Weyl families of essentially unitary pairs

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 $(\mathcal{H},\Gamma)$ belong to the class $\tilde{\mathcal{R}}(\mathcal{H})$ 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 $\tilde{\mathcal{R}}(\mathcal{H})$. Thus bounded Weyl functions of essentially unitary boundary pairs are of class $\mathcal{R}[\mathcal{H}]$.

Equations

$$A_{\ast }:=\mathrm{dom}\Gamma ,A:=A_{\ast }^{\ast }\equiv (A_{\ast }{)}^{\ast }=(\mathrm{dom}\Gamma {)}^{[\perp ]}=\mathrm{mul}{\Gamma }^{[\ast ]}.$$

$$A_{\star }:=\mathrm{dom}\overline{\Gamma }.$$

$$[\hat{f},\hat{g}{]}_{{\mathfrak{H}}^2}:={⟨\hat{f},J_{\mathfrak{H}}\hat{g}⟩}_{{\mathfrak{H}}^2}=-\mathrm{i}({⟨f,g^{\prime }⟩}_{\mathfrak{H}}-{⟨f^{\prime },g⟩}_{\mathfrak{H}})$$

$${\Gamma }^{[\ast ]}:=\{(\hat{k},\hat{g})\in {\mathcal{H}}^2\times {\mathfrak{H}}^2|(\forall (\hat{f},\hat{h})\in \Gamma )[\hat{f},\hat{g}{]}_{{\mathfrak{H}}^2}=[\hat{h},\hat{k}{]}_{{\mathcal{H}}^2}\}.$$

$$[\hat{f},\hat{g}{]}_{{\mathfrak{H}}^2}=[\hat{h},\hat{k}{]}_{{\mathcal{H}}^2},(\hat{f},\hat{h})\in \Gamma ,(\hat{g},\hat{k})\in \Gamma .$$

$${\Gamma }_z:=\Gamma |{}_{{\hat{\mathfrak{N}}}_z(A_{\ast })}:=\Gamma \cap ({\hat{\mathfrak{N}}}_z(A_{\ast })\times {\mathcal{H}}^2),z\in \mathbb{C}$$

$${\mathfrak{N}}_z(A_{\ast }):=\ker (A_{\ast }-z),{\hat{\mathfrak{N}}}_z(A_{\ast }):=\{{\hat{f}}_z=(f_z,z{f}_z)|f_z\in {\mathfrak{N}}_z(A_{\ast })\}$$

$${\Gamma }_w^{-1}\subseteq {\Gamma }^{-1}\subseteq {\Gamma }^{[\ast ]}\subseteq {\Gamma }_z^{[\ast ]}$$

$${\hat{\mathfrak{N}}}_z(A_{\ast }{)}^{[\perp ]}=(\mathrm{dom}{\Gamma }_z{)}^{[\perp ]}=\mathrm{mul}{\Gamma }_z^{[\ast ]}.$$

$${\hat{\mathfrak{N}}}_z(A_{\ast }{)}^{[\perp ]}=\overline{z}{I}_{\mathfrak{H}}\widehat{+}(\{0\}\times {\mathfrak{N}}_z(A_{\ast }{)}^{\perp })$$

$${\mathfrak{N}}_z(A_{\ast }{)}^{\perp }={\mathfrak{N}}_z(A^{\ast }{)}^{\perp }={\mathfrak{M}}_{\overline{z}}:=\overline{\mathrm{ran}}(A-\overline{z}).$$

$$A:=\mathrm{mul}{\Gamma }^{[\ast ]},A_{\star }:=\mathrm{dom}\overline{\Gamma },A_{\#}:=\mathrm{dom}{\Gamma }_{\#},{\Gamma }_{\#}:=({\Gamma }^{[\ast ]}{)}^{-1},$$

$$M_{{\Gamma }_{\#}}(z):={\Gamma }_{\#}({\hat{\mathfrak{N}}}_z(A_{\#})),z\in \mathbb{C}.$$

$$\Gamma^{[*]}_{z}=\bigl{(}\Gamma\cap(\widehat{\mathfrak{N}}_{z}(A_{*})\times\mathcal{H}^{2})\bigr{)}^{[*]}=\overline{\Gamma^{[*]}\,\widehat{+}\,\mathfrak{L}_{z}}\,,\quad\mathfrak{L}_{z}:=\{0\}\times\widehat{\mathfrak{N}}_{z}(A_{*})^{[\bot]}\,.$$

$$\ker {\Gamma }_z^{[\ast ]}=\ker \overline{{\Gamma }^{[\ast ]}\widehat{+}{\mathfrak{L}}_z}=\overline{\ker }({\Gamma }^{[\ast ]}\widehat{+}{\mathfrak{L}}_z).$$

$$\ker {\Gamma }_z^{[\ast ]}=\ker ({\Gamma }^{[\ast ]}\widehat{+}{\mathfrak{L}}_z)\iff \ker ({\Gamma }^{[\ast ]}\widehat{+}{\mathfrak{L}}_z)=\overline{\ker }({\Gamma }^{[\ast ]}\widehat{+}{\mathfrak{L}}_z).$$

$$\ker(\Gamma^{[*]}\,\widehat{+}\,\mathfrak{L}_{z})=\Gamma_{\#}\bigl{(}\widehat{\mathfrak{N}}_{\overline{z}}(A_{\#})\,\widehat{+}\,\Delta_{\overline{z}}\bigr{)}\,,\quad\widehat{\mathfrak{N}}_{\overline{z}}(A_{\#})=A_{\#}\cap\overline{z}I_{\mathfrak{H}}\,,$$

$${\Delta }_{\overline{z}}:=A_{\#}\cap (\{0\}\times {\mathfrak{M}}_{\overline{z}})=\{0\}\times (\mathrm{mul}{A}_{\#}\cap {\mathfrak{M}}_{\overline{z}}).$$

