2 D(A ) with its graph norm n := (I + A ) and by n the space − k · k k · k H 2n 1/2 obtained by completing with respect to the norm n := (I +A )− H k·k− k · . Put 0 := . Then, the family of spaces n; n Z is totally ordered; k H H {H ∈ } namely,

n+1 n = 0 n n 1 ···H ⊂H ⊂···⊂H H ⊂H− ⊂H− − · · · Let us put S = A↾ and take F0 = n; n Z . The operator A (or its D {H ∈ } extension by duality denoted by the same symbol) maps n in n 1, n Z H H − ∀ ∈ continuously; hence S C( n, n 1), for every n Z. Let us denote by ∈ H H − ∈ ̺ (A) the usual resolvent of A. For shortness, we will put ̺n,m(S) := H ̺ n, m (S). H H Proposition 4.1. Let A be a self-adjoint operator, and F as above. Then D 0 ̺F0 (S)= ̺ (A). H 1 Proof. Let λ ̺n,n 1(S), n 0. Then, (A λI)− ( n 1, n), indeed ∈ − ≥ − ∈B H − H 1 (A λI)− ξ n C ξ n 1, ξ n 1 k − k ≤ k k − ∀ ∈H − i.e. 1 Rn(A λI)− ξ C Rn 1ξ , ξ n 1, k −1 k≤ k − k ∀ ∈H − 2n where Rn = (I + A ) 2 . Then 1 (A λI)− Rnξ C Rn 1ξ C Rnξ , ξ n. k − k≤ k − k≤ k k ∀ ∈H This implies that (A λI) 1 is bounded w.r. to the norm of on the − − H subspace R , and, since R is invertible with bounded inverse, it follows nHn n that λ belongs to the usual resolvent, ̺ (A), of A. Let µ ̺ n+1, n(S), H ∈ − − n 0. Then, by Remark 3.16, µ ̺n,n 1(S). The first part of the proof ≥ ∈ − then shows that µ ̺ (A). This, in turn, implies that µ ̺ (A). H H ∈ 1 ∈ Let now λ ̺ (A); then (A λI)− ( ). Now we want to prove H ∈ − ∈ B H1 that λ ̺n,n 1(S), n 1. Since (A λI)− maps n into n 1 and ∈ − ∀ ≥ − H H − it is is clearly injective, we only need to prove the surjectivity. Let η ∈ n 1 and ξ D(A) be such that (A λI)ξ = η, then necessarily H − ⊆ H ∈ − ξ = (A λI) 1η . − − ∈Hn Finally, it is easy to see that ̺ (S)= if m = n 1. n,m ∅ 6 − OPERATORSINRHS:SPECTRALPROPERTIES 17

In conclusion, ̺F0 (S)= ̺ (A).  H The situation becomes more involved if S is a symmetric operator pos- sessing many self-adjoint extensions. In this case in fact, we have to deal with a true multivalued resolvent function.

Example 4.2. Let S be a closed symmetric operator with equal and finite defect indices. Again we put

n ∞(S)= (S ) D D n 0 \≥ ′ and, also in this case, (S) is dense in [28, Prop. 1.6.1]. If S is a D∞ H self-adjoint extension of S, we clearly have ′ n (Sn) (S ), n 1 D ⊂ D ∀ ≥ and then ′ ∞(S) ∞(S ). D ⊂ D Let assume that S has a family Sα α I of self-adjoint extensions. We put { } ∈ = (Sn) endowed with the graph norm as before and consider Hα,n D α F0 = α,n; α I,n N {H ∈ ∈ } Then S ( , ) if and only if α = β and m n 1. By the previous ∈C Hα,n Hβ,m ≤ − result, it follows that

