Operator-Lipschitz estimates for the singular value functional calculus

TL;DR

This paper establishes precise conditions under which a singular value-based functional calculus for compact operators is Lipschitz continuous, providing sharp constants and advancing understanding in applied mathematics contexts.

Contribution

It introduces necessary and sufficient conditions for Lipschitz continuity of the singular value functional calculus with sharp constants, a novel theoretical result.

Findings

Identifies conditions for Lipschitz continuity

Provides sharp Lipschitz constants

Enhances understanding of operator functional calculus

Abstract

We consider a functional calculus for compact operators, acting on the singular values rather than the spectrum, which appears frequently in applied mathematics. Necessary and sufficient conditions for this singular value functional calculus to be Lipschitz-continuous with respect to the Hilbert-Schmidt norm are given. We also provide sharp constants.