$$\mathrm{mul}{A}_{\#}\cap {\mathfrak{M}}_{\overline{z}}=\mathrm{mul}A\text{so that}{\Delta }_{\overline{z}}=\{0\}\times \mathrm{mul}A\subseteq \ker {\Gamma }_{\#}=A$$

$$\ker ({\Gamma }^{[\ast ]}\widehat{+}{\mathfrak{L}}_z)={\Gamma }_{\#}({\hat{\mathfrak{N}}}_{\overline{z}}(A_{\#}))=:M_{{\Gamma }_{\#}}(\overline{z}).$$

$$\overline{{\Gamma }^{[\ast ]}\widehat{+}{\mathfrak{L}}_z}={\Gamma }^{[\ast ]}\widehat{+}{\mathfrak{L}}_z$$

$$\overline{\Gamma}\,\widehat{+}\,\bigl{(}\widehat{\mathfrak{N}}_{z}(A^{*})\times\mathcal{H}^{2}\bigr{)}=\bigl{(}A_{\star}\,\widehat{+}\,\widehat{\mathfrak{N}}_{z}(A^{*})\bigr{)}\times\mathcal{H}^{2}$$

$$\overline{A_{\star }\widehat{+}{\hat{\mathfrak{N}}}_z(A^{\ast })}=\overline{A^{\ast }\widehat{+}{\hat{\mathfrak{N}}}_z(A^{\ast })}=A^{\ast }$$

$$A_{\star }\widehat{+}{\hat{\mathfrak{N}}}_z(A^{\ast })=\overline{A_{\star }\widehat{+}{\hat{\mathfrak{N}}}_z(A^{\ast })}=A^{\ast }.$$

$$\ker {\Gamma }_z^{[\ast ]}=M_{{\Gamma }_{\#}}(\overline{z})\iff A^{\ast }=A_{\star }\widehat{+}{\hat{\mathfrak{N}}}_z(A^{\ast }).\text{qed}$$

$$M_{\Gamma }(z):=\Gamma ({\hat{\mathfrak{N}}}_z(A_{\ast })),z\in {\mathbb{C}}_{\ast }.$$

$$M_{\Gamma }(z)=\mathrm{ran}{\Gamma }_z,M_{\Gamma }(z{)}^{\ast }=(\mathrm{ran}{\Gamma }_z{)}^{[\perp ]}=\ker {\Gamma }_z^{[\ast ]}$$

$$\mathfrak{K}:=\mathrm{span}\{g_{\alpha }:=g_{\sigma }(z_j)|\alpha =(\sigma ,j)\in \mathcal{S}\times J\},J:=\{1,2,\dots ,m\}.$$

$$\mathcal{G}:=({⟨g_{\alpha },g_{{\alpha }^{\prime }}⟩}_{\mathfrak{H}})\in \mathrm{B}({\mathbb{C}}^{md})$$

$${\mathfrak{K}}^{\prime }:={\mathfrak{H}}_{2m+2}\dotplus {\mathfrak{M}}_z,z\in \mathrm{res}L\setminus \mathcal{Z},\mathcal{Z}:=\{z_j|j\in J\}$$

Full text

1. Introduction

Throughout $\mathfrak{H}$ and $\mathcal{H}$ denote Hilbert spaces. Let $\Gamma\subseteq\mathfrak{H}^{2}\times\mathcal{H}^{2}$ be a linear relation from a $J_{\mathfrak{H}}$-space to a $J_{\mathcal{H}}$-space [AI89, Section 1], where the canonical symmetry $J_{\mathfrak{H}}$ ($J_{\mathcal{H}}$) acts on $\mathfrak{H}^{2}$ ($\mathcal{H}^{2}$) 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 $\Gamma^{[*]}$ denote the Krein space adjoint of $\Gamma$ [DHM20, Equation (2.4)], [DHMdS12, Section 7.2]. Then $\Gamma$ is said to be $(J_{\mathfrak{H}},J_{\mathcal{H}})$-isometric if $\Gamma^{-1}\subseteq\Gamma^{[*]}$ and $(J_{\mathfrak{H}},J_{\mathcal{H}})$-unitary if $\Gamma^{-1}=\Gamma^{[*]}$ [DHM20, Definition 2.1], and essentially $(J_{\mathfrak{H}},J_{\mathcal{H}})$-unitary if $\overline{\Gamma}^{\;-1}=\Gamma^{[*]}$ [DHMdS06, Section 2.3], where the overbar denotes the closure (with respect to the Hilbert space topology). In what follows we refer to $\Gamma$ simply as isometric/(essentially) unitary.

Let $\Gamma$ be a linear relation from a $J_{\mathfrak{H}}$-space to a $J_{\mathcal{H}}$-space, and put ($[\bot]$ denotes the $J_{\mathfrak{H}}$-orthogonal complement; the superscript $*$ denotes the Hilbert space adjoint)

[TABLE]

Then $A$ is a closed, not necessarily symmetric, linear relation in $\mathfrak{H}$, and $A_{*}$ is dense in $A^{*}=\overline{\operatorname{dom}}\Gamma$ (with respect to the Hilbert space topology on $\mathfrak{H}^{2}$). If $\Gamma$ is unitary, then $A=S:=\ker\Gamma$ is automatically symmetric; if $\Gamma$ is essentially unitary, then $A=\ker\overline{\Gamma}\subseteq A^{*}=S^{*}$ is also symmetric. Let

[TABLE]

Then $\overline{A_{\star}}=\overline{A_{*}}=A^{*}$ but $A_{\star}\supseteq A_{*}$. For a unitary $\Gamma$, $A_{\star}=A_{*}$.