̺ α,n, α,n−1 (S)= ̺ (Sα). H H H F0 Hence ̺ (S)= α I ̺ (Sα). ∪ ∈ H Let Sα and Sβ be two different self-adjoint extensions of S; then the essential spectra σess(Sα) and σess(Sβ) are equal [35, Theorem 8.18], while the point spectra σp(Sα) and σp(Sβ) are different, in general. A well known concrete example is provided by the differential operator on an interval of the real line. Let us consider, in fact, x 2 2 D(S∗)= f L ([0, 1]) : f(x)= f(0) + g(t)dt; g L ([0, 1]) , { ∈ ∈ } Z0 D(S)= f D(S∗) : f(0) = f(1) = 0 , { ∈ } D(Sα)= f D(S∗) : f(1) = αf(0), α C with α = 1 { ∈ ∈ | | } and S f := ig. ∗ − Then S is closed and symmetric but not self-adjoint, Sα is a self-adjoint extension of S for every α C with α = 1. The point spectrum of S ∈ | | 18 GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI is empty and its whole spectrum is σ(S) = C; as for the Sα’s, one has σp(Sα)= arg(α) + 2kπ,k Z = σ (Sα). Hence { ∈ } H F0 ̺ (S)= ̺ (Sα)= (C arg(α) + 2kπ,k Z )= C. H \ { ∈ } α: α =1 α: α =1 [| | [| | It is worth remarking that the result ̺F0 (S) = C does not change if we take as I a proper subset J of the unit circle α C; α = 1 consisting of { ∈ | | } at least two different (modulo 2π) points. Now we consider the “global” resolvent multivalued function

F0 1 1 λ ̺ (S) (S λ I)− = (S λI)− . ∈ → − { α − } Let us first introduce some notation. We recall that the operator S has defect indices (1, 1). For every λ C R, the subspace Mλ of solutions of ∈ \ the equations

S∗Φλ = λΦλ, has dimension 1. iλx It is easily seen that Φλ(x)= Ke .

The Krein formula [1], allows us to compute:

1 1 (12) ((S λI)− (S λI)− )g = µ(λ) g Φ Φ , α − − β − λ λ where µ(λ) = 0 and the functions λ µ(λ) and λ Φ are analytic in 6 → → λ ̺ (Sα) ̺ (Sβ). H H This formula shows that, in general, (S λI) 1 and (S λI) 1 are not T α − − β − − analytic continuations of each other, since the r.h.s. in (12) is zero if and only if Sα = Sβ. More precisely, in our case

1 1 (((S λI)− (S λI)− )g)(x) α − − β − 1 1 1 i(x+1)λ iλτ = ie g(τ)e− dτ α eiλ − β eiλ  − −  Z0 which vanishes if and only if α = β.

d2 Example 4.3. Let us consider the operator H = 2 . Our aim here is to 0 − dx provide a family of intermediate spaces for H0 and find the corresponding resolvent. Of course, there are several possible domains such that H (or, D 0 better, its restriction to ) can be considered as an element of L( , ). D D D× Let us examine shortly two cases. OPERATORSINRHS:SPECTRALPROPERTIES 19

(1) First, we take = := (R). In this case, H0 is the so-called free D S S Hamiltonian of quantum mechanics and the simplest Schr¨odinger operator. It is easily seen that H L( , ) (more precisely, H 0 ∈ S S× 0 ∈ ( )) and it is essentially self-adjoint on ; its closure, H := H , is L† S S 1 0 defined on the W 2,2(R). As discussed at the beginning of Section 4.1, a natural choice for the family F0 consists in taking the scale of Hilbert spaces generated by H1. This is, actually, a chain 2m,2 of Sobolev spaces; i.e., F0 = W (R)[ m]; m Z (as before, 2n 1/2 { k · k ∈ } n := (I + H )± , with n N). Hence, by Proposition k · k± k · k ∈ 4.1, F0 + σ (H0)= σ (H1)= R 0 . H ∪ { } (2) A second case of interest arises if we impose a boundary condition

by taking, for instance, := y := f : f(y) = 0 , y R, with D S { ∈ S } ∈ the topology induced by . This domain is used when perturbing the S free Hamiltonian with a δ-interaction centered at y, [2]. The domain 2,2 2,2 of the closure H1 of H0 is Wy (R) = f W (R); f(y) = 0 . As { ∈ } shown in [2, Th. 3.1.1], the operator H1 is no longer self-adjoint; it has, in fact, defect indices (1,1) and, for each α R, it possesses a ∈ self-adjoint extension Hα. The domain of Hα is

