Partial inner product spaces, metric operators and generalized hermiticity

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 extent 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 ametric operator. Finally, we reformulate the notion of pseudo-Hermitian operators in the preceding formalism.