Let $A$ be a closed symmetric linear relation in $\mathfrak{H}$. The pair $(\mathcal{H},\Gamma)$ is an isometric (unitary) boundary pair for $A^{*}$ if $A_{*}\subseteq A^{*}$ densely with respect to the topology on $\mathfrak{H}^{2}$, and if $\Gamma$ is isometric (unitary) [DHM20, Definition 3.1]. Note that $\overline{A_{*}}=A^{*}$ implies that $A=\operatorname{mul}\Gamma^{[*]}$. If $(\mathcal{H},\Gamma)$ is an isometric boundary pair for $A^{*}$, then $(\mathcal{H},\overline{\Gamma})$ is also an isometric boundary pair for $A^{*}$.

Let $(\mathcal{H},\Gamma)$ be a unitary boundary pair for $A^{*}=S^{*}$. In the terminology of [DHMdS06, Definition 3.1], a unitary $\Gamma$ is called a boundary relation for $S^{*}$; see also [DHMdS06, Proposition 3.2]. If $\Gamma$ is essentially unitary, we say that the pair $(\mathcal{H},\Gamma)$ is an essentially unitary boundary pair for $A^{*}=S^{*}$.

The first part of [DHMdS06, Theorem 3.9] states that the Weyl family $M_{\Gamma}(z)$, $z\in\mathbb{C}_{*}:=\mathbb{C}\smallsetminus\mathbb{R}$, corresponding to a boundary relation $\Gamma$ for $S^{*}$ is a Nevanlinna family; that is, it belongs to the class $\widetilde{\mathcal{R}}(\mathcal{H})$ (Definition 4.1). Here we prove an analogue of this statement for an essentially unitary $\Gamma$.

Theorem 1.1**.**

Let $M_{\Gamma}$ be the Weyl family corresponding to an essentially unitary boundary pair $(\mathcal{H},\Gamma)$ for $A^{*}$. Then the closure $\mathbb{C}_{*}\ni z\mapsto\overline{M_{\Gamma}(z)}=M_{\overline{\Gamma}}(z)$ belongs to a Nevanlinna family.

Here $M_{\overline{\Gamma}}$ is the Weyl family corresponding to a unitary boundary pair $(\mathcal{H},\overline{\Gamma})$. For $\Gamma$ unitary (hence closed), the theorem clearly reduces to the first part of [DHMdS06, Theorem 3.9]. By assuming additionally that $M_{\Gamma}(z)\in\mathrm{B}(\mathcal{H})$ is a bounded (hence closed) operator, one deduces another corollary.

Corollary 1.2**.**

Let $(\mathcal{H},\Gamma)$ be an essentially unitary boundary pair for $A^{*}$, and assume in addition that $M_{\Gamma}(z)\in\mathrm{B}(\mathcal{H})$, $z\in\mathbb{C}_{*}$. Then the Weyl function $M_{\Gamma}=M_{\overline{\Gamma}}$ belongs to the subclass $\mathcal{R}[\mathcal{H}]$ of Nevanlinna functions.∎

According to [DHMdS06, Proposition 5.9] a Nevanlinna function of class $\mathcal{R}[\mathcal{H}]$ can be realized as the Weyl function of a $B$-generalized boundary pair $(\mathcal{H},\Gamma)$ [DHM20, Definition 3.5]. Let us recall that ordinary, $B$-generalized, $S$-generalized, $ES$-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 $\operatorname{ran}\overline{\Gamma}=\mathcal{H}^{2}$, one concludes that the Weyl function $M_{\Gamma}=M_{\overline{\Gamma}}$ belongs to the subclass $\mathcal{R}^{u}[\mathcal{H}]$ of uniformly strict Nevanlinna functions. The single-valued linear relation (i.e. operator) $\Gamma$ 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 $M_{\Gamma}(z)^{*}$ for an isometric boundary pair $(\mathcal{H},\Gamma)$; it follows that $M_{\Gamma}(z)^{*}=M_{\overline{\Gamma}}(\overline{z})$ for $\Gamma$ essentially unitary. Since $M_{\overline{\Gamma}}$ is a Nevanlinna family for $\overline{\Gamma}$ unitary, this leads to Theorem 1.1; see Section 4.

Throughout we use the standard symbols $\operatorname{dom}$, $\operatorname{ran}$, $\operatorname{mul}$, and $\ker$ 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 $\Gamma$ from a $J_{\mathfrak{H}}$-space to a $J_{\mathcal{H}}$-space. The $J_{\mathfrak{H}}$-metric $[\cdot,\cdot]_{\mathfrak{H}^{2}}$ is written in terms of the $\mathfrak{H}^{2}$-scalar product $\braket{\cdot,\cdot}_{\mathfrak{H}^{2}}$ according to

[TABLE]

for $\widehat{f}=(f,f^{\prime})\in\mathfrak{H}^{2}$ and $\widehat{g}=(g,g^{\prime})\in\mathfrak{H}^{2}$, provided that the $\mathfrak{H}$-scalar product $\braket{\cdot,\cdot}_{\mathfrak{H}}$ is conjugate-linear in the first argument. The same applies to the $J_{\mathcal{H}}$-metric $[\cdot,\cdot]_{\mathcal{H}^{2}}$.

The Krein space adjoint $\Gamma^{[*]}$ of $\Gamma$ is defined by

[TABLE]

In particular, the inclusion $\Gamma^{-1}\subseteq\Gamma^{[*]}$ implies that the Green identity holds:

[TABLE]

Put $A_{*}:=\operatorname{dom}\Gamma$ and

[TABLE]

and let $\Gamma^{[*]}_{z}$ denote the Krein space adjoint of $\Gamma_{z}$. As usual, the eigenspaces of $A_{*}$ are given by

[TABLE]

and similarly for other linear relations.

If $\Gamma$ is isometric, then $\Gamma_{z}\subseteq\Gamma$ implies that

[TABLE]

for $w\in\mathbb{C}$; that is, $\Gamma_{z}$ is also isometric.

