Metric Operators, Generalized Hermiticity, and Partial Inner Product Spaces

Abstract

A quasi-Hermitian operator is an operator in a Hilbert space that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Motivated by the recent developments of pseudo-Hermitian quantum mechanics, we analyze the structure of metric operators, bounded or unbounded. We introduce several generalizations of the notion of similarity between operators and explore to what extent they preserve spectral properties. Next we consider canonical lattices of Hilbert space s generated by unbounded metric operators. Since such lattices constitute the simplest case of a partial inner product space (pip-space), we can exploit the technique of pip-space operators. Thus we apply some of the previous results to operators on a particular pip-space, namely, the scale of Hilbert space s generated by a single metric operator. Finally, we reformulate the notion of pseudo-hermitian operators in the preceding formalism.