Generalized boundary triples, I. Some classes of isometric and unitary boundary pairs and realization problems for subclasses of Nevanlinna functions

Funding information Volkswagen Foundation; Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science and Letters Abstract With a closed symmetric operator A in a Hilbert space H a triple Π = {,Γ0,Γ1} of a Hilbert space  and two abstract trace operators Γ0 and Γ1 from A∗ to  is called a generalized boundary triple for A∗ if an abstract analogue of the second Green’s formula holds. Various classes of generalized boundary triples are introduced and corresponding Weyl functions M(⋅) are investigated. The most important ones for applications are specific classes of boundary triples for which Green’s second identity admits a certain maximality property which guarantees that the corresponding Weyl functions are Nevanlinna functions on , i.e.M(⋅) ∈ (), or at least they belong to the class ̃() of Nevanlinna families on . The boundary condition Γ0f = 0 determines a reference operator A0 ( =ker Γ0 ) . The case where A0 is selfadjoint implies a relatively simple analysis, as the joint domain of the trace mappings Γ0 and Γ1 admits a von Neumann type decomposition via A0 and the defect subspaces of A. The case where A0 is only essentially selfadjoint is more involved, but appears to be of great importance, for instance, in applications to boundary value problems e.g. in PDE setting or when modeling differential operators with point interactions. Various classes of generalized boundary triples will be characterized in purely analytic terms via the Weyl functionM(⋅) and close interconnections between different classes of boundary triples and the corresponding transformed/renormalized Weyl functions are investigated. These characterizations involve solving direct and inverse problems for specific classes of operator functions M(⋅). Most involved ones concern operator functions M(⋅) ∈ () for which τM(λ)(f, g) = (2i Im λ)−1[(M(λ)f, g) − (f,M(λ)g)], f , g ∈ domM(λ),


Abstract
With a closed symmetric operator in a Hilbert space ℌ a triple Π = { , Γ 0 , Γ 1 } of a Hilbert space  and two abstract trace operators Γ 0 and Γ 1 from * to  is called a generalized boundary triple for * if an abstract analogue of the second Green's formula holds. Various classes of generalized boundary triples are introduced and corresponding Weyl functions (⋅) are investigated. The most important ones for applications are specific classes of boundary triples for which Green's second identity admits a certain maximality property which guarantees that the corresponding Weyl functions are Nevanlinna functions on , i.e. (⋅) ∈ (), or at least they belong to the class() of Nevanlinna families on . The boundary condition Γ 0 = 0 determines a reference operator 0 ( = ker Γ 0 ) . The case where 0 is selfadjoint implies a relatively simple analysis, as the joint domain of the trace mappings Γ 0 and Γ 1 admits a von Neumann type decomposition via 0 and the defect subspaces of . The case where 0 is only essentially selfadjoint is more involved, but appears to be of great importance, for instance, in applications to boundary value problems e.g.
in PDE setting or when modeling differential operators with point interactions. Various classes of generalized boundary triples will be characterized in purely analytic terms via the Weyl function (⋅) and close interconnections between different classes of boundary triples and the corresponding transformed/renormalized Weyl functions are investigated. These characterizations involve solving direct and inverse problems for specific classes of operator functions (⋅). Most involved ones concern operator functions (⋅) ∈ () for which on the Lions-Magenes trace theory ( [39,56]) they regularized the classical Dirichlet and Neumann trace mappings to get a proper version of Definition 1.1.
The operator Γ in Definition 1.1 is called the reduction operator (in the terminology of [21]). Definition 1.1 immediately yields a parametrization of the set of all selfadjoint extensions̃of by means of abstract boundary conditions viã where Θ ranges over the set of all selfadjoint extensions of when Θ ranges over the set of all selfadjoint relations in  (subspaces in  × , see [4]). This correspondence is bijective and in this case Θ ∶= Γ (̃) . The following two selfadjoint extensions of are of particular interest: 0 ∶= ker Γ 0 = Θ ∞ and 1 ∶= ker Γ 1 = Θ 1 ; here Θ ∞ = {0} ×  and Θ 1 = . These extensions are disjoint, i.e. 0 ∩ 1 = , and transversal, i.e. they are disjoint and 0+ 1 = * . Here the symbol+ means the componentwise sum of two linear relations, see (2.1).
In what follows 0 is considered as a reference extension of . Let The main analytical tool in the description of spectral properties of selfadjoint extensions of is the abstract Weyl function, introduced and investigated in [30][31][32]. Notice that when the symmetric operator is densely defined its adjoint is a single-valued operator and Definitions 1.1 and 1.2 can be used in a simpler form by treating Γ 0 and Γ 1 as operators from dom * to , see [32,37,46]. In what follows this convention will be tacitly used in most of our examples. The defect subspace is spanned by the Weyl solution (⋅, ) of the equation  = which is given by The function (⋅) is called the Titchmarsh-Weyl coefficient of . In this case a boundary triple Π = { ℂ, Γ 0 , Γ 1 } can be defined as Γ 0 = (0), Γ 1 = ′ (0). The corresponding Weyl function ( ) coincides with the classical Titchmarsh-Weyl coefficient, ( ) = ( ).
In this connection let us mention that the role of the Weyl function ( ) in the extension theory of symmetric operators is similar to that of the classical Titchmarsh-Weyl coefficient ( ) in the spectral theory of Sturm-Liouville operators. For instance, it is known (see [32,52]) that if is simple, i.e. does not admit orthogonal decompositions with a selfadjoint summand, then the Weyl function ( ) determines the boundary triple Π, in particular, the pair { , 0 } , uniquely up to unitary equivalence. Besides, when is simple, the spectrum of Θ coincides with the singularities of the operator function (Θ − ( )) −1 ; see [32].
As was shown in [32,33] and [58] the Weyl function (⋅) and the -field (⋅) both are well defined and holomorphic on the resolvent set  . (1.3) This means that (⋅) is a -function of the operator in the sense of Kreȋn and Langer [51]. The converse is also true.

