On isometry groups of self-adjoint traceless and skew-symmetric matrices

TL;DR

This paper characterizes the isometry groups of self-adjoint traceless matrices under various norms, extending previous results and suggesting methods applicable to skew-symmetric matrices.

Contribution

It provides a complete description of isometry groups for self-adjoint traceless matrices under any unitary similarity invariant norm, extending Nagy's work on Schatten p-norms.

Findings

Determined the isometry groups for self-adjoint traceless matrices.

Extended the classification to all unitary similarity invariant norms.

Indicated applicability to skew-symmetric matrices.

Abstract

This note is concerned with isometries on the spaces of self-adjoint traceless matrices. We compute the group of isometries with respect to any unitary similarity invariant norm. This completes and extends the result of Nagy on Schatten $p$-norm isometries. Furthermore, we point out that our proof techniques could be applied to obtain an old result concerning isometries on skew-symmetric matrices.