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 $r\times r$ complex matrix with polar decomposition $T=U|T|$. Then, the $\la$- Aluthge transform is defined by $$ \Delta_\lambda \left(T \right)= |T|^{\lambda} U |T |^{1-\lambda}. $$ Let $\Delta_\lambda^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, %, i.e.$\Delta_\lambda^{0}(T)=T$ and $\Delta_\lambda^{n}(T)=\Delta_\lambda(\Delta_\lambda^{n-1}(T))$, $n\in\mathbb{N}$. We prove that the sequence $\{\Delta_\lambda^{n}(T)\}_{n\in\mathbb{N}}$ converges for every $r\times r$ {\bf 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 $(0,1)\ni \lambda \mapsto \Delta_\lambda^{\infty}( T)$ is constant.