On a spectral version of Cartan's theorem

TL;DR

This paper investigates conditions under which holomorphic self-maps of matrix spectral domains preserve spectra, especially near fixed points with specific matrix properties, extending classical spectral theorems.

Contribution

It establishes new spectral preservation results for holomorphic maps on matrix domains, particularly around fixed points with diagonalizable or non-derogatory matrices.

Findings

Spectrum-preserving maps near diagonalizable matrices

Spectrum-preserving maps near non-derogatory matrices

Spectrum preservation on analytic subsets for arbitrary matrices

Abstract

For a domain $\Omega$ in the complex plane, we consider the domain $S_n(\Omega)$ consisting of those $n\times n$ complex matrices whose spectrum is contained in $\Omega$. Given a holomorphic self-map $\Psi$ of $S_n(\Omega)$ such that $\Psi(A)=A$ and the derivative of $\Psi$ at $A$ is identity for some $A\in S_n(\Omega)$, we investigate when the map $\Psi$ would be spectrum-preserving. We prove that if the matrix $A$ is either diagonalizable or non-derogatory then for most domains $\Omega$, $\Psi$ is spectrum-preserving on $S_n(\Omega)$. Further, when $A$ is arbitrary, we prove that $\Psi$ is spectrum-preserving on a certain analytic subset of $S_n(\Omega)$.