Partial inner product spaces, metric operators and generalized hermiticity

TL;DR

This paper investigates unbounded metric operators in Hilbert spaces, revealing their role in generating partial inner product spaces and exploring generalized similarity notions relevant to pseudo-hermitian quantum mechanics.

Contribution

It introduces a framework connecting unbounded metric operators with PIP spaces and generalizes similarity concepts to analyze spectral properties in pseudo-hermitian contexts.

Findings

Unbounded metric operators generate a lattice of Hilbert spaces.

Generalized similarity notions can preserve spectral properties.

Reformulation of pseudo-hermitian operators within PIP space formalism.

Abstract

Motivated by the recent developments of pseudo-hermitian quantum mechanics, we analyze the structure of unbounded metric operators in a Hilbert space. It turns out that such operators generate a canonical lattice of Hilbert spaces, that is, the simplest case of a partial inner product space (PIP space). Next, we introduce several generalizations of the notion of similarity between operators and explore to what extend they preserve spectral properties. Then we apply some of the previous results to operators on a particular PIP space, namely, a scale of Hilbert spaces generated by a metric operator. Finally, we reformulate the notion of pseudo-hermitian operators in the preceding formalism.