Convergence of iterated Aluthge transform sequence for diagonalizable matrices II: L-Aluthge transform

J. ANTEZANA, E. PUJALS Y D. STOJANOFF

INTEGRAL EQUATIONS AND OPERATOR THEORY

Birkhäuser

Lugar: Tel Aviv; Año: 2008 vol. 62 p. 465 - 488

0378-620X

Let $lambda in (0,1)$ and let $T$ be a $r imes r$ complex matrix with polar decomposition $T=U|T|$. Then, the $la$- Aluthge transform is defined by $$ Delta_lambda left(T ight)= |T|^{lambda} U |T |^{1-lambda}. $$ Let $Delta_lambda^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, %, i.e.$Delta_lambda^{0}(T)=T$ and $Delta_lambda^{n}(T)=Delta_lambda(Delta_lambda^{n-1}(T))$, $ninmathbb{N}$. We prove that the sequence ${Delta_lambda^{n}(T)}_{ninmathbb{N}}$ converges for every $r imes r$ {f diagonalizable} matrix $T$. We show regularity results for the two parameter map $(la , T) mapsto alulit{infty}{T}$, and we study for which matrices the map $(0,1) i lambda mapsto Delta_lambda^{infty}( T)$ is constant.