A family of matrix flows converging to normal matrices

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.

💡 Research Summary

The paper presents a unified continuous‑time framework that encompasses both the Aluthge transform and the Haagerup flow, two well‑known operator transformations that are known to drive matrices toward normality. By introducing a pair of increasing functions φ = (φ₁, φ₂) defined on a compact interval