1,2 2,2 + D(Hα)= g W (R) W (R y ) : g′(y ) g′(y−)= αg(y) . ∈ ∩ \ { } − As for the spectrum, we have

R+ 0 if α 0 σ (Hα)= ∪ { } 2 ≥ H R+ α , 0 if α< 0, ( ∪ {− 4 } 2 since for α< 0, α is an eigenvalue of H . − 4 α Then, proceeding as in Example 4.2, we get that

F0 + ̺ (H)= ̺ (Hα)= C R 0 H \ { ∪ { }} α R [∈ n where F0 = α,n; α I,n N (with α,n = (H ) endowed with {H ∈ ∈ } H D α the graph norm, as before). Hence, also in this case, we get

F0 σ (H)= R+ 0 . ∪ { } 4.2. Generalized eigenvalues and generalized eigenvectors. As an- nounced in Section 3, we consider now the problem of the existence of a complete set of generalized eigenvalues of a self-adjoint operator and give H a slight improvement of Gelfand theorem on this subject. 20 GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI

Let A be a self-adjoint operator with A I in Hilbert space . Let us 1 ≥ H assume that A− is a Hilbert-Schmidt operator. Let us take, as in Section 4.1, = ∞(A) and F0 = n,n Z the chain of Hilbert spaces generated D D {H ∈ } by the powers of A (in this case, the graph norm of n can be equivalently be n H taken as = A ). Since is a Fr´echet space, LB( , )= L( , ) k·kn k ·k D D D× D D× and, for every X L( , ×), there exists n N such that X ( n, n). ∈ D D ∈ ∈C H H− Hence, we can define an operator X0 in the following way

D(X )= ξ : Xξ , 0,n { ∈H∩Hn ∈ H} X ξ = Xξ, ξ D(X ). 0,n ∈ 0 In general, D(X ) is not dense in and may reduce to the null subspace 0 H only.

Theorem 4.4. Let X L( , ) be symmetric, with = (A) where ∈ D D× D D∞ A is a self-adjoint operator with A I, in Hilbert space , whose inverse 1 ≥ H A− is Hilbert-Schmidt. Assume that X ( n, n), for some n N, ∈ C H H− ∈ and that X0,n is densely defined and essentially selfadjoint. Then X has a complete set of generalized eigenvectors.

Proof. Let E(λ) be the spectral family of X and ζ a unit vector in . Put { } 0 H σ(λ) = E(λ)ζ ζ . Then σ defines a measure on R and almost everywhere h | i with respect to σ there exists the derivative dE(λ)ζ =: χ(λ). dσ(λ) As shown in [17, Sect. 4.3, Theorem 1], χ(λ) is a continuous linear functional on . This set of functionals is complete in the sense that, for every vector D φ M(ζ), the closed subspace generated by E(λ)ζ; λ R , one has [17, ∈ { ∈ } Sect. 4.3, Theorem 2]

(13) φ = χ(λ) φ χ(λ)dσ(λ) and φ 2 = χ(λ) φ χ(λ) 2dσ(λ). h | i k k |h | i | ZR ZR We want to prove that χ(λ) n. We denote by ∆ the interval [α, β] ∈ H− containing the point λ and by E(∆) the operator E(β) E(α). Assume − that the interval ∆ contracts to the point λ. Then, for every ξ , we have ∈ D E(∆)ζ χ(λ) φ = lim φ h | i ∆ σ(∆) |   n E(∆)ζ n = lim A− A φ . ∆ σ(∆) |   OPERATORSINRHS:SPECTRALPROPERTIES 21

Now, we observe that E(∆)ζ/σ(∆) = 1; hence by the compactness of k k A 1, there exists a subnet E(∆ )ζ/σ(∆ ) such that A nE(∆ )ζ/σ(∆ ) − { ′ ′ } − ′ ′ converges in . This implies that H χ(λ) φ C Anφ = C φ , φ . | h | i|≤ k k k kn ∈ D Thus χ(λ) extends to a continuous conjugate linear functional on . We Hn denote this extension by the same symbol. Since X ( n, n), denoting ∈C H H− by X the corresponding extension we get, for every φ , in complete (n) ∈ D analogy to [17, Sect. 5.2, Theorem 1], X χ(λ) φ = χ(λ) Xφ (n) | h | i E(∆)ζ = lim Xφ ∆ σ(∆) |   = λ χ(λ) φ . h | i Hence χ(λ) is an F0-generalized eigenvector. The final step of the proof consists in decomposing into an orthogonal sum of subspaces of the type H M(ζα), as in [17]. 