-generalized and -generalized boundary triples
In BVP's for Sturm-Liouville operators with an operator potential, for partial differential operators [26], and in point interaction theory it seems natural to consider more general boundary triples by weakening the surjectivity assumption 1.1.2 in Definition 1.1. The following notion was introduced in [33] with the name generalized boundary-value space, see also [25], where the term generalized boundary triplet was used.  . (1.5) For every ∈ ( ) the Weyl function ( ) takes values in () and this justifies the present usage of the term -generalized boundary triple, where " " stands for a Weyl function whose values are "bounded" operators. It is well known (see e.g. [39,56]) that the mappings ∶ 3∕2 Δ (Ω) → 1 ( Ω) and ∶ 3∕2 Δ (Ω) → 0 ( Ω) = 2 ( Ω) are well defined and surjective.
As was shown in [27] for every -generalized boundary triple there exists an ordinary boundary triple  .
It should be noted that the Weyl function (⋅) of a -generalized boundary triple satisfies the properties (1.2)-(1.4). However, instead of the property 0 ∈ (Im ( )) one has a weaker condition 0 ∉ (Im ( )). This motivates the following definition. Denote by  [] the class of strict Nevanlinna functions In fact, it was also shown in [33,Chapter 5] that every (⋅) ∈  [] can be realized as the Weyl function of a certain -generalized boundary triple and hence the following statement holds.

Theorem 1.7 ([33]). The set of Weyl functions corresponding to -generalized boundary triples coincides with the class  [] of strict Nevanlinna functions.
This realization result as well as the technique of -generalized boundary triples have recently been applied also e.g. to problems in scattering theory, see [13], in the analysis of discrete and continuous time system theory, and in the boundary control theory; for some recent achievements, see e.g. [5,6,8,9,40,53,54,59,66].
In the present paper we introduce the new class of -generalized boundary triples which is obtained by a weakening of the surjectivity condition 1.5.3 in Definition 1.5.

Definition 1.8. A collection
{ , Γ 0 , Γ 1 } is said to be an almost -generalized boundary triple, or briefly, an -generalized boundary triple for * , if * ∶= dom Γ is dense in * , the conditions 1.5.1, 1.5.2 are satisfied and 1.8.1 ran Γ 0 is dense in .
The Weyl function corresponding to an -generalized boundary triple is again defined by (1.5). One of the main results of the paper is Theorem 4.4 which states that every -generalized boundary triple can be regularized to produce a -generalized boundary triple in the spirit of (1.6). Another result -Theorem 4.6 gives a characterization of the set of the Weyl functions of -generalized boundary triples in the form and is a densely defined symmetric operator in , such that ker Im 0 ( ) ∩ dom = {0}. The class of -generalized boundary triples contains the class of so-called quasi boundary triples, which has been studied in J. Behrndt and M. Langer [11]. In the definition of a quasi boundary triple Assumption 1.5.3 is replaced by the assumption that ran Γ is dense in  × .
A connection between quasi boundary triples and -generalized boundary triples is given in Corollary 4.9. A joint feature in -generalized boundary triples and quasi boundary triples is that without additional assumptions on the mapping Γ = { Γ 0 , Γ 1 } these boundary triples are not unitary in the sense of Definition 1.9 presented below. Consequently, their Weyl functions need not be Nevanlinna functions, i.e., the values ( ) need not be maximal dissipative (accumulative) in ℂ + (ℂ − ); see definitions in Section 2.1. Special type of isometric boundary triples that will appear in Part II of the present paper are so-called essentially unitary boundary triples/pairs. As shown therein (cf. [29,Section 7]) quasi boundary triples studied in [11,12] for elliptic operators are either special type of unitary boundary triples or they are essentially unitary boundary triples, depending on the choice of the underlying regularity index of the space used as the domain * for the boundary triple. For a very recent contribution and some further development on essentially unitary boundary pairs see also [43]. Different applications of quasi boundary triples in boundary value problems including applications to elliptic theory and trace formulas can be found e.g. in [11,14,15,40,62].

Unitary boundary triples
A general class of boundary triples, to be called here unitary boundary triples, was introduced in [25]. This concept was motivated by the realization problem for the most general class of Nevanlinna functions: realize each Nevanlinna function as the Weyl function of an appropriate type generalized boundary triple.
To this end denote by () the class of all operator valued holomorphic Nevanlinna functions on ℂ + (in the resolvent sense) with values in the set of maximal dissipative (not necessarily bounded) linear operators in . Each (⋅) ∈ () is extended to ℂ − by symmetry with respect to the real line ( ) = (̄) * ; see [25,51] In order to present the definition of a unitary boundary triple, introduce the fundamental symmetries , (1.9) and the associated Kreȋn spaces ( [7,17]) obtained by endowing the Hilbert spaces ℌ 2 and  2 with the following indefinite inner products .
is a linear operator from ℌ 2 to  2 such that: 1.9.1 * ∶= dom Γ is dense in * with respect to the topology on ℌ 2 ; The Weyl function ( ) and the -field corresponding to a unitary boundary triple Π are defined again by the same formula (1.5). The transposed boundary triple Π ⊤ ∶= { , Γ 1 , −Γ 0 } associated with a unitary boundary triple Π is also a unitary boundary triple, the corresponding Weyl function takes the form ⊤ ( ) = − ( ) −1 .
The main realization theorem in [25] gives a solution to the inverse problem mentioned above. In fact, in [25,Theorem 3.9] a stronger result is stated showing that the class  () can be replaced by the class () or even by the class() of Nevanlinna pairs when one allows multi-valued linear mappings Γ in Definition 1.9; see Theorem 3.3 in Section 3.2. Theorem 1.10 plays a key role in the construction of generalized resolvents in the framework of coupling method that was originally introduced in [24] and developed in its full generality in [26]. It is worth to mention that in [6] it is shown that a counterpart of the main transform of a unitary boundary triple (with some extra properties) naturally appears in impedance conservative continuous time input/state/output systems, and, moreover, that the transfer function of such systems is directly connected with the Weyl function of the unitary boundary triplet. A systematic study of so-called conservative state/signal systems has been initiated in [5] and, as shown in [6], conservative state/signal systems have a close connection to general unitary boundary triples in Theorem 1.10; see also Remark 5.7. Ordinary and -generalized boundary triples give examples of unitary boundary triples; see [25], and as noted above the conditions defining -generalized or quasi boundary triples do not guarantee their unitarity; for a criterion see Corollary 4.7. Some necessary and sufficient conditions which characterize unitary boundary triples and which differ from the purely analytic criterion in Theorem 1.10 can be found in [25,Proposition 3.6], [27,Theorem 7.51], some general criteria of geometric nature have been established in [68,69], and a further characterization, useful e.g. in applications to elliptic equations, can be found in Part II of the present paper.
In connection with Definition 1.9 we wish to make some comments on a seminal paper [21] by J. W. Calkin, where a concept of the reduction operator is introduced and investigated. Although no proper geometric machinery appears in the definition of Calkin's reduction operator this notion in the case of a densely defined operator essentially coincides with concept of a unitary operator between Kreȋn spaces as in Definition 1.9. An overview on the early work of Calkin and some connections to later developments can be found in the papers in the monograph [40]; for a further discussion see also Section 3.5.
In Theorem 5.8 we extend Kreȋn's resolvent formula to the general setting of unitary boundary triples. Namely, for any proper extension Θ ∈ Ext satisfying Θ ⊂ dom Γ the following Kreȋn-type formula holds: It is emphasized that in this formula Θ is not necessarily closed and it is not assumed that ∈