The $J_{\mathfrak{H}}$-orthogonal complement (recall e.g. [AI89, Definition 1.11]) $\widehat{\mathfrak{N}}_{z}(A_{*})^{[\bot]}$ of $\widehat{\mathfrak{N}}_{z}(A_{*})=\operatorname{dom}\Gamma_{z}$ is written in terms of $\Gamma_{z}$ thus

[TABLE]

On the other hand, $\widehat{\mathfrak{N}}_{z}(A_{*})^{[\bot]}$ consists of $(f,f^{\prime})\in\mathfrak{H}^{2}$ such that $f^{\prime}-\overline{z}f\in\mathfrak{N}_{z}(A_{*})^{\bot}$ ($\bot$ denotes the orthogonal complement in $\mathfrak{H}$); hence

[TABLE]

where $\,\widehat{+}\,$ denotes the componentwise sum [HdSS09, Section 2.4] and $I_{\mathfrak{H}}$ (the graph of) the identity operator in $\mathfrak{H}$. Because $A_{*}$ is dense in $A^{*}=\overline{\operatorname{dom}}\Gamma$, one has that (see also [DHM20, Proposition 3.9(i)])

[TABLE]

Lemma 2.6**.**

Let $\Gamma$ be an isometric linear relation from a $J_{\mathfrak{H}}$-space to a $J_{\mathcal{H}}$-space, and define

[TABLE]

[TABLE]

The following statements are equivalent:

• $\mathrm{(i)}$

$\ker\Gamma^{[*]}_{z}=M_{\Gamma_{\#}}(\overline{z})$.

• $\mathrm{(ii)}$

$M_{\Gamma_{\#}}(\overline{z})$* is a closed linear relation in $\mathcal{H}$.*

• $\mathrm{(iii)}$

$A^{*}=A_{\star}\,\widehat{+}\,\widehat{\mathfrak{N}}_{z}(A^{*})$.

Proof.

(i) $\Rightarrow$ (ii) is clear, since the kernel of a closed linear relation $\Gamma^{[*]}_{z}$ is closed.

(ii) $\Rightarrow$ (i) By using (2.2) we have

[TABLE]

Then

[TABLE]

From here we see that

[TABLE]

By using (2.4) and (2.5)

[TABLE]

[TABLE]

But $A\subseteq A_{\#}$ implies that

[TABLE]

and it therefore follows that

[TABLE]

(i) $\Leftrightarrow$ (iii) By arguing as in [HdSS09, Lemma 2.10], the componentwise sum of two closed linear relations ($\Gamma^{[*]}$ and $\mathfrak{L}_{z}$) 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 $\overline{A_{\star}}=A^{*}$ and

[TABLE]

we get that $\Gamma^{[*]}\,\widehat{+}\,\mathfrak{L}_{z}$ 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 $\Gamma$ from a $J_{\mathfrak{H}}$-space to a $J_{\mathcal{H}}$-space is assumed to be isometric, unless explicitly stated otherwise. Whenever we speak of an isometric boundary pair $(\mathcal{H},\Gamma)$ for $A^{*}$, we assume that $A$ is a closed symmetric linear relation in $\mathfrak{H}$.

The Weyl family of $A$ corresponding to an isometric boundary pair $(\mathcal{H},\Gamma)$ for $A^{*}$ is defined by (see e.g. [DHM20, Definition 3.2])

[TABLE]

In terms of $\Gamma_{z}$ the linear relation $M_{\Gamma}(z)$ and its adjoint in $\mathcal{H}$ can be described by

[TABLE]

It follows from the Green identity (2.1) and (3.1) that $M_{\Gamma}(\overline{z})\subseteq M_{\Gamma}(z)^{*}$. By (3.1), the intersection $M_{\Gamma}(z)\cap M_{\Gamma}(z)^{*}$ is a subset of the set of neutral vectors [AI89, Definition 1.3] of a $J_{\mathcal{H}}$-space. Thus, by applying (2.1) to $(\widehat{f}_{z},\widehat{h})\in\Gamma_{z}$ such that $[\widehat{h},\widehat{h}]_{\mathcal{H}^{2}}=0$, one finds that $M_{\Gamma}(z)\cap M_{\Gamma}(z)^{*}=\operatorname{mul}\Gamma_{z}=\operatorname{mul}\Gamma$. 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 $(\mathcal{H},\Gamma)$. By using Lemma 2.6, one can find other invariance results for $M_{\Gamma}$, which we do not repeat here.

The following corollary is an application of Lemma 2.6 and (3.1).

Corollary 3.2**.**

Let $(\mathcal{H},\Gamma)$ be an essentially unitary boundary pair for $A^{*}$, with the Weyl family $M_{\Gamma}$, and let $M_{\overline{\Gamma}}$ be the Weyl family corresponding to a unitary boundary pair $(\mathcal{H},\overline{\Gamma})$. Then $M_{\Gamma}(z)^{*}=M_{\overline{\Gamma}}(\overline{z})$, $z\in\mathbb{C}_{*}$.

Proof.

Put $\overline{\Gamma}=\Gamma_{\#}$ in the lemma; then $A_{\star}=A_{\#}$. Since $M_{\Gamma_{\#}}(\overline{z})=M_{\overline{\Gamma}}(\overline{z})$ and $M_{\overline{\Gamma}}(\overline{z})$ is closed by [DHMdS06, Theorem 3.9], it follows that $M_{\Gamma}(z)^{*}=M_{\overline{\Gamma}}(\overline{z})$. ∎

Clearly if $\Gamma$ is unitary, we get that $M_{\Gamma}(z)^{*}=M_{\Gamma}(\overline{z})$ for $z\in\mathbb{C}_{*}$.