Remark 4.5. We recall a well known situation where Theorem 4.4 can be applied. This is the case where X ( 1, 1) and [A0, X] ( 1, 1), ∈C H H− ∈C H H− (here [A0, X] denotes the commutator of A0 := A ↾ †( ) and X which D∈L D is well-defined since A has a continuous extension Aˆ to and, therefore, 0 0 D× one can define [A , X]ξ = Aˆ Xξ XA ξ, ξ ). 0 0 − 0 ∈ D Then by the commutator theorem [26, Sec. X.5], X0 is densely defined and it is essentially self-adjoint on every core for A.

Remark 4.6. Once a notion of spectrum is at hand, it is natural to pose the question as to whether other aspects of the beautiful for oper- ators in Hilbert space extend to the different environment we have considered here. Thus, one first asks oneself if a given operator X = X L( , ) † ∈ D D× gives rise to a resolution of the identity in a possibly generalized sense. Some hints come from the previous discussion on generalized eigenvectors and eigenvalues we have done in this section. However, in the case consid- ered here, for instance in Theorem 4.4, we have, since from the very be- ginning, a well behaved operator X and both the resolution of the identity χ(λ); λ R of the rigged Hilbert space obtained in (13) and the mea- { ∈ } sure σ are determined by the spectral family of the essentially self-adjoint operator X0. In the general case, we conjecture that it is not possible to associate a resolution of the identity to a symmetric element of L( , ). D D× 22 GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI

These operators in fact can be so far from being Hilbert space operators (see, for instance the example in Section 5) as to make impossible the use of the powerful tools at our disposal in a Hilbert space theory. Thus, a first step in the direction of getting more results on the existence of a resolution of the identity should consist in considering operators with a sufficiently large restriction to the central Hilbert space . We hope to consider this problem H in a future paper.

Remark 4.7. In the language of Quantum Mechanics the resonant spectrum of the Hamiltonian operator H of a physical system (a selfadjoint operator in Hilbert space) consists of complex nonreal generalized eigenvalues when the operator (or, more precisely, a restriction of it) is considered as acting in a rigged Hilbert space. The smallest space of the rigged Hilbert space is D determined by physical conditions (outgoing boundary conditions, labeled observables, etc.) [15, 12, 7] so that the whole construction depends on the model under consideration. There are several nonequivalent definitions of the resonant spectrum (some of them are discussed also in [7, Sect. 7.2.2, 7.2.3]) but it mostly stems out from the spectrum of some extension of H, in the very same spirit of what we have done in this paper. The resonant (Gamow) states are eigenvectors of the extension H of H to (which × D× certainly exists if H ). These eigenvectors certainly do not belong to D ⊆ D since H can only have real eigenvalues; thus they are true objects in . H D× As shown before, we can perform some spectral analysis of H by choosing a convenient family of interspaces F0. The resonant spectrum however is F0 not contained in σ (H) if F0 contains D(H), considered as Hilbert space with its graph norm, and . Indeed, in this case, σF0 (H) σ (H) R. H ⊆ H ⊆ However, the complex eigenvalues can certainly be found in one of the sets F0 σ , (H) that determine the spectrum σ (H), if the family of interspaces E F F0 is sufficiently rich. This situation shows that the whole family of spec- tra σ , ; , F0 should be taken into account in order to get enough { E F E F ∈ } information on H.