-generalized boundary triples
Following [25] we consider a special class of unitary boundary triples singled out by the condition that 0 ∶= ker Γ 0 is a selfadjoint extension of .

12)
where = * is a selfadjoint (in general unbounded) operator in .
In Theorem 5.17 this result is extended to the case of -generalized boundary pairs {, Γ}, where Γ ∶ * →  ×  is allowed to be multi-valued (see Definitions 3.1 and 5.11).
Notice that, for instance, the implications (i) ⇒ (ii), (iii) are immediate from the following decomposition of * ∶= dom Γ: * = 0+̂( * ), . (1.13) In accordance with (1.12) the Weyl function corresponding to an -generalized boundary triple is an operator valued Nevanlinna function with domain invariance property: dom ( ) = dom = ran Γ 0 , ∈ ℂ ± . It takes values in the set () of closed (in general unbounded) operators while the values of the imaginary parts Im ( ) are bounded operators.

-generalized boundary triples and form domain invariance
Next we discuss one of the main new objects appearing in the present work. To characterize the class of -generalized boundary triples in terms of the corresponding Weyl functions we associate with each (⋅) a family of nonnegative quadratic forms ( ) in : (1.14) The forms ( ) are not necessarily closable. However, it is shown that if ( 0 ) is closable at one point 0 ∈ ℂ + ( 0 ∈ ℂ − ), then ( ) is closable for every ∈ ℂ + (resp. ∈ ℂ − ); for an analytic treatment of this fact see also [28]. In the latter case the domain of the closure ( ) does not depend on ∈ ℂ + ( ∈ ℂ − ) and therefore the Weyl function ( ) is said to be form domain invariant in ℂ + (resp. in ℂ − ). In general ( ) (iv) the forms ( ) and (− ) are closable; The result relies on Theorem 5.5, which contains some important invariance results that unitary boundary triples are shown to satisfy. If { , Γ 0 , Γ 1 } is an -generalized, but not an -generalized, boundary triple for * , then the equality (1.13) fails to hold and turns out to be an inclusion Indeed, since 0 is not selfadjoint (while it is essentially selfadjoint), the decomposition * = 0+̂( * ) doesn't hold; cf. [25,Theorem 4.13]. Then there clearly existŝ∈ * which does not belong to 0+̂( * ), so that Γ 0̂≠ 0 as well as Γ 0̂∉ Γ 0 (̂( * ) ) = dom ( ). In particular, in this case a strict inclusion dom ( ) ⊊ ran Γ 0 holds and, consequently, the Weyl function ( ) can loose the domain invariance property. However, the domain of the closure Γ 0 contains the selfadjoint relation 0 and admits the decomposition This implies the equality dom ( ) = Γ 0 ( dom ( Γ 0 ) ∩̂( * ) ) = ran Γ 0 , which combined with dom ( ) = dom ( ) yields the form domain invariance property for : Passing from the case of an -generalized boundary triple to the case of an -generalized boundary triple (which is not -generalized) means that 0 ≠ * 0 . Then, in particular, conditions (ii) and (iii) in Theorem 1.12 are necessary violated. We split the situation into two different cases: at least for two points 1 , 2 ∈ ℂ + , 1 ≠ 2 , while it is form domain invariant, i.e. dom ( ) = dom (± ) , ∈ ℂ ± .  Let also H be a minimal operator associated with the expression − d 2 . Then H is a symmetric operator . Consider in 2 ( ℝ + ) the direct sum of symmetric operators H , ] .
It is easily seen that a boundary triple Π = for H * can be chosen as ] .
The corresponding Weyl function is given by Clearly, H = H min is a closed symmetric operator in 2 (ℝ + ). Next we put and note that in accordance with the definition of the direct sum of linear mappings We also put Γ ∶= ⊕ ∞ =1 Γ ( ) and note that it is a closure of Γ = Γ ↾ dom Γ, = 1, 2. It can be seen that the orthogonal sum Π ∶= ⊕ ∞ =1 Π of the boundary triples Π determines an -generalized boundary triple. Moreover, in the case that * = 0 the Weyl function (⋅) corresponding to the triple Π = ⊕ ∞ =1 Π satisfies Assumption 1.16, i.e. it is domain invariant, dom ( ) = dom ( ), ∈ ℂ ± , while dom ( ) ⫋ ran Γ 0 . Hence, by Theorem 1.12, 0 ≠ * 0 and Π = ⊕ ∞ =1 Π being -generalized, is not an -generalized boundary triple for H * . In fact, with * = 0 the Weyl function (⋅) as well as its imaginary part Im (⋅) take values in the set of unbounded operators. For the details in this example we refer to Part II of the present work, where also analogous results for moment and Dirac operators with local point interactions are established.
Notice that the minimal operator H as well as the corresponding triple Π for H * in Example 1.17 naturally arise when treating the Hamiltonian H , with -interactions in the framework of extension theory. The latter have appeared in various physical problems as exactly solvable models that describe complicated physical phenomena (see e.g. [2,3,34,48,49] for details).
Theorem 5.32 offers a renormalization procedure which produces from a form domain invariant Weyl function a domain invariant Weyl function, whose imaginary part becomes a well-defined bounded operator function on ℂ ⧵ ℝ, i.e., the renormalized boundary triple is -generalized. Some related results, showing how -generalized boundary triples give rise to -generalized boundary triples, are established in Part II of the present work, where these results are applied in the analysis of regularized trace operators for Laplacians.
Before closing this subsection we wish to mention that other type of examples for -generalized boundary triples are the Kreȋn -von Neumann Laplacian and the Zaremba Laplacian for a mixed boundary value problem treated in Part II of the present work.

