A generalization of the Aluthge transformation in the viewpoint of operator means

TL;DR

This paper generalizes the Aluthge transformation using operator means and double operator integrals, showing convergence to normal matrices and exploring numerical range relations.

Contribution

It introduces a new generalized Aluthge transformation framework based on operator means, extending previous mean transformations and analyzing their properties.

Findings

The n-th iteration of the mean transformation converges to a normal matrix.

Inclusion relations among numerical ranges are established for the generalized transformations.

The framework unifies and extends existing transformations in operator theory.

Abstract

The Aluthge transformation is generalized in the viewpoint of the axiom of operator means by using double operator integrals. It includes the mean transformation which is defined by S. H. Lee, W. Y. Lee and Yoon. Next we shall give some properties of it. Especially, we shall show that the $n$-th iteration of mean transformation of an invertible matrix converges to a normal matrix. Inclusion relations among numerical ranges of generalized Aluthge transformations respect to some operator means are considered.