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A family of matrix flows converging to normal matrices

Masaki Izumi

TL;DR

The paper develops a unified ODE-based framework for matrix and operator flows that converge to normal elements, encompassing the Aluthge flow and the Haagerup gradient flow as special cases. By imposing regularity conditions on a pair of functions $\varphi=(\varphi_1,\varphi_2)$ and exploiting a decomposition into diagonal and strictly upper-triangular components, it establishes global existence, unitary-invariant behavior, and Lyapunov-type convergence results, with exponential convergence under suitable hypotheses. It further extends the theory to bounded Hilbert space operators with unitarily invariant norms, yielding both convergence and non-convergence phenomena depending on the chosen ideal and regularity, and it highlights concrete dynamics (e.g., two-eigenvalue Haagerup flow) and potential limitations in infinite dimensions. The work connects finite-dimensional matrix dynamics to operator-algebraic contexts, offering a flexible template for analyzing similarity-orbit flows and their normal limiting behavior. Overall, it provides a robust, continuous analogue to iterated Aluthge transforms and clarifies when and how such flows stabilize to normal structures.

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.

A family of matrix flows converging to normal matrices

TL;DR

The paper develops a unified ODE-based framework for matrix and operator flows that converge to normal elements, encompassing the Aluthge flow and the Haagerup gradient flow as special cases. By imposing regularity conditions on a pair of functions and exploiting a decomposition into diagonal and strictly upper-triangular components, it establishes global existence, unitary-invariant behavior, and Lyapunov-type convergence results, with exponential convergence under suitable hypotheses. It further extends the theory to bounded Hilbert space operators with unitarily invariant norms, yielding both convergence and non-convergence phenomena depending on the chosen ideal and regularity, and it highlights concrete dynamics (e.g., two-eigenvalue Haagerup flow) and potential limitations in infinite dimensions. The work connects finite-dimensional matrix dynamics to operator-algebraic contexts, offering a flexible template for analyzing similarity-orbit flows and their normal limiting behavior. Overall, it provides a robust, continuous analogue to iterated Aluthge transforms and clarifies when and how such flows stabilize to normal structures.

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 factors. We also examine the same type of flows in the setting of Hilbert space operators equipped with unitarily invariant norms.
Paper Structure (9 sections, 28 theorems, 185 equations)

This paper contains 9 sections, 28 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. The basic properties of the solutions
  3. Convergence results
  4. The case of Hilbert space operators
  5. Appendix
  6. Aluthge flow
  7. Haagerup flow
  8. Divided differences
  9. Interpolation polynomials

Key Result

Lemma 2.3

Let $A\in \mathbb{M}_n$ be a self-adjoint matrix, and let $f$ be an increasing function on $\sigma(A)$. Let $U\in \mathbb{M}_n$ be a unitary matrix. Then and equality holds if and only if $U$ commutes with $f(A)$.

Theorems & Definitions (67)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • proof
  • ...and 57 more