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

TL;DR

This paper proves that the iterated $ ext{Aluthge}$ transform sequence converges for all diagonalizable matrices and explores the regularity and constancy of the limit map with respect to parameters.

Contribution

It establishes convergence of the $ ext{Aluthge}$ transform sequence for diagonalizable matrices and analyzes the regularity of the limit with respect to the transform parameter.

Findings

The sequence $\u03b4_ ext{Aluthge}$ transforms converges for all diagonalizable matrices.

The regularity of the limit map $(mbda, T) o luthge^{ ext{infty}}(T)$ is characterized.

Conditions under which the limit map is constant with respect to $mbda$ are identified.

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.