The Iterated Aluthge Transform of an Operator

The Aluthge transform 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 T, and this study was continued in [7], in which relations between the spectral pictures of T and T were obtained. This article is a continuation of [6] and [7]. Here we pursue the study of the sequence of Aluthge iterates T^(n)} associated with an arbitrary operator T. In particular, we verify that in certain cases the sequence T^(n)} converges to a normal operator, which partially answers Conjecture 1.11 in [6] and its modified version below (Conjecture 5.6).