
TL;DR
This paper extends the classification of Weyl families associated with boundary pairs from unitary to essentially unitary cases, showing their closures belong to the Nevanlinna class, thus broadening the understanding of their spectral properties.
Contribution
It generalizes the known classification of Weyl families from unitary to essentially unitary boundary pairs, establishing their membership in the Nevanlinna class.
Findings
Closures of Weyl family members belong to the Nevanlinna class
Bounded Weyl functions of essentially unitary pairs are in class []
Extension of classification results to a broader class of boundary pairs
Abstract
It is known that the Weyl families corresponding to unitary boundary pairs belong to the class of Nevanlinna families. Here we extend the theorem to the case of essentially unitary boundary pairs by showing that the closures of members of the Weyl families belong to the class . Thus bounded Weyl functions of essentially unitary boundary pairs are of class .
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Weyl
families of essentially unitary pairs
Rytis Juršėnas
Institute of Theoretical Physics and Astronomy, Vilnius University, Sauletekio 3, LT-10257, Vilnius, Lithuania
Abstract.
It is known that the Weyl families corresponding to unitary boundary pairs belong to the class of Nevanlinna families. Here we extend the theorem to the case of essentially unitary boundary pairs by showing that the closures of members of the Weyl families belong to the class . Thus bounded Weyl functions of essentially unitary boundary pairs are of class .
Key words and phrases:
Krein space, isometric relation, unitary relation, essentially unitary relation, boundary pair, Weyl family, Nevanlinna family.
2010 Mathematics Subject Classification:
Primary 47A06, 47A56, 47B25; Secondary 47B50, 35P05.
1. Introduction
Throughout and denote Hilbert spaces. Let be a linear relation from a -space to a -space [AI89, Section 1], where the canonical symmetry () acts on () as the operator of multiplication by the matrix \bigl{(}\begin{smallmatrix}0&-\mathrm{i}I_{\mathfrak{H}}\\ \mathrm{i}I_{\mathfrak{H}}&0\end{smallmatrix}\bigr{)}. Let denote the Krein space adjoint of [DHM20, Equation (2.4)], [DHMdS12, Section 7.2]. Then is said to be -isometric if and -unitary if [DHM20, Definition 2.1], and essentially -unitary if [DHMdS06, Section 2.3], where the overbar denotes the closure (with respect to the Hilbert space topology). In what follows we refer to simply as isometric/(essentially) unitary.
Let be a linear relation from a -space to a -space, and put ( denotes the -orthogonal complement; the superscript denotes the Hilbert space adjoint)
[TABLE]
Then is a closed, not necessarily symmetric, linear relation in , and is dense in (with respect to the Hilbert space topology on ). If is unitary, then is automatically symmetric; if is essentially unitary, then is also symmetric. Let
[TABLE]
Then but . For a unitary , .
Let be a closed symmetric linear relation in . The pair is an isometric (unitary) boundary pair for if densely with respect to the topology on , and if is isometric (unitary) [DHM20, Definition 3.1]. Note that implies that . If is an isometric boundary pair for , then is also an isometric boundary pair for .
Let be a unitary boundary pair for . In the terminology of [DHMdS06, Definition 3.1], a unitary is called a boundary relation for ; see also [DHMdS06, Proposition 3.2]. If is essentially unitary, we say that the pair is an essentially unitary boundary pair for .
The first part of [DHMdS06, Theorem 3.9] states that the Weyl family , , corresponding to a boundary relation for is a Nevanlinna family; that is, it belongs to the class (Definition 4.1). Here we prove an analogue of this statement for an essentially unitary .
Theorem 1.1**.**
Let be the Weyl family corresponding to an essentially unitary boundary pair for . Then the closure belongs to a Nevanlinna family.
Here is the Weyl family corresponding to a unitary boundary pair . For unitary (hence closed), the theorem clearly reduces to the first part of [DHMdS06, Theorem 3.9]. By assuming additionally that is a bounded (hence closed) operator, one deduces another corollary.
