A functional model associated with a generalized Nevanlinna pair

Let L be a Hilbert space and let H be a Pontryagin space. For every selfadjoint linear relation Ã in H⊕L the pair {I+λψ(λ), ψ(λ)}, where ψ(λ) is the compressed resolvent of Ã, is a normalized generalized Nevanlinna pair. Conversely, every normalized generalized Nevanlinna pair is shown to be associ...

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Дата:	2010
Автор:	Neiman, E.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2010
Назва видання:	Український математичний вісник
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/124386
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Цитувати:	A functional model associated with a generalized Nevanlinna pair / E. Neiman // Український математичний вісник. — 2010. — Т. 7, № 2. — С. 197-211. — Бібліогр.: 16 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine

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spelling	irk-123456789-1243862017-09-25T03:03:12Z A functional model associated with a generalized Nevanlinna pair Neiman, E. Let L be a Hilbert space and let H be a Pontryagin space. For every selfadjoint linear relation Ã in H⊕L the pair {I+λψ(λ), ψ(λ)}, where ψ(λ) is the compressed resolvent of Ã, is a normalized generalized Nevanlinna pair. Conversely, every normalized generalized Nevanlinna pair is shown to be associated with some selfadjoint linear relation Ã in the above sense. A functional model for this selfadjoint linear relation Ã is constructed. 2010 Article A functional model associated with a generalized Nevanlinna pair / E. Neiman // Український математичний вісник. — 2010. — Т. 7, № 2. — С. 197-211. — Бібліогр.: 16 назв. — англ. 1810-3200 2010 MSC. 47B38, 47B25, 47B32, 47B50. http://dspace.nbuv.gov.ua/handle/123456789/124386 en Український математичний вісник Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	Let L be a Hilbert space and let H be a Pontryagin space. For every selfadjoint linear relation Ã in H⊕L the pair {I+λψ(λ), ψ(λ)}, where ψ(λ) is the compressed resolvent of Ã, is a normalized generalized Nevanlinna pair. Conversely, every normalized generalized Nevanlinna pair is shown to be associated with some selfadjoint linear relation Ã in the above sense. A functional model for this selfadjoint linear relation Ã is constructed.
format	Article
author	Neiman, E.
spellingShingle	Neiman, E. A functional model associated with a generalized Nevanlinna pair Український математичний вісник
author_facet	Neiman, E.
author_sort	Neiman, E.
title	A functional model associated with a generalized Nevanlinna pair
title_short	A functional model associated with a generalized Nevanlinna pair
title_full	A functional model associated with a generalized Nevanlinna pair
title_fullStr	A functional model associated with a generalized Nevanlinna pair
title_full_unstemmed	A functional model associated with a generalized Nevanlinna pair
title_sort	functional model associated with a generalized nevanlinna pair
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2010
url	http://dspace.nbuv.gov.ua/handle/123456789/124386
citation_txt	A functional model associated with a generalized Nevanlinna pair / E. Neiman // Український математичний вісник. — 2010. — Т. 7, № 2. — С. 197-211. — Бібліогр.: 16 назв. — англ.
series	Український математичний вісник
work_keys_str_mv	AT neimane afunctionalmodelassociatedwithageneralizednevanlinnapair AT neimane functionalmodelassociatedwithageneralizednevanlinnapair
first_indexed	2025-07-09T01:21:20Z
last_indexed	2025-07-09T01:21:20Z