A short description of the contents
For the convenience of the reader in this Introduction we have restricted the exposition of the main definitions and results to the case of generalized boundary triples, i.e. to boundary triples with a single-valued linear mapping Γ ∶ * →  ×  which admits a decomposition Γ = { Γ 0 , Γ 1 } , where Γ 0 and Γ 1 give rise to a pair of boundary conditions in (the boundary space)  typically occurring in boundary value problems in ODE and PDE setting. In the paper itself these results are mostly presented in a more general setting of boundary pairs, where Γ is allowed to be multi-valued. This generality unifies the presentation in later Sections and, in fact, often simplifies the description of the particular analytic properties of Weyl functions associated with different classes of generalized boundary triples and boundary pairs.
In Section 2 we recall basic concepts of linear relations (sums of relations, componentwise sums, defect subspaces, etc.) as well as unitary and isometric relations in Kreȋn space. We also introduce the concepts of Nevanlinna functions and families.
In Section 3 we discuss unitary and isometric boundary pairs and triples. We introduce the notions of Weyl functions and families and discuss their properties. A general version of the main realization result, Theorem 3.3, is presented therein, too. It completes and improves Theorem 1.10. Besides certain isometric transforms of boundary triples are discussed.
In Section 4 we investigate -generalized boundary pairs and triples. Their main properties can be found in Theorem 4.2 and in various Corollaries appearing in this section. In Theorem 4.4 a connection between -generalized and -generalized boundary triples is established by means of triangular isometric transformations. Connections between -generalized boundary triples and quasi boundary triples are also explained. Moreover, a Kreȋn type formula for -generalized boundary triples can be found in Theorem 4.12.
In Section 5 we consider two further subclasses of unitary boundary triples and pairs: -generalized and -generalized boundary triples and pairs. For deriving some of the main results in this connection we have established also some new facts on the interaction between ( ℌ ,  ) -unitary relations and unitary colligations appearing e.g. in system theory and in the analysis of Schur functions, see Section 5.1; a background for this connection can be found in [10]. In particular, this connection is applied to extend Theorem 1.12 to the case of -generalized boundary pairs (see Theorem 5.17). In this case representation (1.12) for the Weyl function remains valid with 0 ∈  . In Theorem 5.24 the class of Weyl functions of -generalized boundary pairs is characterized. In Theorem 5.8 it is shown that every unitary boundary triple admits a Kreȋn type resolvent formula. Besides, in Theorem 5.32 a connection between -generalized boundary triples and -generalized boundary triples is established via an isometric transform introduced in Lemma 3.12 (see formula (3.23)).

Linear relations in Hilbert spaces
A linear relation from ℌ to ℌ ′ is a linear subspace of ℌ × ℌ ′ . Systematically a linear operator will be identified with its graph. It is convenient to write ∶ ℌ → ℌ ′ and interpret the linear relation as a multi-valued linear mapping from ℌ into ℌ ′ . If ℌ ′ = ℌ one speaks of a linear relation in ℌ. Many basic definitions and properties associated with linear relations can be found in [4,16,22].
The following notions appear throughout this paper. For a linear relation ∶ ℌ → ℌ ′ the symbols dom , ker , ran , mul and stand for the domain, kernel, range, multi-valued part, and closure, respectively. The inverse −1 is a relation The sum 1 + 2 and the componentwise sum 1+ 2 of two linear relations 1 and 2 are defined by If the componentwise sum is orthogonal it will be denoted by 1 ⊕ 2 . If is closed, then the null spaces of − , ∈ ℂ, defined by are also closed. Moreover, ( ) (̂( )) stands for the set of regular (regular type) points of .
By the maximality condition, each relation ( ), ∈ ℂ ⧵ ℝ, is necessarily closed. The class of all Nevanlinna families in a Hilbert space is denoted by(). If the multi-valued part mul ( ) of ∈() is nontrivial, then it is independent of ∈ ℂ ⧵ ℝ, so that , cf. [51,52,55]. Identifying operators in  with their graphs one can consider classes introduced in Section 1 as subclasses of(). In addition, a Nevanlinna family ( ), ∈ ℂ ⧵ ℝ, which admits a holomorphic extrapolation to the negative real line (−∞, 0) (in the resolvent sense as in item (iii) of the above definition) and whose values ( ) are nonnegative (nonpositive) selfadjoint relations for all < 0 is called a Stieltjes family (an inverse Stieltjes family, respectively).
may be multi-valued, nondensely defined, and unbounded. It is the graph of an operator if and only if its range is dense in  2 . In this case it need not be densely defined or bounded; and even if it is bounded it need not be densely defined.

Definitions and basic properties
Let be a closed symmetric linear relation in the Hilbert space ℌ. It is not assumed that the defect numbers of are equal or finite. Following [25,27] a unitary/isometric boundary pair for * is defined as follows.
Definition 3.1. Let be a closed symmetric linear relation in a Hilbert space ℌ, let  be an auxiliary Hilbert space and let Γ be a linear relation from the Kreȋn space . Then {, Γ} is called a unitary/isometric boundary pair for * , if: 3.1.1 * ∶= dom Γ is dense in * with respect to the topology on ℌ 2 ;
If Γ is single-valued then these component mappings decompose Γ, Γ = Γ 0 × Γ 1 , and the triple { , Γ 0 , Γ 1 } will be called a unitary/isometric boundary triple for * . In this case the Weyl function corresponding to the unitary/isometric boundary triple { , Γ 0 , Γ 1 } can be also defined via When admits real regular type points it is useful to extend Definition 3.2 of the Weyl family to the points on the real line by ) or, more precisely,

Unitary boundary pairs and unitary boundary triples
The following theorem shows that the set of all Weyl families of unitary boundary pairs coincides with() (see [25,Theorem 3.9]). Recall that a unitary boundary pair {, Γ} for * is said to be minimal, if Notice that Theorem 1.10 contains a general analytic criterion for an isometric boundary triple to be unitary; the Weyl function should be a Nevanlinna function, cf. Theorem 1.10.

