Selfadjoint extensions of relations whose domain and range are orthogonal

TL;DR

This paper characterizes selfadjoint extensions of relations with orthogonal domain and range in Hilbert spaces, providing explicit parametrizations and exploring their properties using boundary triplets and Weyl functions.

Contribution

It introduces a novel description of selfadjoint extensions of relations with orthogonal domain and range, including explicit formulas and conditions for extremal and semibounded extensions.

Findings

Explicit block formulas for nonnegative selfadjoint extensions.

Characterization of extremal extensions via orthogonality.

Correspondence between semibounded extensions and parameters when operator part is bounded.

Abstract

The selfadjoint extensions of a closed linear relation $R$ from a Hilbert space ${\mathfrak H}_1$ to a Hilbert space ${\mathfrak H}_2$ are considered in the Hilbert space ${\mathfrak H}_1\oplus{\mathfrak H}_2$ that contains the graph of $R$. They will be described by $2 \times 2$ blocks of linear relations and by means of boundary triplets associated with a closed symmetric relation $S$ in ${\mathfrak H}_1 \oplus {\mathfrak H}_2$ that is induced by $R$. Such a relation is characterized by the orthogonality property ${\rm dom\,} S \perp {\rm ran\,} S$ and it is nonnegative. All nonnegative selfadjoint extensions $A$, in particular the Friedrichs and Kre\u{\i}n-von Neumann extensions, are parametrized via an explicit block formula. In particular, it is shown that $A$ belongs to the class of extremal extensions of $S$ if and only if ${\rm dom\,} A \perp {\rm ran\,} A$. In addition, using asymptotic properties of an associated Weyl function, it is shown that there is a natural correspondence between semibounded selfadjoint extensions of $S$ and semibounded parameters describing them if and only if the operator part of $R$ is bounded.