On <i>k</i>-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* ) T k ≥ 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.