Preprint A570/2007
The iterated Aluthge transforms of a matrix converge

Demetrio Stojanoff | Antezana, Jorge | Pujals, Enrique

**Keywords: **
Aluthge transform | stable manifold theorem | similarity orbit | polar decomposition.

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 |^{1/2} U |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. We prove that the sequence ${∆_n (T )}_{n\in N}$
converges for every $r × r$ matrix T . This result was conjecturated by Jung, Ko
and Pearcy in 2003. We also analyze the regularity of the limit function.