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fulltext	Український математичний вiсник Том 7 (2010), № 2, 197 – 211 A functional model associated with a generalized Nevanlinna pair Evgen Neiman (Presented by M. M. Malamud) Abstract. Let L be a Hilbert space and let H be a Pontryagin space. For every selfadjoint linear relation Ã in H⊕L the pair {I+λψ(λ), ψ(λ)}, where ψ(λ) is the compressed resolvent of Ã, is a normalized generalized Nevanlinna pair. Conversely, every normalized generalized Nevanlinna pair is shown to be associated with some selfadjoint linear relation Ã in the above sense. A functional model for this selfadjoint linear relation Ã is constructed. 2010 MSC. 47B38, 47B25, 47B32, 47B50. Key words and phrases. selfadjoint relation, reproducing kernel Pon- tryagin space, generalized Nevanlinna pair, generalized Fourier trans- form. Introduction In 1946 M. G. Krein introduced in [12] the notion of the Q-function of a symmetric operator A in a Hilbert space with finite deficiency indices (m,m), which plays an important role in the description of generalized resolvents of A. Later on M. G. Krein and H. Langer in [14] have gen- eralized this notion to the case of a symmetric operator A with infinite indices acting in a Pontryagin space. In that paper it was shown that the Q-function uniquely determines a simple symmetric operator A up to unitary equivalence. Moreover, in [14] a functional model for a symmetric operator relied on ε-construction was introduced and investigated. This Received 13.04.2010 The author thanks the Department of Mathematics of Technical University of Berlin for supporting this research and hospitality during his stay in Berlin in May 2009 ISSN 1810 – 3200. c© Iнститут математики НАН України 198 A functional model... model has allowed them to solve an inverse problem for the Q-function, that is to find a criterion for a generalized Nevanlinna operator valued function to be the Q-function of a π-Hermitian operator. Functional models for symmetric operators in Hilbert spaces in terms of the Q-function were constructed in [1, 15]. Different functional mod- els for symmetric operators have been used by V. A. Derkach and M. M. Malamud in [10] (see also [16]) to solve the inverse problem for the Weyl function. Namely, starting with a uniformly strict R-function M(·) (i.e. R-function satisfying 0 ∈ ρ ( ImM(i) ) ) the authors in [10] constructed a model symmetric operator A and a boundary triplet for A∗ such that the corresponding Weyl function coincides with M(·). This result has also been extended to a wider class of strict R-functions (that is R-functions with ker ImM(i) = {0}) in order to realize any such R- function as the Weyl function corresponding to a generalized boundary triplet (see [10]). Later on a concept a generalized boundary triplet was generalized in [7] where a notion of a boundary relation and the corresponding Weyl family was introduced. Using this notion the authors of [7] have realized arbitrary Nevanlinna pair {ϕ,ψ} as the Weyl family of some symmetric operator corresponding to a boundary relation (a realization theorem). The proof in [7] was based on the Naimark dilation theorem and the so called main transform. Later on another proof of the realization theorem from [7] have been presented in [4, 5] where more general models for symmetric operators were introduced. In the present paper given a generalized Nevanlinna pair {ϕ,ψ} we construct a functional model for a selfadjoint linear relation Ã in Pon- tryagin space such that ϕ, ψ are recovered from Ã via (2.4). To make the paper clear for a wide audience we follow the scheme of [6] and use the notion of the selfadjoint linear