Operator (quasi-)similarity, quasi-Hermitian operators and all that

TL;DR

This paper investigates the structure and spectral properties of quasi-Hermitian and quasi-similar operators in Hilbert spaces, motivated by pseudo-Hermitian quantum mechanics, and extends the analysis to partial inner product spaces.

Contribution

It introduces a comprehensive framework for analyzing unbounded metric operators, quasi-Hermitian operators, and their spectral properties, extending to partial inner product spaces.

Findings

Spectral properties are preserved under certain quasi-similarity relations.

Quasi-Hermitian operators can be bounded or unbounded and are related to their adjoints.

The framework applies to operators in partial inner product spaces, broadening the scope of pseudo-Hermitian quantum mechanics.

Abstract

Motivated by the recent developments of pseudo-Hermitian quantum mechanics, we analyze the structure generated by unbounded metric operators in a Hilbert space. To that effect, we consider the notions of similarity and quasi-similarity between operators and explore to what extent they preserve spectral properties. Then we study quasi-Hermitian operators, bounded or not, that is, operators that are quasi-similar to their adjoint and we discuss their application in pseudo-Hermitian quantum mechanics. Finally, we extend the analysis to operators in a partial inner product space (\pip), in particular the scale of Hilbert spaces generated by a single unbounded metric operator.