Corollary 1.2**.**
Let be an essentially unitary boundary pair for , and assume in addition that , . Then the Weyl function belongs to the subclass of Nevanlinna functions.∎
According to [DHMdS06, Proposition 5.9] a Nevanlinna function of class can be realized as the Weyl function of a -generalized boundary pair [DHM20, Definition 3.5]. Let us recall that ordinary, -generalized, -generalized, -generalized boundary pairs are all unitary boundary pairs; see [DHM20] for more details. Yet Corollary 1.2 shows that one can find a non-unitary boundary pair with the same Weyl function.
Assuming the hypotheses in Corollary 1.2 and in addition , one concludes that the Weyl function belongs to the subclass of uniformly strict Nevanlinna functions. The single-valued linear relation (i.e. operator) with such properties arises, for example, in the study of triplet extensions of self-adjoint operators [Jur18]; see also example in Section 5.
The proof of Theorem 1.1 is organized as follows: In Section 2 we state and prove the main technical lemma (Lemma 2.6). In Section 3 we compute the adjoint for an isometric boundary pair ; it follows that for essentially unitary. Since is a Nevanlinna family for unitary, this leads to Theorem 1.1; see Section 4.
Throughout we use the standard symbols , , , and to denote the domain, the range, the multivalued part, and the kernel of a linear relation. For more details related to the theory of linear relations and Nevanlinna families the reader may consult the papers in [BBM*+*18, DM17, BMN15, BHdS*+*13, dSWW11, DHMdS09, HdSS09, BHdS08, HSdSS07, HdS96, DM91] and also an extensive list of references therein.
2. Main lemma
Consider a linear relation from a -space to a -space. The -metric is written in terms of the -scalar product according to
[TABLE]
for and , provided that the -scalar product is conjugate-linear in the first argument. The same applies to the -metric .
The Krein space adjoint of is defined by
[TABLE]
In particular, the inclusion implies that the Green identity holds:
[TABLE]
Put and
[TABLE]
and let denote the Krein space adjoint of . As usual, the eigenspaces of are given by
[TABLE]
and similarly for other linear relations.
If is isometric, then implies that
[TABLE]
for ; that is, is also isometric.
The -orthogonal complement (recall e.g. [AI89, Definition 1.11]) of is written in terms of thus
[TABLE]
On the other hand, consists of such that ( denotes the orthogonal complement in ); hence
[TABLE]
where denotes the componentwise sum [HdSS09, Section 2.4] and (the graph of) the identity operator in . Because is dense in , one has that (see also [DHM20, Proposition 3.9(i)])
[TABLE]
Lemma 2.6**.**
Let be an isometric linear relation from a -space to a -space, and define
[TABLE]
[TABLE]
The following statements are equivalent:
.
* is a closed linear relation in .*
.
Proof.
(i) (ii) is clear, since the kernel of a closed linear relation is closed.
(ii) (i) By using (2.2) we have
[TABLE]
Then
[TABLE]
From here we see that
[TABLE]
[TABLE]
[TABLE]
But implies that
[TABLE]
and it therefore follows that
[TABLE]
(i) (iii) By arguing as in [HdSS09, Lemma 2.10], the componentwise sum of two closed linear relations ( and ) is closed iff the componentwise sum of their adjoints (and hence of their Krein space adjoints) is a closed linear relation, i.e.
[TABLE]
iff
[TABLE]
is closed, where we also use (2.5). Because and
[TABLE]
we get that is closed iff
[TABLE]
By the proof of (i) it therefore follows that
[TABLE]
3. Weyl family corresponding to an essentially
unitary boundary pair
Here and elsewhere below, a linear relation from a -space to a -space is assumed to be isometric, unless explicitly stated otherwise. Whenever we speak of an isometric boundary pair for , we assume that is a closed symmetric linear relation in .
The Weyl family of corresponding to an isometric boundary pair for is defined by (see e.g. [DHM20, Definition 3.2])
[TABLE]
In terms of the linear relation and its adjoint in can be described by
[TABLE]
It follows from the Green identity (2.1) and (3.1) that . By (3.1), the intersection is a subset of the set of neutral vectors [AI89, Definition 1.3] of a -space. Thus, by applying (2.1) to such that , one finds that . This result is stated without proof in [DHM20, Lemma 3.6(i)], [DHMdS12, Lemma 7.52(i)], and is shown in [DHMdS06, Lemma 4.1(i)] for a unitary boundary pair . By using Lemma 2.6, one can find other invariance results for , which we do not repeat here.