relation Ã rather then the notion of the boundary relation Γ. In fact, one can treat Ã as the main transform of a boundary relation Γ and then the main result can be reformulated in terms of Γ. The paper is organized as follows. In Section 1 definitions of Nκ-pairs and normalized Nκ-pairs are given. In Section 2 we consider a pair {ϕ,ψ} generated by a selfadjoint relation Ã in a Pontryagin space and show that it is a normalized Nκ-pair. In Theorem 3.1 we prove the converse result. Moreover, a functional model for the selfadjoint relation Ã is constructed. In the rest of the paper properties of a generalized Fourier transform, associated with this model are studied. We also proved the unitary equivalence of an arbitrary L-minimal selfadjoint linear relation Ã to the model relation A(ϕ,ψ) in the reproducing kernel Pontryagin space. E. Neiman 199 1. Generalized Nevanlinna pairs Let L be a Hilbert space. By a kernel is meant a function Kω(λ) on Ω×Ω with values in the space of continuous operators on a Hilbert space L (Ω ⊂ C). We say that the kernel Kω(λ) has κ negative squares and write sq−K = κ if for any choice set of points ω1, . . . , ωn in Ω, vectors u1, . . . , un in L and ξj in space C n the quadratic form n∑ i,j=1 ( Kωj (ωi)uj , ui ) L ξjξi has at most κ negative eigenvalues, and for some choice of n, ωj , uj such matrix has exactly κ negative squares ([2]). Definition 1.1. A pair {Φ,Ψ} of [L]-valued functions Φ(·), Ψ(·) mero- morphic on C \R with a common domain of homomorphy hΦΨ is said to be a Nκ-pair (a generalized Nevanlinna pair) if: (i) the kernel N ΦΨ ω (λ) = Ψ(λ̄)∗Φ(ω̄) − Φ(λ̄)∗Ψ(ω̄) λ− ω̄ , has κ negative square on hΦΨ; (ii) Ψ(λ̄)∗Φ(λ) − Φ(λ̄)∗Ψ(λ) = 0 for all λ ∈ hΦΨ; (iii) for all λ ∈ hΦΨ ∩ C+ there is µ ∈ C+ such that 0 ∈ ρ ( Φ(λ) − µΨ(λ) ) and 0 ∈ ρ ( Φ(λ) − µΨ(λ) ) . Two Nκ-pairs {Φ,Ψ} and {Φ1,Ψ1} are said to be equivalent, if Φ1(λ) = Φ(λ)χ(λ) and Ψ1(λ) = Ψ(λ)χ(λ) for some operator function χ(·) ∈ [H], which is holomorphic and invertible on hΦΨ. The set of all equivalence classes of Nκ-pairs in L will be denoted by Ñκ(L). We will write, for short, {Φ,Ψ} ∈ Ñκ(L) for the generalized Nevanlinna pair {Φ,Ψ}. If Φ(λ) ≡ IL where IL is the identity operator in the space L then the Definition 1.1 means that Ψ(λ) is an Nκ(L)-function in the sense of [13]. Recall that the class Nκ(L) consists of meromorphic in C+∪C− operator valued functions Ψ(λ) such that Ψ(λ) = Ψ(λ)∗, and the kernel N Ψ ω (λ) = Ψ(λ) − Ψ(ω)∗ λ− ω has κ negative squares on hΨ — the domain of holomorphic Ψ. In this case the condition (iii) is satisfied automatically. Clearly, if {Φ,Ψ} is Nκ-pair such that 0 ∈ ρ ( Φ(λ) ) λ ∈ hΦΨ, then it is equivalent to the pair{ IL,Ψ(λ)Φ(λ)−1 } , where ΨΦ−1 ∈ Nκ(L). 200 A functional model... Definition 1.2. An Nκ-pair {φ, ψ} is said to be a normalized Nκ-pair if: (iii′) ϕ(λ) − λψ(λ) ≡ IL for all λ ∈ hϕψ. Clearly, every Nκ-pair {Φ,Ψ} such that 0 ∈ ρ ( Φ(λ) − λΨ(λ) ) for λ ∈ hΦΨ is equivalent to a unique normalized Nκ-pair {ϕ,ψ} given by ϕ(λ) = Φ(λ) ( Φ(λ)−λΨ(λ) )−1 , ψ(λ) = Ψ(λ) ( Φ(λ)−λΨ(λ) )−1 . (1.1) 2. Nκ-pair corresponding to a selfadjoint linear relation and a scale Let H be a vector space with a Hermitian form [·, ·]H : H × H → C. Two elements u and v of H are said to be orthogonal if [u, v]H = 0. Similarly, two subspaces of H are said to be orthogonal if every element of the first is orthogonal to every element of the second. The linear space( H, [·, ·] ) is called a Pontryagin space if there exists a direct orthogonal decomposition H = H+ ⊕ H−, where H+ with the form [·, ·]H is a Hilbert space and H− with the form −[·, ·]H is a Hilbert space of