Remark 3.3*.*

In the proof of Corollary 3.2 we use the fact that the Weyl family $M_{\overline{\Gamma}}$ corresponding to a unitary boundary pair $(\mathcal{H},\overline{\Gamma})$ for $A^{*}$ is a Nevanlinna family, and hence is in particular defined by a closed linear relation $M_{\overline{\Gamma}}(z)$, $z\in\mathbb{C}_{*}$. 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: $M_{\Gamma}(z)^{*}=M_{\overline{\Gamma}}(\overline{z})$ for an essentially unitary boundary pair $(\mathcal{H},\Gamma)$ such that $(\mathcal{H},\overline{\Gamma})$ is an $S$-generalized boundary pair $(\mathcal{H},\overline{\Gamma})$, i.e. such that $A_{0}:=\ker(\overline{\Gamma})_{0}$ is additionally assumed to be self-adjoint ([DHM20, Definition 5.11]). Here the linear relation $(\overline{\Gamma})_{0}:=\{(\widehat{f},h)\,|\,(\exists h^{\prime})\,(\widehat{f},(h,h^{\prime}))\in\overline{\Gamma}\}$. The argumentation is due to von Neumann formula $A^{*}=A_{0}\,\widehat{+}\,\widehat{\mathfrak{N}}_{z}(A^{*})$, $z\in\mathbb{C}_{*}$, from which one has $A_{\star}=A_{0}\,\widehat{+}\,\widehat{\mathfrak{N}}_{z}(A_{\star})$, since $A_{0}\subseteq A_{\star}$; for the latter representation of $A_{\star}$, statement (iii) is true (see also [DHM20, Theorem 5.17]). Vice verse, because by Corollary 3.2 $M_{\Gamma_{\#}}(\overline{z})=M_{\overline{\Gamma}}(\overline{z})$ is closed, we conclude from Lemma 2.6 that (iii) must be true for $\Gamma$ essentially unitary.

Corollary 3.4**.**

Let $(\mathcal{H},\Gamma)$ be an essentially unitary boundary pair for $A^{*}$, and let $A_{\star}:=\operatorname{dom}\overline{\Gamma}$. Then $A^{*}=A_{\star}\,\widehat{+}\,\widehat{\mathfrak{N}}_{z}(A^{*})$ for all $z\in\mathbb{C}_{*}$.

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 $M(z)$, $z\in\mathbb{C}_{*}$, of linear relations in $\mathcal{H}$ belongs to the class $\widetilde{\mathcal{R}}(\mathcal{H})$ of Nevanlinna families, or is said to be a Nevanlinna family, if:

• $(a)$

For $\Im z>0$ ($\Im z<0$), the relation $M(z)$ is maximal dissipative (accumulative), and the operator family $(M(z)+w)^{-1}\in\mathrm{B}(\mathcal{H})$, $w\in\mathbb{C}_{+}$ ($\mathbb{C}_{-}$), is analytic;

• $(b)$

$M(z)^{*}=M(\overline{z})$.

Moreover: $M\in\mathcal{R}[\mathcal{H}]$ if $M\in\widetilde{\mathcal{R}}(\mathcal{H})$ and $\operatorname{dom}M(z)=\mathcal{H}$; $M\in\mathcal{R}^{s}[\mathcal{H}]$ if $M\in\mathcal{R}[\mathcal{H}]$ and $\ker\Im M(z)=\{0\}$; $M\in\mathcal{R}^{u}[\mathcal{H}]$ if $M\in\mathcal{R}^{s}[\mathcal{H}]$ and $0\in\operatorname{res}\Im M(z)$ ($\operatorname{res}$ labels the resolvent set). These subclasses are all covered by the subclass $\mathcal{R}(\mathcal{H})\subset\widetilde{\mathcal{R}}(\mathcal{H})$ of Nevanlinna families $M$ such that $M(z)$ is an operator. For more on the classification of Nevanlinna families the reader may refer to [DHMdS06].

Recall that $\mathbb{C}_{+}$ ($\mathbb{C}_{-}$) is the set of $z\in\mathbb{C}$ such that $\Im z>0$ ($\Im z<0$). A linear relation $M(z)$ is dissipative (resp. accumulative) if $(\forall(h,h^{\prime})\in M(z))$ $\Im\braket{h,h^{\prime}}_{\mathcal{H}}\geq 0$ (resp. $\leq 0$); we emphasize that the $\mathcal{H}$-scalar product is conjugate-linear in the first argument. A dissipative (resp. accumulative) $M(z)$ is maximal dissipative (resp. maximal accumulative) if $M(z)$ has no proper dissipative (resp. accumulative) extensions.

The Weyl family $M_{\Gamma}(z)$, $z\in\mathbb{C}_{*}$, corresponding to an isometric boundary pair $(\mathcal{H},\Gamma)$ for $A^{*}\supseteq A=\overline{A}$ is dissipative (accumulative) for $\Im z>0$ ($\Im z<0$). Indeed, in view of (3.1), $\widehat{h}=(h,h^{\prime})\in M_{\Gamma}(z)$ implies that $(\widehat{f}_{z},\widehat{h})\in\Gamma_{z}$ for some $\widehat{f}_{z}\in\widehat{\mathfrak{N}}_{z}(A_{*})$. Then, by the Green identity (2.1), $\Im\braket{h,h^{\prime}}_{\mathcal{H}}=(\Im z)\lVert f_{z}\rVert^{2}_{\mathfrak{H}}$; hence the claim. But then $(M_{\Gamma}(z)+w)^{-1}$ is an operator family by [DdS74, Theorem 3.1(i)].

If in addition $M_{\Gamma}(z)^{*}=M_{\Gamma}(\overline{z})$, then $M_{\Gamma}(\overline{z})^{*}=M_{\Gamma}(z)$, and therefore each member of the Weyl family is closed in this case: $M_{\Gamma}(z)^{**}=M_{\Gamma}(\overline{z})^{*}=M_{\Gamma}(z)$. But then the operator $(M_{\Gamma}(z)+w)^{-1}\in\mathrm{B}(\mathcal{H})$ by [DdS74, Theorem 3.1(vi)], and the relation $M_{\Gamma}(z)$ is maximal dissipative (accumulative) by [DdS74, Theorem 3.4(ii)].