5. Examples

The following examples give some motivations to the ideas developed in the paper. OPERATORSINRHS:SPECTRALPROPERTIES 23

Example 5.1. Let and be as in Example 3.6. The operator M S S× δ defined by M : , with δ S → S× M f g = δf g = δ f ∗g = f(0)g(0) h δ | i h | i h | i is the multiplication operator by the Dirac δ distribution. It is easily seen that M L( , ). We want to determine the spectrum of M . First we δ ∈ S S× δ look for eigenvalues, i.e. for all λ C such that (Mδ λI)f = 0 has nonzero ∈ − solutions. If λ is an eigenvalue and f is an eigenvector, then M f g = λ f g , g ; h δ | i h | i ∀ ∈ S i.e., (14) f(0)g(0) = λ f g . h | i If we take f = g we obtain f(0) 2 = λ f 2, hence λ 0. | | k k2 ≥ It is clear that λ = 0 is an eigenvalue. The corresponding eigenspace := f , f(0) = 0 . This subspace is closed in with its usual S0 { ∈ S } S topology, but it is dense in L2(R) with respect to its norm. If λ> 0, from (14), we get f(0)g(0) = λ f g λ f g , g . | | | h | i|≤ k k2k k2 ∀ ∈ S n 2 2 If we choose gn(x) = exp n x /2 , n N, then gn = 1 and we √π {− } ∈ k k should have n f(0) λ f , n N; √π | |≤ k k ∈ then f(0) = 0. This in turn implies that λ = 0, a contradiction. Hence, 0 is the unique eigenvalue of Mδ. Thus, if λ C 0 , the operator Mδ λI is injective. It is easy to check ∈ \ { } − that the range (M λI) is the following subspace of R δ − S× (M λI)= f(0)δ λf; f . R δ − { − ∈ S} There are no values of λ for which this space contains L2(R). So that 1 (Mδ λI)− cannot be defined on the whole ×. − S k,2 Let us consider as F0 the family of Sobolev spaces W (R) and their k,2 0,2 2 duals; i.e., F0 = W (R), k Z , where W (R) = L (R). From now on { ∈ } we shorten W k,2(R) as W k,2, etc. k,2 (k) If k > 0, Mδ has a continuous extension to W denoted by Mδ and defined by M (k)f g = δf g = f(0)g(0), f, g W k,2. δ | h | i ∀ ∈ D E 24 GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI

The continuity of every function in W k,2 ensures that the right hand side (k) of the previous equality is well-defined. It is easily seen that Mδ has no nonzero eigenvalues. By [11, Th. VIII.7], every f W k,2 is bounded and ∈ then one has, for some C > 0, (k) M f g = f(0)g(0) f g C f k,2 g r,2. δ | | | ≤ k k∞k k∞ ≤ k k k k D E ( k) r,2 k,2 r,2 This implies that Mδ f W − . Hence Mδ (W ,W − ), for every (k) ∈ ∈ C k,r > 0. Since (M λI), λ C never contains L2, it follows that R δ − ∈ ̺k, r(Mδ) = , where, for short, ̺k, r(Mδ) = ̺W k,2,W −r,2 (Mδ). On the − − ∅ k,2 r,2 k,2 r,2 other hand, Mδ (W ,W ) (W − ,W − ), k,r > 0 since, otherwise 6∈ C ∪C F0 δ should be regular distribution. In conclusion, σ (Mδ) = C and 0 is the unique eigenvalue.

More generally, one can consider the distribution C = n Z δn, where ∈ δ f = f(n), the so-called δ-comb, and the operator M defined on by h n | i C P S M f g = C f ∗g = f(n)g(n). h C | i h | i n Z X∈ The series on the r.h.s. converges for every f, g . Every n Z is an ∈ S ∈ eigenvalue with corresponding eigenspace n = f ; f(n) = 0 . If we k,2 S { ∈ S } take again F0 = W , k Z , one finds, also in this case, by an argument { ∈ } F0 similar to the previous one, that σ (MC )= C.

