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Motivated by the recent developments of pseudo-hermitian quantum mechanics,we analyze the structure of unbounded metric operators in a Hilbert space. Itturns 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 betweenoperators and explore to what extend they preserve spectral properties. Then weapply some of the previous results to operators on a particular PIP space,namely, a scale of Hilbert spaces generated by a metric operator. Finally, wereformulate the notion of pseudo-hermitian operators in the precedingformalism.

Partial inner product spaces, metric operators and generalized hermiticity