Corollary 3.4. The class of Weyl functions corresponding to unitary boundary triples coincides with the class  () of (in general unbounded) strict Nevanlinna functions.
Proof. The statement is immediate when combining Theorem 3.3 with Proposition 4.5 from [25]. □ As a consequence of (3.1) and (3.3) the following identity holds (cf. where ℎ ∈ dom ( ) and ∈ dom ( ), , ∈ ℂ ⧵ ℝ.
As was already mentioned in Section 1 every operator valued function can be realized as a Weyl function of some ordinary boundary triple ( -generalized boundary triple, respectively).
The multi-valued analog for the notion of -generalized boundary triple was introduced in [25, Section 5.3], a formal definition reads as follows.

Isometric boundary pairs and isometric boundary triples
. In view of (1.9)-(1.11) this just means that the abstract Green's identity (3.1) holds. It follows from (3.1) that compare Proposition 2.2. Let Γ 0 and Γ 1 be the linear relations determined by (3.4). The kernels 0 ∶= ker Γ 0 and 1 ∶= ker Γ 1 need not be closed, but they are symmetric extensions of ker Γ which are contained in the domain * = dom Γ of Γ; cf. [25,Proposition 2.13]. If * = dom Γ is dense in * then the pair {, Γ} is viewed as an isometric boundary pair for * ; cf. Definition 3.1. In general ∶= ( * ) * = (dom Γ) [⟂] is an extension of ker Γ which need not belong to dom Γ; for some sufficient conditions for the equality = ker Γ, see [26, Section 2.3] and [27, Section 7.8]. With In particular, with = (3.7) implies that Here equality does not hold if Γ is not unitary. However, with the Weyl family the multi-valued part of Γ can be described explicitly; see [27,Lemma 7.57], cf. also [25,Lemma 4.1].

12)
where the adjoint (̄) * of (̄) is in general a linear relation. In particular, and if, in addition, mul Γ 1 = {0}, then Furthermore, the following statements hold: This identity can be rewritten equivalently in the form { , ′ − ′′ } ∈ mul Γ implies that ′ = ′′ and therefore the above argument shows that It remains to prove the statements (i)-(iii).
is densely defined then clearly (̄) * is a closed operator and if Γ 1 is single-valued then (3.14) shows that Γ 1 ( ) is closable as a restriction of (̄) * .
The Weyl function of an isometric or a unitary boundary pair takes values which need not be invertible, and in general can be unbounded, possibly multi-valued, operators. In what follows Weyl functions ( ), whose domain (or form domain) does not dependent on ∈ ℂ ⧵ ℝ are of special interest. Here a characterization for domain invariant Weyl families will be established. We start with the next lemma concerning the domain inclusion dom ( ) ⊂ dom ( ).
Finally, observe that the assumption (3.16) implies (3.19). Since 0 is symmetric, in this case the -field (⋅) satisfies Proof. The assertions (i) and (ii) follow directly from Lemma 3.10. To see (iii) one can use the same argument that is presented in [25,Corollary 4.12]. □

Some transforms of boundary triples
In this subsection a specific transform of isometric boundary triples is treated. In what follows such transforms are used repeatedly and, in fact, they appear also in concrete boundary value problems in ODE and PDE setting. To formulate a general result in the abstract setting consider in the Kreȋn space the transformation operator whose action is determined by the triangular operator By assumptions on one has ker * = mul * = {0}, so that the adjoint * is an injective operator in . To keep a wider generality, is not assumed to be a closed operator, while in applications that will often be the case. In particular, it is possible that * is not densely defined and also its range need not be dense. Since is a densely defined symmetric operator, it is closable and its closure ⊂ * is also symmetric. With the assumptions on in (3.22) a direct calculation shows that ( Hence, is an isometric operator in the Kreȋn space . Moreover, is injective. These observations lead to the following (unbounded) extension of [26, Proposition 3.18].

Lemma 3.12. Let
{ , Γ 0 , Γ 1 } be an isometric boundary triple for * such that ker Γ = , let ( ) and ( ) be the corresponding -field and the Weyl function, and let be as defined in (3.22). Then is isometric in the Kreȋn space (  2 ,  ) and moreover: defines an isometric boundary triple with domaiñ * ∶= domΓ and kernel kerΓ = ; (ii) the -field and the Weyl function ofΓ are in general unbounded nondensely defined operators given bỹ Proof.
(i) By the assumptions in (3.22) is an isometric operator in the Kreȋn space . Since is injective, one has kerΓ = ker Γ = . In general is not everywhere defined, so that̃ * is typically a proper linear subset of * = dom Γ which is not necessarily dense in * .
(ii) As indicated need not be closable. An extreme situation appears when is a singular operator; cf. [47]. By definition this means that dom ⊂ ker or, equivalently, that ran ⊂ mul . Thus, in this case dom * = ran * = {0}. If, for instance, Γ is an ordinary boundary triple for * then 0 = ker Γ 0 and 1 = ker Γ 1 are selfadjoint. It is easy to check that * = Moreover, ranΓ = ↾ dom is a symmetric operator in  and dom̃( ) = dom̃( ) is trivial.

Some additional remarks
Despite of the fact that the paper [21] has been quoted by M. G. Kreȋn [50] and a discussion on [21] appears in the monograph [23] the actual results of Calkin on reduction operators remained widely unknown among experts in extension theory. Apparently this was caused by the fact that the paper [21] was ahead of time -it was using the new language of binary linear relations with hidden ideas on geometry of indefinite inner product spaces, concepts which were not well developed at that time. The concept of a bounded reduction operator investigated therein (see [21, Chapter IV]) essentially covers the notion of an ordinary boundary triple in Definition 1.1 as well as the notion of -boundary triple introduced in [60] for symmetric operators with unequal defect numbers. An overview on the early work of Calkin and more detailed description on its connections to boundary triples and unitary boundary pairs (boundary relations) can be found in the monograph [40]. In fact, [40] contains a collection of articles reflecting various recent activities in different fields of applications with related realization results for Weyl functions, including analysis of differential operators, continuous time state/signal systems and boundary control theory with interconnection analysis of port-Hamiltonian systems involving Dirac and Tellegen structures etc.

