Perturbation theory for Hermitian matrix-functions based on vector-fields

TL;DR

This paper extends spectral matrix-function theory for Hermitian matrices by allowing vector-field functions instead of scalar functions, providing new approximation formulas and Lipschitz estimates.

Contribution

It generalizes the Daleskii-Krein theorem to vector-field functions, offering novel first-order approximation formulas and Lipschitz bounds for these matrix-functions.

Findings

Derived first-order approximation formulas for vector-field spectral functions.

Established Lipschitz estimates for the new class of matrix-functions.

Generalized classical spectral function theorems to vector-field cases.

Abstract

We consider "spectral" matrix-functions for Hermitian matrices, where the novelty is that the function applied to the spectrum is allowed to be a vector-field rather than a scalar function (a.k.a isotropic matrix functions). We prove first order approximation formulas, generalizing the classical Daleskii-Krein theorem, as well as Lipschitz estimates.