Iterated Aluthge transforms: a brief survey

Abstract

Given an r × r complex matrix T, if T = U|T| is the polar de- composition 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 2 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 conjectured by Jung, Ko and Pearcy in 2003.