Operators similar to unitary or selfadjoint ones

Abstract

Let T be a bounded linear operator on a Hilbert space. General necessary and sufficient conditions are given in order that VTV~ is unitary for some bounded linear operator V with bounded everywhere defined inverse. Similarly let B be a closed and densely defined linear operator in a Hilbert space. General necessary and sufficient conditions are given in order that VBV~X is self adjoint for some bounded linear operator V with bounded everywhere defined inverse.