Iterated Aluthge transforms: a brief survey

Abstract

Abstract. Given an r × r complex matrix T, if T = U|T| is the polar decomposition of T, then, the Aluthge transform is defined by ∆ (T) = |T| 1/2U|T| 1/2 . Let ∆n(T) denote the n-times iterated Aluthge transform of T, i.e. ∆0 (T) = T and ∆n(T) = ∆(∆n−1 (T)), n ∈ N. In this paper we make a brief survey on the known properties and applications of the Aluthge trasnsorm, particularly the recent proof of the fact that the sequence {∆n(T)}n∈N converges for every r×r matrix T. This result was conjecturated by Jung, Ko and Pearcy in 2003.