Convergence of iterated Aluthge transform sequence for diagonalizable matrices

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 $r\times r$ {\bf diagonalizable} matrix $T$. We show that the limit ${\Delta }^{\infty }(\cdot )$ is a map of class $C^{\infty }$ on the similarity orbit of a diagonalizable matrix, and %of class $C^{\infty }$ on the (open and dense) set of $r\times r$ matrices with $r$ different eigenvalues.

25 pages