By the above we conclude the following:

Lemma 4.2**.**

Let $M_{\Gamma}$ be the Weyl family corresponding to an isometric boundary pair $(\mathcal{H},\Gamma)$ for $A^{*}$. If $M_{\Gamma}(z)^{*}=M_{\Gamma}(\overline{z})$, $z\in\mathbb{C}_{*}$, then $M_{\Gamma}$ 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 $A$ and of its simple part coincide. Indeed, let $A_{s}$ be the simple part [LT77, Proposition 1.1] of $A$ and let $\Gamma_{s}$ be the restriction to $\mathfrak{H}^{2}_{s}$ of $\Gamma$, where $\mathfrak{H}_{s}$ is the closed linear span of $\{\mathfrak{N}_{z}(A^{*})\,|\,z\in\mathbb{C}_{*}\}$. Put $A_{s*}:=\operatorname{dom}\Gamma_{s}=A_{*}\cap\mathfrak{H}^{2}_{s}$. Then $\widehat{\mathfrak{N}}_{z}(A_{s*})=\widehat{\mathfrak{N}}_{z}(A_{*})\cap\mathfrak{H}^{2}_{s}=\widehat{\mathfrak{N}}_{z}(A_{*})$. Thus, since $\Gamma$ is isometric, $\Gamma_{s}\subseteq\Gamma$ is also isometric, and the corresponding Weyl family of $A_{s}$ is given by $M_{\Gamma_{s}}(z)=M_{\Gamma}(z)$, $z\in\mathbb{C}_{*}$, by noting that $\Gamma_{s}\cap\Gamma_{z}=\Gamma_{z}$. In addition, given an isometric $\Gamma$, assume that $\Gamma_{s}$ is essentially unitary. Then $\Gamma$ is also essentially unitary, whose closure $\overline{\Gamma}=\overline{\Gamma_{s}}$.

5. Example of an essentially unitary

boundary triple

An isometric boundary pair $(\mathcal{H},\Gamma)$ for $A^{*}\supseteq A=\overline{A}$ having the property $\operatorname{mul}\Gamma=\{0\}$ is named by an isometric boundary triple $(\mathcal{H},\Gamma_{0},\Gamma_{1})$ ([DHM20, Section 3.1]). Then a linear relation $\Gamma$ is identified with an operator $\widehat{f}\mapsto(\Gamma_{0}\widehat{f},\Gamma_{1}\widehat{f})$ from $A_{*}:=\operatorname{dom}\Gamma$ to $\mathcal{H}^{2}$, and the corresponding Weyl function $M_{\Gamma}$ satisfies $M_{\Gamma}(z)\Gamma_{0}=\Gamma_{1}$ on $\widehat{\mathfrak{N}}_{z}(A_{*})$, $z\in\mathbb{C}_{*}$. When $\Gamma$ is in addition essentially unitary, the triple $(\mathcal{H},\Gamma_{0},\Gamma_{1})$ is called an essentially unitary boundary triple. We recall from [DHMdS06, Lemma 4.1(ii)] that $\operatorname{mul}\Gamma=\{0\}$ implies $\operatorname{mul}M_{\Gamma}(z)=\{0\}$, i.e. $M_{\Gamma}\in\mathcal{R}(\mathcal{H})$, but note that $\operatorname{dom}M_{\Gamma}(z)=\Gamma_{0}(\widehat{\mathfrak{N}}_{z}(A_{*}))$ is in general a proper subset of $\mathcal{H}$. In case the triple $(\mathcal{H},\Gamma_{0},\Gamma_{1})$ is such that $A_{*}=A^{*}$ and $\Gamma$ is unitary, one has $\operatorname{ran}\Gamma=\mathcal{H}^{2}$ by [DHMdS06, Corollary 2.4], and the triple is called an ordinary boundary triple for $A^{*}$. In case $A$ is densely defined (i.e. $A^{*}$ is an operator), $A^{*}$ 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 $A$ be a densely defined, closed, symmetric operator in a Hilbert space $\mathfrak{H}$ with defect numbers $(d,d)$. Let $L$ be a self-adjoint extension of $A$ in $\mathfrak{H}$. For simplicity, we assume that $L$ is lower semibounded. By the von Neumann formula the adjoint $A^{*}\supseteq L$ is described by $\operatorname{dom}A^{*}=\operatorname{dom}L\dotplus\mathfrak{N}_{z}(A^{*})$, where the eigenspace is spanned by the deficiency elements $g_{\sigma}(z)$, $z\in\operatorname{res}L$, with $\sigma$ ranging over an index set $\mathcal{S}$ of cardinality $d$. That is, $\mathfrak{N}_{z}(A^{*})=g_{z}(\mathbb{C}^{d})$, where $g_{z}(c):=\sum_{\sigma\in\mathcal{S}}c_{\sigma}g_{\sigma}(z)$ for $c=(c_{\sigma})\in\mathbb{C}^{d}$.

Let $\mathfrak{H}_{n+1}\subset\mathfrak{H}_{n}$, $n\in\mathbb{Z}$, be the scale of Hilbert spaces associated with $L$ (see e.g. [AK00]); in particular, $\mathfrak{H}_{2}=\operatorname{dom}L$ and $\mathfrak{H}_{0}=\mathfrak{H}$. Then a deficiency element $g_{\sigma}(z)\in\mathfrak{H}_{0}\smallsetminus\mathfrak{H}_{1}$ can be defined in the generalized sense as $g_{\sigma}(z)=(L-z)^{-1}\varphi_{\sigma}$ for the functional $\varphi_{\sigma}\in\mathfrak{H}_{-2}\smallsetminus\mathfrak{H}_{-1}$. Thus, $A$ is the symmetric restriction of $L$ to the domain of $u\in\mathfrak{H}_{2}$ such that $\braket{\varphi,u}=0$; we use the vector notation $\braket{\varphi,\cdot}=(\braket{\varphi_{\sigma},\cdot})\colon\mathfrak{H}_{2}\rightarrow\mathbb{C}^{d}$.