finite dimension. The space H is called Pontryagin space with κ negative squares (Πκ-space) if the dimension of H− is κ <∞ ([2]). We will use the notion of a linear relation in a space H. Recall, that a subspace T of H2 is called the linear relation in H. For a linear relation T in H the symbols domT , kerT , ranT , and mulT stand for the domain, kernel, range, and the multivalued part, respectively. The adjoint T+ is the closed linear relation in H defined by (see [2]) T+ = { {h, k} ∈ H2 : [k, f ]H = [h, g]H, {f, g} ∈ T }. (2.2) Recall that a linear relation T in H is called symmetric (selfadjoint) if T ⊂ T+ (T = T+, respectively). Let H be a Pontryagin space and L be a Hilbert space. Definition 2.1. A linear relation Ã = Ã∗ in H ⊕ L is said to be L- minimal if H0 := span {PH(Ã− λ)−1L : λ ∈ ρ(Ã)} = H, (2.3) where PH is the orthogonal projection onto the Pontryagin space H. Let Ã be a selfadjoint linear relation in H ⊕ L and let PL be the orthogonal projection onto the scale space L. Define the operator valued functions ϕ(λ) := IL + λPL(Ã− λ)−1 ↾L, ψ(λ) := PL(Ã− λ)−1 ↾L (λ ∈ ρ(Ã)). (2.4) E. Neiman 201 Clearly, ϕ(λ)∗ = ϕ(λ), ψ(λ)∗ = ψ(λ) (λ ∈ ρ(Ã)). (2.5) Proposition 2.1. Let H be a Πκ-space, let L be a Hilbert space and let Ã be a selfadjoint linear relation in H⊕ L. The pair of operator valued functions {ϕ,ψ} associated with Ã via (2.4) is a normalized Nκ′-pair where 0 ≤ κ′ ≤ κ. If, additionally, the linear relation Ã is L-minimal then κ′ = κ. Proof. In view of the properties (2.5) the kernel N ϕψ ω (λ) for the pair {ϕ,ψ} takes the form N ϕψ ω (λ) = ψ(λ)φ(ω̄) − φ(λ)ψ(ω̄) λ− ω̄ , λ, ω ∈ ρ(Ã). (2.6) It follows from Definition (2.4) that N ϕψ ω (λ) = ψ(λ) − ψ(ω)∗ λ− ω̄ − ψ(λ)ψ(ω)∗ = PL Rλ −Rω̄ λ− ω̄ ↾L −PLRλPLRω̄ ↾L = PLRλPHRω̄ ↾L, (2.7) where Rλ = (Ã−λ)−1 is a resolvent of lineal relation Ã. Let ωj belongs to ρ(A), uj belongs to space L and ξj belongs to space C n for j = 1, . . . , n. Then n∑ j,k=1 ( N ϕψ ωj (ωk)uj , uk ) L ξj ξk = n∑ j,k=1 ( (PLRωk PHRωj \|L)uj , uk ) L ξj ξk = n∑ j,k=1 [PHRωj uj , PHRωk uk]H ξj ξk = n∑ j,k=1 [gj , gk]H ξj ξk, (2.8) where gj = PHRωj uj . Since H is Πk-space and uj (j = 1, . . . , n) are arbitrary vectors in L then the quadratic form (2.8) has κ′ negative squares, where κ′ ≤ κ. Thus property (i) of Definition 1.1 is proved. The property (ii) is easily checked. Obviously ϕ(λ)− λψ(λ) ≡ IL for all λ ∈ ρ(Ã) and, hence, the pair {φ, ψ} is a normalized Nκ′-pair. If the relation Ã is L-minimal then the set span { PHRωu : ω ∈ ρ(Ã), u ∈ L } is dense in the space H. In this case the quadratic form (2.8) has exactly κ negative squares and hence the kernel N ϕψ ω (λ) has κ negative squares. Thus the pair {φ, ψ} is a normalized Nκ-pair. 202 A functional model... Definition 2.2. The pair of operator valued functions {ϕ,ψ} determined by (2.4) will be called the Nκ-pair corresponding to the selfadjoint linear relation Ã and the scale L. Note that if the vector values functions ϕ(λ) and ψ(λ) are defined by (2.4) then hϕψ = hϕ = hψ. 3. Functional model of a selfadjoint linear relation Consider the reproducing kernel Pontryagin space H(φ, ψ), which is characterized by the properties: (1) N φψ ω (·)u ∈ H(φ, ψ) for all ω ∈ hϕψ and u ∈ L; (2) for every f ∈ H(φ, ψ) the following identity holds [ f(·),Nφψω (·)u ] H(φ,ψ) = (f(ω), u)L, ω ∈ hϕ,ψ, u ∈ L. (3.9) It follows from (3.9) that the evaluation operator E(λ) : f 7→ f(λ) (f ∈ H(φ, ψ)) is a bounded operator from H(φ, ψ) to L. Also note that the set of functions {Nφψω (·)u : ω ∈ hϕψ, u ∈ L} is total in H(ϕ,ψ) ([2]). In the next theorem we give functional model of a selfadjoint linear relation Ã recovered