-GENERALIZED BOUNDARY PAIRS AND BOUNDARY TRIPLES
In this section we present a new generalization of the class of -generalized boundary triples from [33] (cf. Definition 1.5). A single-valued -generalized boundary pair is also said to be an almost -generalized boundary triple, shortly, an -generalized boundary triple for * .
If Γ is an -generalized boundary pair for * , then the same is true for its closure. Indeed, since Γ is an extension of . Hence, the closure satisfies Green's identity (3.1) and this implies that the corresponding kernels ker
(v) Since dom ( ) = dom ( ) = ran Γ 0 does not depend on ∈ ℂ ⧵ ℝ, the following equality holds ( by Proposition 3.11. According to (iii) ( ) is bounded and densely defined, so that its closure ( ) is bounded and defined everywhere on . The formula in (iv) is obtained by taking closures in (4.4).

Corollary 4.3.
Let Γ be an -generalized boundary pair for * and let (⋅) and (⋅) = + 0 (⋅) be the corresponding -field and Weyl function as in Theorem 4.2 with = Re ( ) for some fixed ∈ ℂ ⧵ ℝ. Then: (i) with a fixed ∈ ℂ ⧵ ℝ the graph of Γ admits the following representation: (ii) the range of Γ satisfies  Proof.
(i) Using the representation of Γ ( ) in (4.1), the inclusion mul Γ ⊂ ( ) in Lemma 3.6, and the fact that by Theorem 4.2 ( ) is an operator, one concludes that the representation of Γ given in Proposition 3.9 (ii) can be rewritten in the form as stated in (i). (iii) In view of (i) this follows from mul Γ 0 = ker Im ( ) = ker ( ); see Lemma 3.6. □ Corollary 4.3 shows that for an -generalized boundary pair the inclusion mul Γ ⊂ (ran Γ) [⟂] is in general strict. In particular, the range of Γ for a single-valued -generalized boundary pair, i.e., an -generalized boundary triple, need not be dense in  × . Notice that an -generalized boundary pair with the surjectivity condition ran Γ 0 =  is called a -generalized boundary pair for * ; see Definition 3.5. The next result gives a connection between -generalized boundary pairs and -generalized boundary pairs. Theorem 4.4. Let {, Γ} be a -generalized boundary pair for * , and let (⋅) and (⋅) be the corresponding Weyl function and -field. Let also be a symmetric densely defined operator in  and let Γ = { Γ 0 , Γ 1 } where Γ = Γ, = 0, 1, be the corresponding components of Γ as in (3.4). Then the transform defines an -generalized boundary pair for * . The corresponding Weyl functioñ(⋅) and̃(⋅)-field are connected bỹ .6) is closed if and only if is a closed symmetric operator in , in particular, the closure of Γ is given by (4.6) with replaced by its closure .
Conversely, if { ,Γ } is an -generalized boundary pair for * then there exists a -generalized boundary pair {, Γ} for * and a densely defined symmetric operator in  such thatΓ is given by (4.6).
(iv) The first equivalence is contained in [25,Proposition 5.3]. To prove the second criterion, we apply Corollary 4.3, in particular, the representation of ran Γ in (4.5): ran (̄) * ) ⟂ = ker ( ) = ker Im ( ). Therefore, the conditions ran Γ is closed, is bounded, and ker Im ( ) = 0 imply that ran Γ is also dense in  ×  and, thus, Γ is surjective. The converse is clear.

□
The class of -generalized boundary triples contains the class of so-called quasi boundary triples, which has been studied in J. Behrndt and M. Langer [11].

Definition 4.8 ([11]). Let be a densely defined symmetric operator in
is said to be a quasi boundary triple for * , if * ∶= dom Γ is dense in * and the following conditions are satisfied:

the range of Γ is dense in  × .
For isometric boundary pairs mul Γ ⊂ (ran Γ) [⟂] and thus the condition 4.8.3 implies that Γ is single-valued. Since the con-  for some or, equivalently, for every ∈ ℂ ⧵ ℝ.
Proof. Item (ii) of Corollary 4.3 shows that ran Γ is dense in  if and only if dom * ∩ ker ( ) = {0} for some or, equivalently, for every ∈ ℂ ⧵ ℝ. This is equivalent to the conditions in (4.11), since ker ( ) = ker Im ( ) = ker Im 0 ( ); see . It should be also noted that a condition which is equivalent to (4.11) appears in [70, Section 5.1]; see also [69]. For some further related facts, see Corollary 5.18 and Remark 5.20 in Section 5.

A Kreȋn type formula for -generalized boundary triples
In this section a Kreȋn type (resolvent) formula for -generalized boundary triples will be presented. We refer to [27, Proposition 7.27] where a special case of -generalized boundary triples was treated, and [11,12] for a special case of quasi boundary triples. The form of the formula as given in Theorem 4.12 below is new even in the standard case of ordinary boundary triples.
If 0 = ker Γ 0 is selfadjoint, then it follows from the first von Neumann's formula that for each ∈ ( 0 ) the domain of Γ can be decomposed as follows: ) .

