The combinatorics of permuting and preserving curve-bound spectra

TL;DR

This paper characterizes spectrum- and commutativity-preserving maps on matrices with spectra in a continuous interval, extending prior results and exploring new possibilities based on spectral geometry.

Contribution

It generalizes existing theorems to broader classes of matrices and spectra, identifying new involution-preserving maps influenced by spectral geometry.

Findings

Maps are conjugations, transpose conjugations, or spectrum orderings.

Continuous spectrum preservers on unitary groups are conjugations or transpose conjugations.

New involution-preserving maps depend on the geometry of the spectral set.

Abstract

We prove that continuous spectrum- and commutativity-preserving maps to $\mathcal{M}_n(\mathbb{C})$ from the space of normal (real or complex) $n\times n$, $n\ge 3$ matrices with spectra contained in a given continuous-injection interval image $\Lambda\subseteq \mathbb{C}$ or $\mathbb{R}$ are (a) conjugations; (b) transpose conjugations, or (c) orderings of spectra according to an orientation of $\Lambda$, with fixed eigenspaces. This generalizes results of Petek's (self-maps of real or complex Hermitian matrices) and the author's (complex Hermitian matrices as the domain, $\mathcal{M}_n(\mathbb{C})$ as the codomain). An application rules out possibility (c) for normal matrices with spectra constrained to a simple closed curve, extending a result by the author, Gogi\'c and Toma\v{s}evi\'c to the effect that continuous commutativity and spectrum preservers on unitary groups are (transpose) conjugations. The involution preserving eigenspaces and complex-conjugating eigenvalues is a novel possibility beyond (a), (b) and (c) if the domain consists of all semisimple operators with $\Lambda$-bound spectra instead; its continuity (or lack thereof) and whether or not that map furthermore extends continuously to arbitrary $\Lambda$-constrained-spectrum matrices hinge on the geometry and regularity of $\Lambda$.