from a Nκ-pair. Theorem 3.1. Let L be a Hilbert space and let {φ, ψ} be a normalized Nκ-pair. Then the linear relation A(φ, ψ) = {{[ f u ] , [ f ′ u′ ]} : f, f ′ ∈ H(φ, ψ), u, u′ ∈ L, f ′(λ) − λf(λ) = φ(λ)u− ψ(λ)u′, λ ∈ hϕψ } (3.10) is a selfadjoint linear relation in H(φ, ψ) ⊕ L and the normalized pair {φ, ψ} is the Nκ-pair corresponding to A(φ, ψ) and L. Proof. Step 1. Let us show that A(φ, ψ) contains vectors of the form {Fωv, F ′ ωv} := {[ Nω(·)v ψ(ω̄)v ] , [ ω̄Nω(·)v φ(ω̄)v ]} , v ∈ L, ω ∈ hϕψ, (3.11) E. Neiman 203 where Nω(·) := N φψ ω (·) and A′ := span { {Fωv, F ′ ωv} : v ∈ L, ω ∈ hϕψ } is a symmetric linear relation. Indeed, it follows from (3.10) and the equality (ω̄ − λ)Nω(λ)v = φ(λ̄)∗ψ(ω̄)v − ψ(λ̄)∗φ(ω̄)v that {Fωv, F ′ ωv} ∈ A(φ, ψ). For arbitrary ωj ∈ hϕψ, vj ∈ L (j = 1, 2) one obtains [ ω̄1Nω1 (·)v1,Nω2 (·)v2 ] H(φ,ψ) − [ Nω1 (·)v1, ω̄2Nω2 (·)v2 ] H(φ,ψ) + ( φ(ω̄1)v1, ψ(ω̄2)v2 ) L − ( ψ(ω̄1)v1, φ(ω̄2)v2 ) L = (ω̄1 − ω2) ( Nω1 (ω2)v1, v2 ) L − (( φ(ω̄2) ∗ψ(ω̄1) − ψ(ω̄2) ∗φ(ω̄1) ) v1, v2 ) L = 0, therefore, A′ is symmetric in H(φ, ψ) ⊕ L. Step 2. Let us show that ran(A′ − λ) is dense in H(φ, ψ)⊕L for λ ∈ hϕψ. Choose the vector {Fωv, F ′ ωv} with ω = λ̄. Since φ(λ)−λψ(λ) = IL then { Fλ̄v, F ′ λ̄ v − λFλ̄v } = {[ Nλ̄(·)v ψ(λ)v ] , [ 0 φ(λ)v − λψ(λ)v ]} = {[ Nλ̄(·)v ψ(λ)v ] , [ 0 v ]} ∈ A′ − λ. (3.12) Hence 0 ⊕ L ⊂ ran(A′ − λ). Taking {Fωv, F ′ ωv} with ω 6= λ̄ one obtains from (3.11) {[ Nω(·)v ψ(ω̄)v ] , [ (ω̄ − λ)Nω(·)v φ(ω̄)v − λψ(ω̄)v ]} ∈ A′ − λ and, hence, [ Nω(·)v 0 ] ∈ ran(A′−λ) for all ω 6= λ̄. Due to the properties (1) and (2) of H(φ, ψ) one obtains the statement. Thus A′ is an essentially selfadjoint lineal relation and hence (A′)+ is a selfadjoint lineal relation in H(ϕ,ψ) ⊕ L. Step 3. Let us show that A(φ, ψ) = (A′)+. Indeed, for every vector F̂ := {F, F ′} = {[ f(·) u ] , [ f ′(·) u′ ]} ∈ A(φ, ψ) 204 A functional model... where f, f ′ ∈ H(φ, ψ) and u, u′ ∈ L and arbitrary ω ∈ hϕφ, v ∈ L it follows from (3.10) that [ F ′, Fωv ] H(φ,ψ)⊕L − [ F, F ′ ωv ] H(φ,ψ)⊕L = [ f ′,Nω(·)v ] H(φ,ψ) − [ f, ω̄Nω(·)v ] H(φ,ψ) + ( u′, ψ(ω̄)v ) L − ( u, φ(ω̄)v ) L = ( f ′(ω) − ωf(ω) + ψ(ω̄)∗u′ − φ(ω̄)∗u, v ) L = 0. Hence F̂ ∈ (A′)+ and A(φ, ψ) ⊂ (A′)+. Conversely, if [ f ′,Nω(·)v ] H(φ,ψ) − [ f, ω̄Nω(·)v ] H(φ,ψ) + ( u′, ψ(ω̄)v ) L − ( u, φ(ω̄)v ) L = 0 for some f, f ′ ∈ H(ϕ,ψ), u, u′ ∈ L and all ω ∈ hϕ,ψ, v ∈ L, then f ′(ω) − ωf(ω) − ( φ(ω)u− ψ(ω)u′ ) = 0 and, hence, F̂ ∈ A(φ, ψ). This proves that (A′)+ ⊂ A(φ, ψ), and, hence, (A′)+ = A(φ, ψ). Therefore, A(φ, ψ) is a selfadjoint lineal relation. Step 4. Finally, we show that {ϕ,ψ} is a pair corresponding to the selfadjoint linear relation Ã and the scale L. Indeed, it follows from (3.12) and Definition 1.2 (iii′) that PL ( Ã(φ, ψ) − λ )−1 ↾L= ψ(λ), IL + λPL ( Ã(φ, ψ) − λ )−1 ↾L= ϕ(λ). Therefore, the pair {ϕ,ψ} is a normalized Nκ-pair corresponding to the linear relation A(φ, ψ) and the scale L. Remark 3.1. It follows from (3.12) that the linear relation A(ϕ,ψ) given by (3.10) is L-minimal. Remark 3.2. For every normalized Nκ pair {ϕ,ψ} and h ∈ H(ϕ,ψ) the following identity holds PL(A(ϕ,ψ) − λ)−1 [ h 0 ] = h(λ), (λ ∈ hϕψ). (3.13) Indeed, it follows from (3.12) that for every v ∈ L one obtains ( PL ( A(ϕ,ψ) − λ )−1 [ h 0 ] , v ) L = [[ h 0 ] , ( A(ϕ,ψ) − λ̄ )−1 [ 0 v ]] H(ϕ,ψ)⊕L E. Neiman 205 = [ h,Nλ(·)v ] H(ϕ,ψ) = ( h(λ), v ) L . Therefore, E(λ) = PL ( A(ϕ,ψ) − λ )−1 ↾H(ϕ,ψ) is the evaluation operator in H(ϕ,ψ). We define the lineal space Ñω via the formula Ñω := { N ϕψ ω (·)u, u ∈ L } . (3.14) Proposition 3.1. Let {ϕ,ψ} be a normalized Nκ-pair in the space L. Then (i) the space Ñω is a positive subspace