5.1. Finite rank perturbation of $L$

Fix $m\in\mathbb{N}$ and consider the set

[TABLE]

The points $z_{j}\in\operatorname{res}L\cap\mathbb{R}$ are such that $z_{j}\neq z_{j^{\prime}}$ for $j\neq j^{\prime}$ and $j,j^{\prime}\in J$. The system $\{g_{\alpha}\}$ is linearly independent, so the Hermitian (Gram) matrix

[TABLE]

is positive definite.

Consider another set

[TABLE]

where the subset $\mathfrak{M}_{z}\subseteq\mathfrak{H}_{2m}\subseteq\mathfrak{H}_{2}$ is defined by

[TABLE]

where the multiplier

[TABLE]

for $m>1$, and $b_{1}(z):=1$ for $m=1$ and $z\in\mathbb{C}$. Define the operator $K$ in $\mathfrak{H}$ by

[TABLE]

for $u+k\in\mathfrak{K}^{\prime}\dotplus\mathfrak{K}$, where $C_{\sigma j,\sigma^{\prime}j^{\prime}}:=z_{j}[\mathcal{G}^{-1}]_{\sigma j,\sigma^{\prime}j^{\prime}}$. Then $K$ is referred to as the rank-$md$ perturbation of $L$. Note that the sum defining $\operatorname{dom}K$ is direct.

5.2. Boundary value space

Let $\mathcal{H}:=\mathbb{C}^{d}$ and define the operator $\Gamma$ (identified with its graph) by

[TABLE]

Here the column-vectors

[TABLE]

and the matrix

[TABLE]

for some matrix-valued Nevanlinna family $R(\cdot)=(R_{\sigma\sigma^{\prime}}(\cdot))$ of class $\mathcal{R}^{s}[\mathcal{H}]$. In fact, if the functional $\braket{\varphi^{\mathrm{ex}},\cdot}=(\braket{\varphi^{\mathrm{ex}}_{\sigma},\cdot})$ extends $\braket{\varphi,\cdot}\colon\mathfrak{H}_{2}\rightarrow\mathcal{H}$ to $\operatorname{dom}A^{*}$ according to (see also [AK00, Section 3.1.3], [HK09])

[TABLE]

then the matrix $R(z)\in\mathrm{B}(\mathcal{H})$, $z\in\operatorname{res}L$, is defined by

[TABLE]

From here one verifies that $\ker\Im R(z)=\{0\}$: By definition the imaginary part $\Im R(z)$ is the matrix with entries $(\Im z)\braket{g_{\sigma}(z),g_{\sigma^{\prime}}(z)}_{\mathfrak{H}}$. Then the kernel $\ker\Im R(z)$ is the set of $c\in\mathcal{H}$ such that $g_{z}(c)\in g_{z}(\mathcal{H})^{\bot}$; hence $c=0$.

An ordinary boundary triple $(\mathbb{C}^{d},\widetilde{\Gamma}_{0},\widetilde{\Gamma}_{1})$ for $A^{*}$ is defined by

[TABLE]

with $u^{\#}\in\mathfrak{H}_{2}$ and $c\in\mathcal{H}$, and with $R(z)$ as described above, while $\Gamma$ defines the boundary value space of operator $K$. Indeed, associate with $\Gamma$ two single-valued linear relations

[TABLE]

Then the boundary form of the operator $K$ is given by

[TABLE]

for $u,v\in\operatorname{dom}K$, provided that $\Gamma_{0}$ (resp. $\Gamma_{1}$) is regarded as the mapping $\operatorname{dom}K\rightarrow\mathcal{H}$.

5.3. Linear relation $\Gamma_{z}$ and

its Krein space adjoint

Let $K^{*}$ be the adjoint of $K$ in $\mathfrak{H}$.

Proposition 5.1**.**

$K^{*}=A$.

Proof.

The (graph of the) adjoint $K^{*}$ consists of $(y,x)\in\mathfrak{H}\times\mathfrak{H}$ such that $(\forall u+k\in\mathfrak{K}^{\prime}\dotplus\mathfrak{K})$

[TABLE]

Let $P$ be an orthogonal projection in $\mathfrak{H}$ onto $\mathfrak{K}$. Then $x=x_{\bot}+k_{x}\in\mathfrak{K}^{\bot}\oplus\mathfrak{K}$ and $y=y_{\bot}+k_{y}\in\mathfrak{K}^{\bot}\oplus\mathfrak{K}$, where $\mathfrak{K}^{\bot}:=\mathfrak{H}\ominus\mathfrak{K}$. Since $\mathfrak{K}^{\prime}\subseteq\mathfrak{K}^{\bot}$ it follows that

[TABLE]

and

[TABLE]

where $Z_{d}$ is the matrix direct sum of $d$ diagonal matrices $\operatorname{diag}\{z_{j};j\in J\}$. Because the set $\mathfrak{K}^{\prime}\supseteq\mathfrak{H}_{2m+2}$, i.e. is a dense subset of $\mathfrak{H}_{2}$, it follows from the above that $y\in\mathfrak{H}_{2}$ and $x=Ly$ and $d(k_{x})=\mathcal{G}^{-1}Z_{d}\mathcal{G}d(k_{y})$. On the other hand, $k_{x}=PLy$ implies that

[TABLE]

Since $\mathcal{X}$ is invertible, one deduces that $\braket{\varphi,y}=0$, and hence $y\in\operatorname{dom}A$. ∎