Now let Γ be single-valued and let Γ be decomposed as
Let̃be an extension of which belongs to the domain of Γ and let Θ be a linear relation in  corresponding tõ: Theorem 4.12. Let be a closed symmetric relation, let Π = { , Γ 0 , Γ 1 } be an -generalized boundary triple for * with 0 = ker Γ 0 , and let (⋅) and (⋅) be the corresponding Weyl function and -field, respectively. Then for any extension Θ ∈ Ext satisfying Θ ⊂ dom Γ the following Kreȋn-type formula holds Here the inverses in the first and last terms are taken in the sense of linear relations.
The proof of this theorem is postponed until Section 5.2, where an analogous resolvent formula is proved for unitary boundary triples. However, some remarks and consequences of Theorem 4.12 are in order already here. . In particular, Θ − need not be invertible; Θ and Θ need not even be closed. Hence, even when Π = { , Γ 0 , Γ 1 } is an ordinary boundary triple for * the formula (4.16) uses only the assumption ∈ ( 0 ) instead of the standard assumption that ∈ ( The following statement is an immediate consequence of Theorem 4.12. . Then: (ii) if (Θ − ( )) −1 is a bounded operator, then the same is true for (iii) if 0 ∈ (Θ − ( )) then ∈ ( Θ ) .

Unitary boundary pairs and unitary colligations
Some formulas from Section 3 can be essentially improved when using the interrelations between unitary relations and unitary colligations, see [10].
Then (Γ) is the graph of a unitary operator  ∶ . The mapping ∶ Γ  →  establishes a one-to-one correspondence between the set of unitary boundary pairs and the set of unitary operators in ℌ ⊕ .
The last statement is obtained from (ii). □ Similarly, as was shown in [6, Theorem 5.35] any unitary boundary triple { , Γ 0 , Γ 1 } with the extra properties ran Γ 0 = ℌ and mul = {0}, where (a skew-adjoint operator) is the analog of the main transform of Γ (see (5.1)), corresponds to some impedance conservative continuous time input/state/output system Realization problems for Schur functions via transfer functions of scattering conservative (and passive) continuous time input/state/output systems were studied in [8,9] and were motivated by the earlier works [65,66]. On the other hand, connections between general unitary boundary pairs and the notion of conservative state/signal system nodes, whose systematic study was initiated in [5] (see also e.g. [53,54]), have been established in [6,Theorem 5.34]. Moreover, the connection between conservative state/signal system nodes and so-called Dirac structures can be found in [6,Proposition 5.38], while the connection between Dirac structures and unitary boundary pairs is made explicit in [41].

A Kreȋn type formula for unitary boundary triples
In this section Kreȋn's resolvent formula is extended to the setting of general unitary boundary triples. It is analogous to the formula established in Section 4.2. Recall from [25] that for a unitary boundary triple the kernel 0 = ker Γ 0 need not be selfadjoint, it is in general only a symmetric extension of which can even coincide with ; see e.g. [25,Example 6.6]. For simplicity the next result is formulated for nonreal points ∈ ℂ ⧵ ℝ; these points are regular type points for 0 . As in Section 4.2, let̃be an extension of which belongs to the domain of Γ and let Θ be a linear relation in  corresponding tõvia (4.15).

Theorem 5.8. Let be a closed symmetric relation, let Π =
{ , Γ 0 , Γ 1 } be a unitary boundary triple for * with 0 = ker Γ 0 , and let (⋅) and (⋅) be the corresponding Weyl function and -field, respectively. Then for any linear relation Θ(⊂ ran Γ) in  and the extension Θ ∈ Ext given by (4.15) the following equality holds where the inverses in the first and last term are taken in the sense of linear relations.
Proof. We first prove the inclusion "⊂" in (5.27). Since 0 is symmetric, ( 0 − ) −1 is a bounded, in general nondensely defined, operator for every fixed ∈ ℂ ⧵ ℝ. Now assume that { , ′′ } ∈ Hence, This proves the reverse inclusion in (5.27) and completes the proof. □ It is useful to make some further comments on the formula (5.27).
(i) Again notice the generality of the formula (5.27); in particular, as in Theorem 4.12 need not belong to ( Θ ) .
(ii) A careful look at the above proof shows that the key elements which in addition to the general properties of -fields and Weyl functions of isometric boundary triples are used in the proof are the following two requirements: (1) the equality Γ 1 ( ) = (̄) * , so that Γ 1̂0 = (̄) * when and̂0 are connected by (5.28); (2) ( ) (hence also ( )) is a densely defined operator or, equivalently, (̄) * is a closed operator. Hence, the formula (5.27) in Theorem 4.12 remains valid for isometric boundary triples which satisfy these two additional properties.
(iii) In the formula (5.27) the operator ( 0 − ) −1 cannot be shifted to the right hand side without loosing the stated equality.
Indeed, in that case only the following inclusion remains valid: Namely, by the equality Γ 1 ( ) = (̄) * one has ran ( 0 − ) = dom (̄) * and thus the term ( 0 − ) −1 can be shifted to the right side of (5.27) without changing the domain on the right side. However, in this case the range of the right side belongs to the span dom 0 + ( * ) and for general unitary boundary triples this would restrict the choice of Θ ; recall that for a unitary boundary triple 0 need not be even essentially selfadjoint, one can even have 0 = .
By considering the multi-valued parts we obtain the following statement for the point spectrum of Θ from Theorem 5.8. We are now ready to prove also Theorem 4.12 from Section 4.2.
Thus, from part (ii) in Remark 5.9 one concludes that the formula (5.27) holds. Furthermore, for an -generalized boundary triple 0 is selfadjoint. Thus dom

-generalized boundary triples
Here we extend Definition 1.11 to the case of boundary pairs. Definition 5.11. A unitary boundary pair {, Γ} is said to be an -generalized boundary pair, if 0 is a selfadjoint linear relation in ℌ.
In the following proposition some special boundary triples/pairs are characterized in terms of their Potapov-Ginzburg transform.
Proof. The equivalence of (i) and (ii) is implied by (5.8).

-generalized boundary triples and form domain invariance
Recall, see Definition 1.13, that a unitary boundary triple { , Γ 0 , Γ 1 } for * is called -generalized, if the extension 0 is essentially selfadjoint in ℌ.
As the main result of this section it will be shown that the class of Weyl functions of -generalized boundary triples coincides with the class of form domain invariant Nevanlinna functions.

Definition 5.21. A Nevanlinna function
∈ () is said to be form domain invariant in ℂ + (ℂ − ), if the quadratic form ( ) in  generated by the imaginary part of ( ) via The following two lemmas are preparatory for the main result.