The following corollary is an application of Lemma 2.6 and (3.1).
Corollary 3.2**.**
Let be an essentially unitary boundary pair for , with the Weyl family , and let be the Weyl family corresponding to a unitary boundary pair . Then , .
Proof.
Put in the lemma; then . Since and is closed by [DHMdS06, Theorem 3.9], it follows that . ∎
Clearly if is unitary, we get that for .
Remark 3.3*.*
In the proof of Corollary 3.2 we use the fact that the Weyl family corresponding to a unitary boundary pair for is a Nevanlinna family, and hence is in particular defined by a closed linear relation , . If we did not rely on the present fact, we would instead obtain from (iii) in Lemma 2.6 a weaker variant of Corollary 3.2: for an essentially unitary boundary pair such that is an -generalized boundary pair , i.e. such that is additionally assumed to be self-adjoint ([DHM20, Definition 5.11]). Here the linear relation . The argumentation is due to von Neumann formula , , from which one has , since ; for the latter representation of , statement (iii) is true (see also [DHM20, Theorem 5.17]). Vice verse, because by Corollary 3.2 is closed, we conclude from Lemma 2.6 that (iii) must be true for essentially unitary.
Corollary 3.4**.**
Let be an essentially unitary boundary pair for , and let . Then for all .
4. Nevanlinna families
The following definition of a Nevanlinna family is due to [DHMdS12, Definition 9.12], [BHdS08, Definition 2.1], [DHMdS06, Section 2.6].
Definition 4.1**.**
A family , , of linear relations in belongs to the class of Nevanlinna families, or is said to be a Nevanlinna family, if:
For (), the relation is maximal dissipative (accumulative), and the operator family , (), is analytic;
.
Moreover: if and ; if and ; if and ( labels the resolvent set). These subclasses are all covered by the subclass of Nevanlinna families such that is an operator. For more on the classification of Nevanlinna families the reader may refer to [DHMdS06].
Recall that () is the set of such that (). A linear relation is dissipative (resp. accumulative) if (resp. ); we emphasize that the -scalar product is conjugate-linear in the first argument. A dissipative (resp. accumulative) is maximal dissipative (resp. maximal accumulative) if has no proper dissipative (resp. accumulative) extensions.
The Weyl family , , corresponding to an isometric boundary pair for is dissipative (accumulative) for (). Indeed, in view of (3.1), implies that for some . Then, by the Green identity (2.1), ; hence the claim. But then is an operator family by [DdS74, Theorem 3.1(i)].
If in addition , then , and therefore each member of the Weyl family is closed in this case: . But then the operator by [DdS74, Theorem 3.1(vi)], and the relation is maximal dissipative (accumulative) by [DdS74, Theorem 3.4(ii)].
By the above we conclude the following:
Lemma 4.2**.**
Let be the Weyl family corresponding to an isometric boundary pair for . If , , then is a Nevanlinna family .∎
By applying Corollary 3.2 and Lemma 4.2 we deduce Theorem 1.1.
Remark 4.3*.*
Recall that the Weyl family of and of its simple part coincide. Indeed, let be the simple part [LT77, Proposition 1.1] of and let be the restriction to of , where is the closed linear span of . Put . Then . Thus, since is isometric, is also isometric, and the corresponding Weyl family of is given by , , by noting that . In addition, given an isometric , assume that is essentially unitary. Then is also essentially unitary, whose closure .
5. Example of an essentially unitary
boundary triple
An isometric boundary pair for having the property is named by an isometric boundary triple ([DHM20, Section 3.1]). Then a linear relation is identified with an operator from to , and the corresponding Weyl function satisfies on , . When is in addition essentially unitary, the triple is called an essentially unitary boundary triple. We recall from [DHMdS06, Lemma 4.1(ii)] that implies , i.e. , but note that is in general a proper subset of . In case the triple is such that and is unitary, one has by [DHMdS06, Corollary 2.4], and the triple is called an ordinary boundary triple for . In case is densely defined (i.e. is an operator), is identified with its graph.
