A family of matrix flows converging to normal matrices

TL;DR

This paper introduces a family of matrix flows defined by differential equations that generalize the Aluthge transform, all of which converge to normal matrices, extending known results to continuous and operator settings.

Contribution

It defines new matrix flows via differential equations that converge to normal matrices, including a continuous analogue of the Aluthge transform and flows in operator spaces.

Findings

The flows converge to normal matrices.

Includes a continuous analogue of the Aluthge transform.

Extends convergence results to Hilbert space operators.

Abstract

The celebrated Antezana-Pujals-Stojanoff Theorem states that the iterated Aluthge transforms of an arbitrary matrix converge to a normal matrix. We introduce a family of matrix flows that share this convergence property by defining them through ordinary differential equations. The family includes a continuous analogue of the Aluthge transform, as well as a differential equation discussed by Haagerup in the context of II$_1$ factors. We also examine the same type of flows in the setting of Hilbert space operators equipped with unitarily invariant norms.