On the polar decomposition of the product of two operators and its applications (Q1882519)

A bounded linear operator \(T\) acting on a complex Hilbert space \(H\) has a polar decomposition if \(T=U| T|\), where \(T=(T^* )^{1/2}\) and \(U\) is a partial isometry uniquely determined by the condition that the kernels \(U\) and \(T\) coincide. There are given necessary and sufficient conditions for the product \(TS\), where \(T=U| T|\), \(S=V| S|\), to be decomposed. In particular, it is shown that \(TS=UV| TS|\) is the polar decomposition if and only if \(| T|\) commutes with \(| S^*|\). This result implies characterizations of binormal and centered operators by means of the polar decomposition.