Example 5.2. Let us consider the operator MΦ of multiplication by a tempered distribution Φ:

M f g = Φ f ∗g , f,g . h Φ | i h | i ∈ S As shown in [33], M L( , ). Φ ∈ S S× To begin with, let us first consider the case Φ = Φh, where Φh is the regu- lar tempered distribution defined by a measurable slowly increasing function h; this means that there exists m N 0 such that ∈ ∪ { } m h(x) (1 + x )− dx < . | | | | ∞ ZR The action of MΦh is given by

M f g = Φ f ∗g = h(x)f(x)g(x)dx, f, g h Φh | i h h | i ∈ S ZR and M L( , ). Φh ∈ S S× The eigenvalue equation (15) M f λf = 0, f Φh − ∈ S OPERATORSINRHS:SPECTRALPROPERTIES 25 has nonzero solutions in if, and only if, h is constant a.e. and h(x) = a, S for almost every x R, with a C and λ = a. In this case, σ(MΦ ) = ∈ ∈ h σp(MΦh )= a . { } 1 The operator (MΦ λI)− exists for every λ h(R), the closure of the h − 6∈ essential range of h. The operator (M λI) 1 can be identified with Φh − − the operator of multiplication by the function g = (h λ) 1. Clearly, − − (M λI) 1 is defined on the subspace Φh − − M := Φ ×; Φ is regular and Φ = Φ(h λ)f , f . { ∈ S − ∈ S} × Since M $ ×, then ̺ , (MΦh )= . S S S ∅ k,2 Let us consider as F0 the family of Sobolev spaces W (R) and their duals as in Example 5.1.

2 2 For shortness we put ̺W k, ,W m, (MΦh ) := ̺k,m(MΦh ), k,m Z. k,2 ∈ Let us assume that h L∞(R). Then, for every f W , g ∈ ∈ ∈ S

(16) MΦh f g h f 2 g 2 h f k,2 g r,2, k, r N. | h | i|≤k k∞k k k k ≤ k k∞k k k k ∀ ∈ k,2 Hence, for every f W , MΦh f is a continuous linear functional on r,2 ∈ k,2 r,2 every W , r 0 and, by (16), MΦ (W ,W − ), for every k, r N. ≥ h ∈ C ∈ The operator (M λI) 1 is defined on the subspace Φh − − r,2 k,2 Mk,r := Φ W − ; Φ is regular and Φ = Φ(h λ)f , f W . { ∈ − ∈ } Then, we have the following situation:

k = 0, r = 0: In this case ̺0,0(MΦ )= C h(R); h \ k> 0, r = 0: The operator MΦh λI is not onto, hence ̺k,0(MΦh )= ; r,2 − ∅ k> 0,r > 0: Mk,r $ W − and ̺k, r(MΦh )= . − ∅ It remains to check the cases of indices both positive or both negative. For this, let us assume that h L W 1,2. In this case M (W p,2,W q,2) ∈ ∞ \ loc Φh 6∈ C with p,q > 0, since, if hf W q,2 then h W q,2 W 1,2 . By duality there ∈ ∈ loc ⊂ loc cannot be a continuous extension of M belonging to (W p,2,W q,2) with Φh C p,q < 0, since, otherwise, we would have M (W q,2,W p,2). Finally, Φh − − ∈ C p,2 q,2 we notice that no continuous extension of MΦh belonging to (W − ,W ) p,2 C p,2 with p,q > 0 may exists, since, otherwise, from W W − it would q,2 p,2 ⊂ follow MΦh g W , for every g W and this is excluded. In conclusion, F0 ∈ ∈ ̺ (MΦ )= C h(R). h \ Let us finally examine a case where h is not bounded but slowly increasing.

For instance, h(x)= x, x R. We consider again as F0 the chain of Sobolev ∈ spaces considered above. First we notice that every real number λ is an F0- generalized eigenvalue: indeed, the distribution δλ, the Dirac delta centered 26 GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI at λ, is in W 1,2 and one has ∈ − (M λI)δ f = δ (x λ)f = λf(λ) λf(λ) = 0, f W 1,2. h x − λ | i h λ | − i − ∀ ∈ Since, for every r 0 ≥ M f g xf g xf g , f , g W r,2, | h x | i|≤k k2k k2 ≤ k k2k kr,2 ∀ ∈ S ∈ M f is, for every f , an element of W r,2, but M (W k,2,W r,2), x ∈ S − x 6∈ C − k > 0. Moreover, M (W k,2,W r,2), for k, r 0, since this would imply x 6∈ C ≥ that the operator of multiplication by x, regarded as an operator in L2 should be everywhere defined on W k,2 and this is not true. By a duality argument r,2 k,2 r,2 k,2 we can also exclude that Mx (W − ,W ) and Mx (W − ,W − ), F0∈C ∈C r,k > 0. Thus in this case σ (Mx)= C. The situation changes if we include in F the extreme spaces and : 0 S S× in this case the spectrum coincides with the usual spectrum σ (Mx) of Mx H since M ( ,L2) and M is essentially selfadjoint on . x ∈C S x S

