Convergence of iterated Aluthge transform sequence for diagonalizable matrices

JORGE ANTEZANA; ENRIQUE PUJALS; DEMETRIO STOJANOFF

ADVANCES IN MATHEMATICS

Año: 2007 p. 255 - 278

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 $r imes r$ {f 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 on the (open and dense) set of $r imes r$ matrices  with $r$ different eigenvalues.