On the unitary part of dominant contractions (Q805996)

Suppose T is a contraction on a Hilbert space H. Then \(H_ u(T):=\{x\in H;\| T^ nx\| =\| T^{*n}x\| =\| x\|\) for all n) is the maximal reducing subspace for T on which T is unitary. The purpose of this note it is to prove that if \(T^*\) is dominant (i.e. for each \(z\in {\mathbb{C}}\) there exists \(M_ z\) such that \(\| (T-zI)x\| \leq \| (T^*-zI)x\|\) for all \(x\in H\), then the nonnegative square root of \(\lim_{n}T^{*n}T^ n\) is a projection onto \(H_ u(T)\) which commutes with T.