On k-quasihyponormal operators II

Abstract

An operator T on a Hilbert space is in the class of k-quasihyponormal operators Q(k), if T*k(T*T−TT*)Tk ≥ 0. It is shown that if T is in Q(k) and S is normal such that TX = XS, where X is one to one with dense range, then T is normal; and is unitarily equivalent to S. It is proved that S can be replaced by a cohyponormal operator, if T in Q(1) is one to one. It is also shown that two quasisimilar operators in Q(k) have equal spectra, and every reductive operator quasisimilar to a normal operator is normal.