Convergence of iterated Aluthge transform sequence for diagonalizable matrices II: $\lambda$-Aluthge transform

Abstract

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 $${\Delta }_{\lambda }(T)=|T{|}^{\lambda }U|T{|}^{1-\lambda }.$$ Let ${\Delta }_{\lambda }^n(T)$ denote the n-times iterated Aluthge transform of $T$, $n\in \mathbb{N}$. 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.

24 pages