The iterated Aluthge transforms of a matrix converge

TL;DR

This paper proves that the iterated Aluthge transforms of any complex matrix always converge, confirming a conjecture from 2003, and also investigates the regularity of the resulting limit function.

Contribution

It establishes the convergence of the iterated Aluthge transform for all matrices, resolving a long-standing conjecture and analyzing the limit's regularity.

Findings

The sequence of iterated Aluthge transforms converges for every complex matrix.

The convergence confirms a conjecture by Jung, Ko, and Pearcy from 2003.

The regularity of the limit function is also analyzed.

Abstract

Given an $r\times r$ complex matrix $T$, if $T=U|T|$ is the polar decomposition of $T$, then, the Aluthge transform is defined by $$ \Delta(T)= |T|^{1/2} U |T |^{1/2}. $$ Let $\Delta^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, i.e. $\Delta^{0}(T)=T$ and $\Delta^{n}(T)=\Delta(\Delta^{n-1}(T))$, $n\in\mathbb{N}$. We prove that the sequence $\{\Delta^{n}(T)\}_{n\in\mathbb{N}}$ converges for every $r\times r$ matrix $T$. This result was conjecturated by Jung, Ko and Pearcy in 2003. We also analyze the regularity of the limit function.