Limit of iteration of the induced Aluthge transformations of centered operators

TL;DR

This paper investigates the convergence of induced Aluthge transformations of centered operators, extending known results to infinite-dimensional settings and various means, and characterizes their limit points.

Contribution

It generalizes the convergence analysis of Aluthge transforms to induced versions, including for matrices and $C^*$-algebras, and describes their limit behavior.

Findings

Induced Aluthge transforms of centered operators converge to normal operators.

Convergence holds for invertible centered matrices and specific means.

Limit points are explicitly characterized for matrices and operators in $II_1$-factors.

Abstract

Aluthge transform is a well-known mapping defined on bounded linear operators. Especially, the convergence property of its iteration has been studied by many authors. In this paper, we discuss the problem for the induced Aluthge transforms which is a generalization of the Aluthge transform defined in 2021. We give the polar decomposition of the induced Aluthge transformations of centered operators and show its iteration converges to a normal operator. In particular, if $T$ is an invertible centered matrix, then iteration of any induced Aluthge transformations converges. Using the canonical standard form of matrix algebras we show that the iteration of any induced Aluthge transformations with respect to the weighted arithmetic mean and the power mean converge. Those observation are extended to the $C^*$-algebra of compact operators on an infinite dimensional Hilbert space, and as an application we show the stability of $\mathcal{AN}$ and $\mathcal{AM}$ properties under the iteration of the induced Aluthge transformations. We also provide concrete forms of their limit points for centered matrices and several examples. Moreover, we discuss the limit point of the induced Aluthge transformation with respect to the power mean in the injective $II_1$-factor $\mathcal{M}$ and determine the form of its limit for some centered operators in $\mathcal{M}$.