The Iterated Aluthge Transform of an Operator

Abstract.

The Aluthge transform $\widetilde{T}$ (defined below) of an operator T on Hilbert space has been studied extensively, most often in connection with p-hyponormal operators. In [6] the present authors initiated a study of various relations between an arbitrary operator T and its associated $\widetilde{T}$, and this study was continued in [7], in which relations between the spectral pictures of T and $\widetilde{T}$ were obtained. This article is a continuation of [6] and [7]. Here we pursue the study of the sequence of Aluthge iterates {$\widetilde{T}$ ⁽ⁿ⁾} associated with an arbitrary operator T. In particular, we verify that in certain cases the sequence {$\widetilde{T}$ ⁽ⁿ⁾} converges to a normal operator, which partially answers Conjecture 1.11 in [6] and its modified version below (Conjecture 5.6).