in H(ϕ,ψ) if and only if N ϕψ ω (ω) is a strictly positive operator in L. (ii) if additionally ⋂ λ ker N ϕψ ω (λ) = {0} then the space Ñω is a de- generate subspace in H(ϕ,ψ) if and only if 0 is an eigenvalue of N ϕψ ω (ω). Proof. Denote Nω(·) := N ϕψ ω (·). Let us prove the first statement. Since [ Nω(·)u,Nω(·)u ] H(ϕ,ψ) = ( Nω(ω)u, u ) L (u ∈ L) then a conditions Nω(ω) > 0 is equivalent to the inequality( Nω(·)u,Nω(·)u ) H(ϕ,ψ) > 0 which holds for all (0 6=)u ∈ L. Now we prove the second statement. Let at first the space Ñω is a degenerate subspace. Then exist (0 6=)u0 ∈ L such that 0 = [ Nω(·)u0,Nω(·)v ] H(ϕ,ψ) = ( Nω(ω)u0, v ) L which holds for all v ∈ L. Therefore Nω(ω)u0 = 0 and hence 0 is an eigenvalue of Nω(ω). Conversely, let Nω(ω)u0 = 0 where (0 6=)u0 ∈ L. Then 0 = ( Nω(ω)u0, v ) L = [ Nω(·)u0,Nω(·)v ] H(ϕ,ψ) , therefore Nω(·)u0 is orthogonal to the space Ñω. Since Nω(·)u0 6≡ 0 then it is a nontrivial isotropic vector in the space Ñω. Proposition 3.2. Let Ã be a selfadjoint linear relation in H ⊕ L and let {ϕ,ψ} be the normalized Nevanlinna pair given by (2.4). Let the operator valued function γ(λ) : L → H be defined by γ(λ) := PH(Ã− λ)−1\|L (λ ∈ ρ(Ã)). (3.15) Then the following identity holds N ϕψ ω (λ) = γ(λ̄)∗γ(ω̄). (3.16) 206 A functional model... Proof. Indeed, it follows from (2.7) that the kernel N ϕψ ω (λ) takes the form N ϕψ ω (λ) = (PLRλPH)(PHRω̄\|L) = γ(λ̄)∗γ(ω̄). Proposition 3.3. Let {ϕ,ψ} be a normalized Nκ-pair in the space L. Then Ñω is a closed space if and only if Nω(ω) is normally solvable. Proof. Denote B := N ϕψ ω (ω) and consider its spectral decomposition B = B+ ⊕B− ⊕B0 (3.17) and the corresponding decomposition of the Hilbert space L L = L+ ⊕ L− ⊕ L0 (3.18) where B+ > 0, B− < 0, and B0 = 0L0 . It follows from (3.12) and (3.15) that γ(ω)v = N ϕψ ω̄ (·)v ∀v ∈ L. Since Ñω = γ(ω)L then Ñω can be decomposed as Ñω = N+ ω [+]N− ω [+]N0 ω (3.19) where N± ω = γ(ω)L± and N0 ω = γ(ω)L0. This decomposition is orthogo- nal since Nω(ω) = Nω̄(ω̄). For instance, if v+ ∈ L+, v− ∈ L− then [ γ(ω)v+, γ(ω)v− ] H(ϕ,ψ) = [ Nω(·)v+,Nω(·)v− ] H(ϕ,ψ) = ( Nω(ω)v+, v− ) L = ( Nω(ω)v+, v− ) L = 0. Let B = N ϕψ ω (ω) be normally solvable ranB is closed in L. Since L− and L0 are finite-dimensional subspaces in L then ranB+ is closed and by Banach Theorem there is c > 0 such that (B+v, v)L ≥ c2\|\|v\|\|2L (v ∈ L+). (3.20) Due to Proposition 3.2 it can be rewritten as [ γ(ω)v, γ(ω)v ] H(ϕ,ψ) ≥ c2‖v‖2 L (v ∈ L+). (3.21) Thus N+ ω = γ(ω)L+ is closed. Since dim N− ω ≤ κ and dimN0 ω ≤ κ this implies that Ñω is closed. Conversely, if Ñω is closed, then N+ ω is also closed. Since γ(ω) ↾L+ is invertible, then there is c > 0 such that (3.21) holds. In view of Proposition 3.2 this implies that ranB+ is closed in L. Since B− is finite-dimensional then ranB is closed. This proves the statement. E. Neiman 207 Remark 3.3. Ifm is a function from the classNκ(L) such that I−λm(λ) is invertible for all λ ∈ C\R then the pair { IL, m(λ) } is equivalent to the normalized Nκ-pair {ϕ,ψ} = {( IL − λm(λ) )−1 , m(λ) ( IL − λm(λ) )−1} and the corresponding model operator can be rewritten as A(φ, ψ)= {{[ f u ] , [ f ′ u′ ]} : f, f ′ ∈ H(φ, ψ), u, u′ ∈ L; f ′(λ) − λf(λ) = u−m(λ)u′, λ ∈ C\R } . (3.22) Considering the projection of this model to the space H(ϕ,ψ), one obtains the model for a symmetric operator S with the abstract Weyl function m(λ), given in [10] in the Hilbert space case and in [8] in the Pontryagin space case. In particular, a model for a selfadjoint extension A0 of S can be derived from (3.22) in the form A0 = { {f, f ′} ∈ H(φ, ψ)2 : f ′(λ) − λf(λ) ≡ u for some u ∈ L } . (3.23) This reproducing kernel space model appeared originally in [1]. 