The iterated Aluthge transforms of a matrix converge

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 Δ(T)=|T|^1/2U|T|^1/2. Let Δⁿ(T) denote the n-times iterated Aluthge transform of T, i.e., Δ⁰(T)=T and Δⁿ(T)=Δ(Δ^n-1(T)), nεN. We prove that the sequence {Δⁿ(T)}_nεN converges for every r×r matrix T. This result was conjectured by Jung, Ko and Pearcy in 2003. We also analyze the regularity of the limit function.