Pseudo‐Hermitian quantum mechanics (QM) is a recent, unconventional, approach to QM, based on the use of non‐self‐adjoint Hamiltonians, whose self‐adjointness can be restored by changing the ambient Hilbert space, via a so‐called metric operator. The PT‐symmetric Hamiltonians are usually pseudo‐Hermitian operators, a term introduced a long time ago by Dieudonné for characterizing those bounded operators A that satisfy a relation of the form GA = A G, where G is a metric operator, that is, a strictly positive self‐adjoint operator. This chapter explores further the structure of unbounded metric operators, in particular, their incidence on similarity. It examines the notion of similarity between operators induced by a bounded metric operator with bounded inverse. The goal here is to study which spectral properties are preserved under such a quasi‐similarity relation. The chapter applies some of the previous results to operators on the scale of Hilbert spaces generated by the metric operator G.