Analysis of Non-normal Operators via Aluthge Transformation

Abstract.

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