Let $A_{*}:=\operatorname{dom}\Gamma=K$. By Proposition 5.1 $A_{*}$ is dense in $A^{*}$, i.e. $\overline{A_{*}}=A^{*}$. One also verifies that the eigenspace $\mathfrak{N}_{z}(A_{*})=\mathfrak{N}_{z}(A^{*})$ for $z\in\operatorname{res}L\smallsetminus\mathcal{Z}$. Then, the single-valued linear relation $\Gamma_{z}$ (recall (2.2)) and its Krein space adjoint $\Gamma^{[*]}_{z}$ are given by

[TABLE]

It follows that $\Gamma_{z}\subseteq\Gamma$ and $\Gamma^{-1}_{w}\subseteq\Gamma^{[*]}_{z}$ for all $z,w\in\operatorname{res}L\smallsetminus\mathcal{Z}$. Moreover

[TABLE]

5.4. Weyl function

Let $\Gamma^{[*]}$ be the Krein space adjoint of $\Gamma$.

Proposition 5.2**.**

$\Gamma^{[*]}=\overline{\Gamma}^{\;-1}$, with the closure $\overline{\Gamma}=(\widetilde{\Gamma}_{0},\widetilde{\Gamma}_{1})$.

Proof.

Step 1. By definition, $\Gamma^{[*]}$ consists of $((c,c^{\prime}),(u,u^{\prime}))\in\mathcal{H}^{2}\times\mathfrak{H}^{2}$ such that

[TABLE]

and

[TABLE]

where $L_{-2}$ is a bounded continuation to $\mathfrak{H}_{-2}$ of $L$, and where an orthogonal projection $P$ and the matrices $Z_{d}$ and $\mathcal{X}$ are as in the proof of Proposition 5.1. Using $u=u_{\bot}+k_{u}\in\mathfrak{K}^{\bot}\oplus\mathfrak{K}$ and

[TABLE]

one concludes that $u\in\operatorname{dom}\tau$ and $u^{\prime}=\tau u$ and $c=c(k_{u})=:\Gamma^{\tau}_{0}u$, where the operator

[TABLE]

Using in addition that

[TABLE]

one finds that

[TABLE]

On the other hand, by the definition of $\tau$

[TABLE]

Thus

[TABLE]

and subsequently

[TABLE]

Step 2. The boundary form of $\tau$ satisfies an abstract Green identity

[TABLE]

for $u,v\in\operatorname{dom}\tau$, and for $\Gamma^{\tau}_{0}$ and $\Gamma^{\tau}_{1}$ as defined in step 1. Therefore, to show that

[TABLE]

it remains to prove that $\tau=A^{*}$ and $\Gamma^{\tau}_{i}=\widetilde{\Gamma}_{i}$, $i\in\{0,1\}$.

By direct computation, and using the equivalence relation $\mathfrak{H}_{2}\cap\mathfrak{K}\ni k\Leftrightarrow c(k)=0$, one finds that the adjoint $\tau^{*}=A$; since $\tau$ and $A$ are closed, this yields $\tau=A^{*}$.

Consider $u=u_{\bot}+k_{u}\in\operatorname{dom}\tau=\operatorname{dom}A^{*}$; then $u=u^{\#}+g_{z}(c)$ with $c:=c(k_{u})$ and $u^{\#}:=u_{\bot}+k_{u}-g_{z}(c)$. Since $u_{\bot}\in\mathfrak{H}_{2}$ and $k_{u}-g_{z}(c)\in\mathfrak{H}_{2}$, it holds $u^{\#}\in\mathfrak{H}_{2}$. Then

[TABLE]

and

[TABLE]

This completes the proof of $(\Gamma^{[*]})^{-1}=(\widetilde{\Gamma}_{0},\widetilde{\Gamma}_{1})$.

Step 3. The closure $\overline{\Gamma}$ is the adjoint of $J_{\mathfrak{H}}\Gamma^{[*]}J_{\mathcal{H}}$; hence it consists of $((y,y^{\prime}),(x,x^{\prime}))\in\mathfrak{H}^{2}\times\mathcal{H}^{2}$ such that

[TABLE]

for all $u\in\operatorname{dom}A^{*}$; i.e. $\overline{\Gamma}=(\Gamma^{[*]})^{-1}$. ∎

Observe that $\Gamma^{-1}\subseteq\Gamma^{[*]}\subseteq\Gamma^{[*]}_{w}$ for all $w\in\operatorname{res}L\smallsetminus\mathcal{Z}$, so (2.3) holds true. Note also that $A=\ker\overline{\Gamma}=\ker\Gamma\subseteq A_{*}=K\subseteq A^{*}=\operatorname{dom}\overline{\Gamma}$.

Since $\Gamma$ is essentially unitary and $\overline{A_{*}}=A^{*}$, the triple $(\mathcal{H},\Gamma_{0},\Gamma_{1})$ is an essentially unitary boundary triple for $A^{*}$. The triple $(\mathcal{H},\widetilde{\Gamma}_{0},\widetilde{\Gamma}_{1})$ is an ordinary boundary triple for $A^{*}$, with the associated Weyl family defined by $M_{\overline{\Gamma}}(z)=R(z)$, $z\in\operatorname{res}L$.

The domain

[TABLE]

so by Corollary 1.2 the Weyl function

[TABLE]

and therefore $M_{\Gamma}$ belongs to a Nevanlinna family of class $\mathcal{R}^{s}[\mathcal{H}]$. That $M_{\Gamma}=R$ can be also checked by computing $\Gamma(\widehat{\mathfrak{N}}_{z}(A_{*}))$ directly. Moreover, since $\operatorname{ran}\overline{\Gamma}=\mathcal{H}^{2}$, one concludes that actually $R\in\mathcal{R}^{u}[\mathcal{H}]$.