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A quasi-Hermitian operator is an operator that is similar to its adjoint insome sense, via a metric operator, i.e., a strictly positive self-adjointoperator. Whereas those metric operators are in general assumed to be bounded,we analyze the structure generated by 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). We introduce several generalizations of the notion of similaritybetween operators, in particular, the notion of quasi-similarity, and weexplore to what extend they preserve spectral properties. Then we apply some ofthe previous results to operators on a particular PIP-space, namely, a scale ofHilbert spaces generated by a metric operator. Finally, motivated by the recentdevelopments of pseudo-Hermitian quantum mechanics, we reformulate the notionof pseudo-Hermitian operators in the preceding formalism.

Metric operators, generalized hermiticity and lattices of Hilbert lpaces