  We show that similarity (or equivalent) transformations enable one to
construct non-Hermitian operators with real spectrum. In this way we can also
prove and generalize the results obtained by other authors by means of a
gauge-like transformation and its generalization. Such similarity
transformations also reveal the connection with pseudo-Hermiticity in a simple
and straightforward way. In addition to it we consider the positive and
negative eigenvalues of a three-parameter non-Hermitian oscillator.