Example 5.3. As it is well known, the Hermite functions defined by φ0(x)= 1/4 x2/2 π− e− and n n 1/2 n 1/4 x2/2 d x2 φ (x) = (2 n!)− ( 1) π− e e− n − dx   constitute an orthonormal basis of L2(R). If f , then f has the expansion ∈ S

∞ m (17) f = cnφn, with sup cn n < , m N n | | ∞ ∀ ∈ nX=0 and the series converges in the topology of . The space of sequences c S { n} satisfying, for a given m N, ∈ m sup cn n < , n | | ∞ will be denoted by sm. We will indicate with s the so-called space of rapidly s s decreasing sequences; i.e., = m N m. ∈ An element F can be represented as ∈ S× T ∞ s (18) F = bnφn, with bn M(1 + n) , for some M > 0, s N, | |≤ ∈ n=0 X the series being weakly convergent. Let now a be a sequence of complex numbers such that { n} s an cn (19) cn , m N such that sup | || m| < . ∀{ }∈ ∃ ∈ n (1 + n) ∞ OPERATORSINRHS:SPECTRALPROPERTIES 27

Then ∞ ∞ f = c φ Af := a c φ n n 7→ n n n n=0 n=0 X X defines a linear map from into . Since is a reflexive Fr´echet space, it is S S× S sufficient to check that A is continuous from [σ( , )] into [σ( , )]. S S S× S× S× S Continuity follows immediately from the fact that the map

∞ ∞ A† : f = d φ A†f := a d φ , d s, n n 7→ n n n { n}∈ Xn=0 nX=0 is the adjoint of A. Hence A L( , ). A natural choice of F consists in ∈ S S× 0 taking the spaces m whose elements are all F S× for which the expansion S s ∈ (18) has coefficients in m and their dual m×. It is easy to check that every S 1 an is an eigenvalue of A. Thus, if λ an, n N , the sequence is 6∈ { ∈ } an λ 1 − bounded. For these values of λ, the operator (A λI)− maps nm intoo m F0 − S S (and × into ×) continuously. Hence σ (A)= an, n N . Sm Sm { ∈ }

Acknowledgement One of us (SDB) gratefully acknowledges the warm hospitality of Prof. Maria Fragoulopoulou and of the Department of Math- ematics of the University of Athens where a part of this work was done.

References

[1] N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert space, Dover Publ., 1993. [2] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden Solvable Models in Quantum Mechanics. Springer Verlag, New York, 1988. [3] J-P. Antoine, G. Bellomonte, C. Trapani Fully representable and *-semisimple topo- logical partial *-algebras Studia Mathematica, 208 (2012), 167-194. [4] J-P. Antoine, A. Inoue, C. Trapani, Partial *-algebras and their operator realizations, Kluwer, Dordrecht, 2002. [5] J.-P. Antoine, C.Trapani and F. Tschinke, Bounded elements in certain topological partial *-algebras, Studia Math. 203 (2011), 223-251. [6] J.-P.Antoine, C.Trapani and F. Tschinke, Spectral properties of partial *-algebras, Mediterranean J. Math. 7 (2010), 123-142. [7] J-P. Antoine, C. Trapani, Partial Inner Product Spaces — Theory and Applications, Springer Lecture Notes in Mathematics, vol. 1986, Berlin, Heidelberg, 2009. [8] H. Baumg¨artel Generalized Eigenvectors for Resonances in the Friedrichs Model and Their Associated Gamov Vectors, Rev. Math. Phys., 18 (2006), 61 –78. [9] G. Bellomonte, C. Trapani, Rigged Hilbert spaces and contractive families of Hilbert spaces Monatshefte f. Math., 164, (2011) 271-285. [10] G. Bellomonte, C. Trapani, Quasi *-algebras and generalized inductive limits of C*- algebras Studia Mathematica, 202 (2011), 165-190. 28 GIORGIA BELLOMONTE, SALVATORE DI BELLA, AND CAMILLO TRAPANI

