Best approximants relative to a C$^*$-subalgebra, joint numerical range and subdifferentials

TL;DR

This paper investigates the minimality of Hermitian matrices relative to a C*-subalgebra in spectral norm, linking moments, joint numerical range, and subdifferentials of the maximum eigenvalue, extending known results.

Contribution

It generalizes the concept of moments of a subspace, relates it to joint numerical range and subdifferentials, and characterizes minimality with new theoretical insights.

Findings

Extended results for diagonal subalgebras to general C*-subalgebras.

Described subdifferential of maximum eigenvalue in terms of eigenspace moments.

Provided examples illustrating the characterization of minimality.

Abstract

We study the minimality of $n\times n$ Hermitian matrices $A$ respect to a $C^*$-subalgebra $\mathcal{B}$ of $M_n(\mathbb{C})$ in the spectral norm, that is \[\|A\|\leq \|A+B\|,\ \text{ for every } B\in \mathcal{B}.\] We generalize the notion of the moment of a subspace and relate it to the joint numerical range and the subdifferentials of the maximum eigenvalue. We extend results previously known for the subalgebra of diagonal operators and describe the subdifferential of the maximum eigenvalue in terms of the moment of the corresponding eigenspace. We also characterize $\mathcal{B}$-minimality via moments and subdifferentials, and provide examples.