4. Generalized Fourier transform In this section we show that every L-minimal selfadjoint linear relation A is unitarily equivalent to its functional model A(ϕ,ψ), constructed in Theorem 3.1. The operator F : H → H(ϕ,ψ) given by the formula h 7→ (Fh)(λ) = γ(λ̄)∗h = PL(Ã− λ)−1h (h ∈ H) (4.24) is called the generalized Fourier transform associated with Ã and the scale L. Theorem 4.1. Let Ã be a selfadjoint linear relation in H ⊕ L and let {ϕ,ψ} be the corresponding Nκ-pair given by (2.4). Then: 1) The generalized Fourier transform F maps isometrically the sub- space H0 onto H(ϕ,ψ) and F is identically equal to 0 on H⊖H0; 2) For every {[ f u ] , [ f ′ u′ ]} ∈ Ã the following identity holds E(λ)F(f ′ − λf) = [ ϕ(λ) −ψ(λ) ] [ u u′ ] . (4.25) 208 A functional model... Proof. 1) For every vector h = γ(ω̄)v (ω ∈ ρ(Ã), v ∈ L) it follows from Proposition 3.2 that (Fh)(λ) = γ(λ̄)∗γ(ω̄)v = N ϕψ ω (λ)v. Therefore, F maps the linear space span{γ(ω̄)L : ω ∈ ρ(Ã)} which is dense in H0 onto the linear space span{Nϕψω (·)L : ω ∈ ρ(Ã)} which is dense in H(ϕ,ψ). Moreover, this mapping is isometric, since [Fh,Fh]H(ϕ,ψ) = [Nϕψω (·)v,Nϕψω (·)v]H(ϕ,ψ) = (Nϕψω (ω)v, v)L = [h, h]H. (4.26) This proves the first statement. It is clear from (4.24) that Fh ≡ 0 for h ∈ H ⊖H0. 2) Let h = γ(ω̄)v = PH(Ã−ω̄)−1v, v ∈ L. It follows from (2.4), (3.15) that [ h ψ(ω̄)v ] = (Ã− ω̄)−1 [ 0 v ] , [ ω̄h ϕ(ω̄)v ] = ( I + ω̄(Ã− ω̄)−1 ) [ 0 v ] and hence {[ h ψ(ω̄)v ] , [ ω̄h ϕ(ω̄)v ]} ∈ Ã. Since Ã = Ã∗ one obtains for all {[ f u ] , [ f ′ u′ ]} ∈ Ã [f ′, h]H − [f, ω̄h]H + ( u′, ψ(ω̄) ) L − ( u, ϕ(ω̄)v ) L = 0, v ∈ L. This implies γ(ω̄)∗(f ′ − ω̄f) = ϕ(ω)u− ψ(ω)u′, ω ∈ ρ(Ã). (4.27) This proves the identity (4.25). Corollary 4.1. In the case, when the linear relation Ã is L-minimal it is unitary equivalent to the linear relation A(ϕ,ψ) via the formula A(ϕ,ψ) = {{[ Ff u ] , [ Ff ′ u′ ]} : {[ f u ] , [ f ′ u′ ]} ∈ Ã } . (4.28) The operator F ⊕ IL establishes this unitary equivalence. Corollary 4.2. It follows from (3.13) that the Fourier transform F as- sociated with the operator A(ϕ,ψ) is identical, since (Fh)(λ) = PL(A(ϕ,ψ) − λ)−1 [ h 0 ] = h(λ) for every h ∈ H(ϕ,ψ). E. Neiman 209 Lemma 4.1. Let Ã be a selfadjoint linear relation in H⊕L, let {ϕ,ψ} be the normalized Nκ-pair given by (2.4). Then the following implications hold (i) kerψ(λ) = {0} for some λ ∈ ρ(Ã) ⇒ PL dom Ã is dense in L; (ii) kerϕ(λ) = {0} for some λ ∈ ρ(Ã) ⇒ PL ran Ã is dense in L. If, in addition, the relation Ã is L-minimal, and N ϕψ ω (ω) > 0 for some ω ∈ ρ(Ã) then kerϕ(ω) = {0}, kerψ(ω) = {0}. Proof. Let us prove the first statement. The set PL dom Ã consists of the vectors u ∈ L such that {[ f u ] , [ f ′ u′ ]} ∈ Ã for some f, f ′ ∈ H, u′ ∈ L. If there is a vector v ∈ L such that v ⊥ u for all u ∈ PL dom Ã then {[ 0 0 ] , [ 0 v ]} ∈ Ã, and then ψ(λ)v = 0, due to (2.4). But kerψ(λ) = {0} therefore v = 0. The proof of the second statement is similar. Assume now that N ϕψ ω (ω) > 0 for some ω ∈ ρ(Ã) and that ψ(ω)v = 0. Then in view of (2.4) ϕ(ω)v = v and ( Nω(ω)v, v ) L = ( Nω(ω)v, v ) L = 1 ω − ω̄ (( ψ(ω̄)ϕ(ω) − ϕ(ω̄)ψ(ω) ) v, v ) L = 1 ω − ω̄ ( v, ϕ(ω)v ) L = 0. This implies v = 0. Criterions for the right parts in (i) and (ii) to be true are given in the following lemma. Lemma 4.2. Let Ã be a L-minimal selfadjoint linear relation in H⊕L, let {ϕ,ψ} be the normalized Nκ-pair given by (2.4). Then (i) ⋂ λ∈ρ(Ã) kerψ(λ) = {0} if and only if PL ran Ã is dense in L; (ii) ⋂ λ∈ρ(Ã) kerϕ(λ) = {0} if and only if PL ran Ã is dense in L. 210 A functional model... Proof. The necessity of (i) and (ii) follows from Lemma 4.1. To prove the sufficiency let us consider the linear relation A(ϕ,ψ) given by (3.10). By Corollary 4.1 A(ϕ,ψ) is unitary equivalent to the linear relation Ã and, hence, we may prove the statement for the linear relation A(ϕ,ψ). Assume that ψ(λ)v = 0 for some v ∈ L and for all λ ∈ ρ ( A(ϕ,ψ) ) . Then in view of (2.4) ϕ(λ)v = v and N ϕψ λ̄ (ω)v = 1 ω − λ ( ψ(ω)ϕ(λ) − ϕ(ω)ψ(λ) ) v = 0 for all λ, ω ∈ ρ ( A(ϕ,ψ) ) . Now it follows from (3.11) that {[ 0 0 ] , [ 0 v ]} = {[ N ϕψ ω (·)v 0 ] , [ ω̄N ϕψ ω (·)v v ]} ∈ A(ϕ,ψ). and, hence, v ⊥ PL dom Ã. Acknowledgments. I express my gratitude to V. A. Derkach for guidance in the work and many useful discussions. I am also grateful to M. M. Malamud for valuable remarks and suggestions. References [1] D. Alpay, P. Bruinsma, A. Dijksma, H. S. V. de Snoo, A Hilbert space associated with a Nevanlinna function // Proceeding MTNS meeting Amsterdam, (1989), 115–122. [2] D. Alpay, A. Dijksma, J. Rovnyak, H. S. V. de Snoo, Schur functions, opera- tor colligations, and reproducing kernel Pontryagin spaces/ Oper. Theory: Adv. Appl., 96, Birkhäuser Verlag, Basel, 1997. [3] T. Ya. Azizov and I. S. Iohvidov, Linear operators in spaces with indefinite metric, Nauka, Moscow, 1986 (Russian) [English translation: John Wiley, New York, 1989]. [4] J. Behrndt, S. Hassi, and H. S. V. de Snoo, Functional models for Nevanlinna families // Opuscula Math., 28 (2008), 233–245. [5] J. Behrndt, S. Hassi, and H. S. V. de Snoo, Boundary relations, unitary colliga- tions, and functional models // Complex Anal. Oper. Theory, 3 (2009), 57–98. [6] V. A. Derkach, Abstract interpolation problem in Nevanlinna classes // Oper. Theory Adv. Appl., 190 (2009), 197–236. [7] V. A. Derkach, S. Hassi, M. M. Malamud, and H. S. V. de Snoo, Boundary relations and their Weyl families // Trans. Amer. Math. Soc., 358 (2006), 5351– 5400. [8] V. A. Derkach, S. Hassi, H. S. V. de Snoo, Operator models associated with Kac subclasses of genaralized Nevanlinna functions // Methods of Functional Analysis and Topology, 5 (1999), 65–87. [9] V. A. Derkach and M. M. Malamud, Generalized resolvents and the boundary value problems for hermitian operators with gaps // J. Functional Analysis, 95 (1991), 1–95. E. Neiman 211 [10] V. A. Derkach and M. M. Malamud, The extension theory of hermitian operators and the moment problem // J. Math. Sciences, 73 (1995), 141–242. [11] A. Dijksma and H. S. V. de Snoo, Symmetric and selfadjoint relations in Krĕın Spaces I // Oper. Theory Adv. Appl., 24 (1987), 145–166. [12] M. G. Krein, On resolvents of Hermitian operators with defect indices (m,m) // Dokl. Akad. Nauk SSSR, 52 (1946), N 8, 657–660. [13] M. G. Krein, G. K. Langer, Defect subspaces and generalized resolvents of an Hermitian operator in the space Πκ // Functional Analysis and Its Applications, 5 (1971), N 2, 136–146; 5 (1971), N 3, 54-69. [14] M. G. Krĕın and H. Langer, Über die Q-functions eines π-hermiteschen Operators im Raume Πκ // Acta Sci. Math. (Szeged), 34 (1973), 191–230. [15] H. Langer, B. Textorius, On generalized resolvents and Q–functions of symmetric linear relations in Hilbert spaces // Pacif. J. Math. 72 (1977), N 1, 135–165. [16] M. M. Malamud, S. M. Malamud, Spectral theory of operator measures in Hilbert space // St.-Petersburg Math. Journal, 15 (2003), N 3, 1–77. Contact information Evgen Neiman Department of Mathematics Donetsk National University Universitetskaya str. 24 83055 Donetsk Ukraine E-Mail: evg_sqrt@mail.ru

A functional model associated with a generalized Nevanlinna pair

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