[11] H. Brezis, Analyse fonctionnele, Th´eorie et applications, Masson, Paris, 1987. [12] O. Civitarese and M. Gadella, Physical and mathematical aspects of Gamow states, Phys. Rep. 396 (2004), 41-113. [13] O. Costin, A. Soffer, Resonance Theory for Schr¨odinger Operators, Comm. Math. Phys. 224 (2001), 133-152. [14] R. de la Madrid, Rigged Hilbert space approach to the Schr¨odinger equation. J. Phys. A 35 (2002), 319-342. [15] R. de la Madrid and M. Gadella A Pedestrian Introduction to Gamow Vectors Am. J. Phys. 70 (2002), 626-638. [16] G. Epifanio and C. Trapani, Quasi *-algebras valued quantized fields, Ann. Inst. H. Poincar´e 46 (1987), 175–185. [17] I.M. Gelfand and G E. Shilov, Generalized functions, vol. 3, Academic Press, New York, 1961. [18] G. K¨othe, Topological Vector Spaces II, Springer, Heidelberg, 1979.

[19] K-D. K¨ursten, The completion of the maximal Op*-algebra on a Fr´echet domain, Publ. Res. Inst. Math. Sci., Kyoto Univ. 22 (1986), 151–175. [20] K-D. K¨ursten, On algebraic properties of partial algebras, Rend. Circ. Mat. Palermo, Ser.II, Suppl. 56 (1998), 111–122. [21] K-D. K¨ursten and M. L¨auter, An extreme example concerning factorization products n on the Schwartz space S(R ), Note Mat. 25 (2005/06), 31–38. [22] G. Lassner, Topological algebras and their applications in Quantum Statistics, Wiss. Z. KMU-Leipzig, Math.-Naturwiss. R. 30 (1981), 572–595. [23] G. Lassner, Algebras of unbounded operators and quantum dynamics, A 124 (1984), 471–480. [24] M.Oberguggenberger, Multiplication of distributions and applications to partial dif- ferential equations, Pitman Research Notes in Mathematics Series, n. 259, Longman, Harlow, 1992. [25] G. Parravicini, V. Gorini, E.C.G. Sudarshan, Resonances, scattering theory, and rigged Hilbert spaces, J. Math. Phys. 21 (1980) 2208–2226. [26] M. Reed and B. Simon, Methods of Modern Mathematical Physics, II vol. Academic Press, New York, 1975. [27] A.Russo and C.Trapani, Quasi *-algebras and multiplication of distributions, J.Math.Anal.Appl. 215, (1997) 423-442. [28] K. Schm¨udgen, algebras and representation theory, Birkh¨auser Verlag, Basel, 1990. [29] K. Schm¨udgen, Unbounded Self-adjoint Operators on Hilbert Space , Springer Verlag , Dordrecht, 2012. [30] C.Trapani, Banach elements and spectrum in Banach quasi *-algebras, Studia Math- ematica, 172 (2006), 249-273. [31] C. Trapani, Quasi *-algebras of operators and their applications, 7 (1995), 1303– 1332. [32] C. Trapani, *-Representations, seminorms and structure properties of normed quasi *-algebras, Studia Mathematica, 186 (2008), 47-75. OPERATORSINRHS:SPECTRALPROPERTIES 29

[33] C. Trapani and F. Tschinke, Partial multiplication of operators in rigged Hilbert spaces, Integral Equations Operator Theory 51 (2005), 583–600. [34] C. Trapani and F. Tschinke, Partial *-algebras of distributions, Publ. RIMS, Kyoto Univ., 41 (2005), 259-279. [35] J. Weidman, Linear operators in Hilbert spaces, Springer Verlag, New York, 1976.

Dipartimento di Matematica e Informatica, Universita` di Palermo, I-90123 Palermo, Italy E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]