The iterated Aluthge transforms of a matrix converge

J. ANTEZANA, E. PUJALS Y D. STOJANOFF

ADVANCES IN MATHEMATICS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2011 vol. 226 p. 1591 - 1620

0001-8708

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