Invertible extensions of symmetric operators and the corresponding generalized resolvents

TL;DR

This paper investigates invertible extensions of symmetric operators in Hilbert spaces, characterizing them via parameters in generalized Neumann formulas and deriving associated resolvents through boundary conditions in the Shtraus formula.

Contribution

It provides a new characterization of invertible extensions of symmetric operators using generalized Neumann formulas and boundary conditions in the context of generalized resolvents.

Findings

Invertible extensions are characterized by parameters in generalized Neumann formulas.

Generalized resolvents are obtained through boundary conditions in the Shtraus formula.

The study offers a systematic approach to understanding invertible extensions and their resolvents.

Abstract

In this paper we study invertible extensions of a symmetric operator in a Hilbert space $H$. All such extensions are characterized by a parameter in the generalized Neumann's formulas. Generalized resolvents, which are generated by the invertible extensions, are extracted by a boundary condition among all generalized resolvents in the Shtraus formula.