SOME CONDITIONS UNDER WHICH SOME OPERATORS ARE NORMAL

Let T 2 B(H) and T = UjTj be its polar decomposition. We prove that (i) if T is log-hyponormal or p-hyponormal and Un = U¤ for some n, then T is normal; (ii) if the spectrum of U is contained in some open semicircle, then T is normal if and only if so is its Aluthge transform ˜ T = jTj 1 2UjTj12