Weyl families of transformed boundary pairs

TL;DR

This paper investigates the properties and transformations of Weyl families associated with boundary pairs in Krein spaces, providing new characterizations, conditions for adjoints, and a scheme for transforming boundary pairs.

Contribution

It introduces novel criteria for the closure and adjoint of Weyl families, and develops a transformation scheme for boundary pairs with explicit formulas and correspondences.

Findings

Characterization of the closure and adjoint of Weyl families.

Conditions for the equality of transformed Weyl families.

A transformation scheme for boundary pairs with explicit formulas.

Abstract

Let $(\mathfrak{L},\Gamma)$ be an isometric boundary pair associated with a closed symmetric linear relation $T$ in a Krein space $\mathfrak{H}$. Let $M_\Gamma$ be the Weyl family corresponding to $(\mathfrak{L},\Gamma)$. We cope with two main topics. First, since $M_\Gamma$ need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation $M_\Gamma(z)$, for some $z\in\mathbb{C}\smallsetminus\mathbb{R}$, becomes a nontrivial task. Regarding $M_\Gamma(z)$ as the (Shmul'yan) transform of $zI$ induced by $\Gamma$, we give conditions for the equality in $\overline{M_\Gamma(z)}\subseteq\overline{M_{\overline{\Gamma}}(z)}$ to hold and we compute the adjoint $M_{\overline{\Gamma}}(z)^*$. As an application we ask when the resolvent set of the main transform associated with a unitary boundary pair for $T^+$ is nonempty. Based on the criterion for the closeness of $M_\Gamma(z)$ we give a sufficient condition for the answer. It follows, for example, that, if $T$ is a standard linear relation in a Pontryagin space then the Weyl family $M_\Gamma$ corresponding to a boundary relation $\Gamma$ for $T^+$ is a generalized Nevanlinna family. In the second topic we characterize the transformed boundary pair $(\mathfrak{L}^\prime,\Gamma^\prime)$ with its Weyl family $M_{\Gamma^\prime}$. The transformation scheme is either $\Gamma^\prime=\Gamma V^{-1}$ or $\Gamma^\prime=V\Gamma$ with suitable linear relations $V$. Results in this direction include but are not limited to: a 1-1 correspondence between $(\mathfrak{L},\Gamma)$ and $(\mathfrak{L}^\prime,\Gamma^\prime)$; the formula for $M_{\Gamma^\prime}-M_\Gamma$, for an ordinary boundary triple and a standard unitary operator $V$; construction of a quasi boundary triple from an isometric boundary triple $(\mathfrak{L},\Gamma_0,\Gamma_1)$ with $\ker\Gamma=T$ and $T_0=T^*_0$.