Iterated Aluthge transforms: a brief survey

Authors

Jorge Antezana, Universidad Nacional de La Plata
Enrique R. Pujals, CUNY Graduate CenterFollow
Demetrio Stojanoff, Universidad Nacional de La Plata

Document Type

Article

Publication Date

2008

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 conjectured by Jung, Ko and Pearcy in 2003.

Comments

This article was originally published in Revista de la Unión Matemática Argentina, available at https://inmabb.criba.edu.ar/revuma/pdf/v49n1/v49n1a04.pdf

This work is distributed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).