Analysis of Non-normal Operators via Aluthge Transformation

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