In particular, if the statements (i)-(v) are satisfied both in ℂ + and ℂ − then Π is an -generalized boundary triple and the Weyl function is form domain invariant with
Proof. The equivalence (i) ⇔ (ii) is obtained from Lemma 5. 22. The fact that the domain of ( ) is dense in  follows from Proposition 5.2. The equivalences (i) ⇔ (iv), (v) and (ii) ⇔ (iii) follow from Lemmas 5.22 and 5.23. In particular, Lemma 5.22 shows that the form ( ) is closable for some (and then for every) ∈ ℂ + and for some (and then for every) ∈ ℂ − if and only if 0 is essentially selfadjoint. In this case the closure of the form ( ) is given by  . Then: (i) for every ∈ ( , ) ( ) admits a single-valued closure ( ) such that (5.35) and (5.36) hold for all , ∈ (ℂ ⧵ ℝ) ∪ ( , ); (ii) for every ∈ ( , ) and , ∈  there exists a limit ( ) The proof of the first statement is precisely the same as the proof of Lemma 5.23. The statement (ii) is implied by the equality (5.39), and the continuity of ( ) with respect to ∈ ( , ); see (5.36).
The assumption that ( ) admits a single-valued closure for some ∈ ℂ − does not imply that ( ) admits a single-valued closure for some ∈ ℂ + . In particular, for a maximal symmetric relation the following extreme situation holds. Proof. First recall that for every closed symmetric relation there is a unitary boundary pair for * ; see [25,Proposition 3.7]. By definition ( ) is a single-valued operator and it is known that ran ( ) = for every ∈ ℂ ⧵ ℝ; see [25,Lemma 2.14]. Hence the statement in the lower half-plane ℂ − is clear. In particular, one has ker ( ) = dom ( ) and consequently ran ( ) * = mul ( ) * , ∈ ℂ − . On the other hand, by Theorem 5.5 ran ( ) * and mul ( ) * do not depend on ∈ ℂ ⧵ ℝ. Consequently, the equality ran ( ) * = mul ( ) * holds also for every ∈ ℂ + . Then equivalently dom ( ) = ker ( ), which shows that ( ) is a singular operator with mul ( ) = ran ( ) = for every ∈ ℂ + . □ Observe that in Proposition 5.26 the corresponding Weyl function is actually domain invariant in each half-plane ℂ + and ℂ − , while it is neither domain nor form domain invariant in ℂ ⧵ ℝ; see Proposition 3.11 (i). For an explicit example demonstrating Proposition 5.26 we refer to [25,Example 6.7], where is the minimal differential operator generated in ℌ = 2 (0, ∞) by the differential expression . ≤ ∞ and − is a maximal symmetric operator in ℌ with + ( − ) = 0 and 0 < − ( − ) ≤ ∞, then = + ⊕ − is a symmetric operator in ℌ ⊕ ℌ with defect numbers { is a unitary boundary pair for * ± then clearly the orthogonal sum is a unitary boundary pair for * . Moreover, the corresponding -field is ( ) = + ( ) ⊕ − ( ), ∈ ℂ ⧵ ℝ. Now Proposition 5.26 shows that ( ) is not closable for any ∈ ℂ ⧵ ℝ. Hence, there exists symmetric operators with arbitrary deficiency indices ± ( ) and a unitary boundary pair for * such that the corresponding -field ( ) is not closable for any ∈ ℂ ⧵ ℝ. This also holds in the case of equal deficiency indices 0 < − ( − ) = + ( However, in this case the boundary pair Π for * is not minimal in general. . Considering the boundary pair Π + ⊕ Π − for * one concludes from Proposition 5.26 that the corresponding -field ( ) = + ( ) ⊕ − ( ) is a bounded operator for every ∈ ℂ − , while ( ) is not closable for any ∈ ℂ + .
In the next example we present a unitary boundary triple whose Weyl function is form domain invariant but not domain invariant.
Example 5.28. Let (⋅) be a scalar Nevanlinna function and  = 2 (0, ∞). Define an operator valued function (⋅) Clearly, ( ) is densely defined, ( ( ) ) ≠ ∅ for each ∈ ℂ + and the family (⋅) is holomorphic in ℂ + in the resolvent sense. Now consider the form generated by the imaginary part of ( ). Integrating by parts one obtains . Hence the form ( ) is nonnegative and ( ) is -dissipative for each ∈ ℂ + . Moreover, (⋅) ∈ () since ker ( ) = {0}. Therefore, by Theorem 1.10, there exists a certain unitary boundary triple such that the corresponding Weyl function coincides with (⋅). Notice that the form associated with ( ) in (1.14) coincides with ( ) up to an inessential renormalization by Im . Clearly, the form ( ) is closable with the closure given by Thus, the form domain dom ( does not depend on ∈ ℂ + while the domain dom ( ) does, i.e. satisfies Assumption 1.15. The operator associated with the form ( ) is given by The operator , can be treated as the imaginary part of the unbounded operator .
A simple example of a unitary boundary triple whose Weyl function is form domain invariant and -field is unbounded can be obtained as follows (see also [25,Example 6.5] These formulas imply that ( ) = 1∕2 and ( ) = , ∈ ℂ. In particular, the Weyl function is a Nevanlinna function. According to [25,Proposition 3.6] this implies that Γ is in fact ℌ -unitary.
In this example the Weyl function is also domain invariant. In fact, domain invariance of a Nevanlinna function implies its form domain invariance. Hence, the operator ( ) ∶= ( )− ( ) * −̄i s nonnegative and densely defined in  ⊖ mul ( ). Therefore, the form ( ) is closable for ∈ ℂ ⧵ ℝ; see [45]. By applying the same reasoning tōit is seen that also the form (̄) is closable. Now by applying Lemma 5.22 it is seen that 0 is essentially selfadjoint and hence by Theorem 5.24 is form domain invariant. □ The converse statement does not hold. In fact, in [28] an example of a form domain invariant Nevanlinna function is constructed, such that the domains of ( ) and ( ) have a zero intersection:

Renormalizations of form domain invariant Nevanlinna functions
The next theorem shows that form domain invariant Nevanlinna functions in  can be renormalized with a bounded operator such that the renormalized function * becomes domain invariant.
where 0 ⊂ is a closed densely defined symmetric restriction of .
Proof. The proof is divided into five steps.