  Motivated by the recent developments of pseudo-Hermitian quantum mechanics,
we analyze the structure generated by unbounded metric operators in a Hilbert
space. To that effect, we consider the notions of similarity and
quasi-similarity between operators and explore to what extent they preserve
spectral properties. Then we study quasi-Hermitian operators, bounded or not,
that is, operators that are quasi-similar to their adjoint and we discuss their
application in pseudo-Hermitian quantum mechanics. Finally, we extend the
analysis to operators in a partial inner product space (\pip), in particular
the scale of Hilbert spaces generated by a single unbounded metric operator.
