On operators with similar positive parts

Let S=US|S| and T=UT|T| be the polar decompositions of S and T in L(H), respectively. We say that S and T have similar positive parts if |S| and |T| are similar. In this paper, we investigate properties of operators that are preserved under this similarity condition. We describe the form for the polar decomposition of an operator when the operator and its adjoint have similar positive parts. For this case, we also study the existence of invariant subspaces under this assumption.