Analysis of non-normal operators via Aluthge transformation

Let T be a bounded linear operator on a complex Hilbert space H. In this paper, we show that T has Bishop's property (beta) if and only if its Aluthge transformation T has property (beta). As applications, we can obtain the following results. Every w-hyponormal operator has property (beta). Quasi-similar w-hyponormal operators have equal spectra and equal essential spectra. Moreover, in the last section, we consider Cho's problem that whether the semi-hyponormality of T implies the spectral mapping theorem Re sigma(T) sigma(ReT) or not.