Metric operators, generalized hermiticity and lattices of Hilbert lpaces

TL;DR

This paper explores the structure of unbounded metric operators in Hilbert spaces, revealing a lattice of Hilbert spaces and extending the concept of pseudo-Hermiticity relevant to quantum mechanics.

Contribution

It introduces a framework for unbounded metric operators, generalizes similarity notions, and applies these to pseudo-Hermitian operators in quantum physics.

Findings

Unbounded metric operators generate a lattice of Hilbert spaces.

Generalized similarity notions can preserve spectral properties.

Reformulation of pseudo-Hermitian operators within this framework.

Abstract

A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by 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). We introduce several generalizations of the notion of similarity between operators, in particular, the notion of quasi-similarity, and we 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, motivated by the recent developments of pseudo-Hermitian quantum mechanics, we reformulate the notion of pseudo-Hermitian operators in the preceding formalism.