For the illustration of Corollary 1.2, we consider an example of an essentially unitary boundary triple (cf. [Jur18]). Let be a densely defined, closed, symmetric operator in a Hilbert space with defect numbers . Let be a self-adjoint extension of in . For simplicity, we assume that is lower semibounded. By the von Neumann formula the adjoint is described by , where the eigenspace is spanned by the deficiency elements , , with ranging over an index set of cardinality . That is, , where for .
Let , , be the scale of Hilbert spaces associated with (see e.g. [AK00]); in particular, and . Then a deficiency element can be defined in the generalized sense as for the functional . Thus, is the symmetric restriction of to the domain of such that ; we use the vector notation .
5.1. Finite rank perturbation of
Fix and consider the set
[TABLE]
The points are such that for and . The system is linearly independent, so the Hermitian (Gram) matrix
[TABLE]
is positive definite.
Consider another set
[TABLE]
where the subset is defined by
[TABLE]
where the multiplier
[TABLE]
for , and for and . Define the operator in by
[TABLE]
for , where . Then is referred to as the rank- perturbation of . Note that the sum defining is direct.
5.2. Boundary value space
Let and define the operator (identified with its graph) by
[TABLE]
Here the column-vectors
[TABLE]
and the matrix
[TABLE]
for some matrix-valued Nevanlinna family of class . In fact, if the functional extends to according to (see also [AK00, Section 3.1.3], [HK09])
[TABLE]
then the matrix , , is defined by
[TABLE]
From here one verifies that : By definition the imaginary part is the matrix with entries . Then the kernel is the set of such that ; hence .
An ordinary boundary triple for is defined by
[TABLE]
with and , and with as described above, while defines the boundary value space of operator . Indeed, associate with two single-valued linear relations
[TABLE]
Then the boundary form of the operator is given by
[TABLE]
for , provided that (resp. ) is regarded as the mapping .
5.3. Linear relation and
its Krein space adjoint
Let be the adjoint of in .
Proposition 5.1**.**
.
Proof.
The (graph of the) adjoint consists of such that
[TABLE]
Let be an orthogonal projection in onto . Then and , where . Since it follows that
[TABLE]
and
[TABLE]
where is the matrix direct sum of diagonal matrices . Because the set , i.e. is a dense subset of , it follows from the above that and and . On the other hand, implies that
[TABLE]
Since is invertible, one deduces that , and hence . ∎
Let . By Proposition 5.1 is dense in , i.e. . One also verifies that the eigenspace for . Then, the single-valued linear relation (recall (2.2)) and its Krein space adjoint are given by
[TABLE]
It follows that and for all . Moreover
[TABLE]
5.4. Weyl function
Let be the Krein space adjoint of .
Proposition 5.2**.**
, with the closure .
Proof.
Step 1. By definition, consists of such that
[TABLE]
and
[TABLE]
where is a bounded continuation to of , and where an orthogonal projection and the matrices and are as in the proof of Proposition 5.1. Using and
[TABLE]
one concludes that and and , where the operator
[TABLE]
Using in addition that
[TABLE]
one finds that
[TABLE]
On the other hand, by the definition of
[TABLE]
Thus
[TABLE]
and subsequently
[TABLE]
Step 2. The boundary form of satisfies an abstract Green identity
[TABLE]
for , and for and as defined in step 1. Therefore, to show that
[TABLE]
it remains to prove that and , .
By direct computation, and using the equivalence relation , one finds that the adjoint ; since and are closed, this yields .
Consider ; then with and . Since and , it holds . Then
[TABLE]
and
[TABLE]
This completes the proof of .
Step 3. The closure is the adjoint of ; hence it consists of such that
[TABLE]
for all ; i.e. . ∎
Observe that for all , so (2.3) holds true. Note also that .
Since is essentially unitary and , the triple is an essentially unitary boundary triple for . The triple is an ordinary boundary triple for , with the associated Weyl family defined by , .
The domain
[TABLE]
so by Corollary 1.2 the Weyl function
[TABLE]
and therefore belongs to a Nevanlinna family of class . That can be also checked by computing directly. Moreover, since , one concludes that actually .
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