Eigenvalue collision and exotic preservers on semisimple operators

TL;DR

This paper classifies spectrum-preserving maps on semisimple matrices, extending previous results and revealing new possibilities depending on the spectral set’s local properties.

Contribution

It generalizes classification results for spectrum preservers to semisimple matrices, highlighting the role of local regularity of complex conjugation.

Findings

Spectrum preservers on normal matrices are conjugations or transpose conjugations.

Distinct possibilities for spectrum preservers depend on the local regularity of conjugation near eigenvalue coincidences.

The results extend to arbitrary matrices with spectra in connected subsets of the complex plane.

Abstract

We classify $n\times n$-matrix-valued continuous commutativity and spectrum preservers defined on spaces of (a) normal, (b) semisimple and (c) arbitrary $n\times n$ matrices with spectra contained in sufficiently connected subsets $\mathcal{X}\subseteq \mathbb{C}$, generalizing a number of results due to \v{S}emrl, Gogi\'{c}, Toma\v{s}evi\'c and the author among others. In case (a) these are always conjugations or transpose conjugations, while in cases (b) and (c) qualitatively distinct possibilities arise depending on the local regularity of the complex-conjugation map close to coincident-eigenvalue loci of $\mathcal{X}^n$.