Partial inner product spaces, metric operators and generalized hermiticity

Motivated by the recent developments of pseudo-Hermitian quantummechanics, we analyze the structure of unbounded metric operators in a Hilbertspace. It turns out that such operators generate a canonical lattice of Hilbertspaces, that is, the simplest case of a partial inner product space (PIP-space).Next, we introduce several generalizations of the notion of similarity betweenoperators and explore to what extent they preserve spectral properties. Then weapply some of the previous results to operators on a particular PIP-space, namelya scale of Hilbert spaces generated by ametric operator. Finally, we reformulatethe notion of pseudo-Hermitian operators in the preceding formalism.