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

Abstract

Let $λ\in (0,1)$ and let $T$ be a $r\times r$ complex matrix with polar decomposition $T=U|T|$. Then, the $\la$- Aluthge transform is defined by $$ Δ_λ(T )= |T|^λ U |T |^{1-λ}. $$ Let $Δ_λ^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, $n\in\mathbb{N}$. We prove that the sequence $\{Δ_λ^{n}(T)\}_{n\in\mathbb{N}}$ 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 $(0,1)\ni λ\mapsto Δ_λ^{\infty}(T)$ is constant.