Operator (Quasi-)Similarity, Quasi-Hermitian Operators and All that

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