On local irreducibility of the spectrum

Abstract:

Let $\mathcal M_n$ be the algebra of $n\times n$ complex matrices, and for $x\in \mathcal M_n$ denote by $\sigma (x)$ and $\rho (x)$ the spectrum and spectral radius of $x$ respectively. Let $D$ be a domain in $\mathcal M_n$ containing $0$, and let $F:D\to \mathcal M_n$ be a holomorphic map. We prove: (1) if $\sigma (F(x))\cap \sigma (x)\ne \emptyset$ for $x\in D$, then $\sigma (F(x))=\sigma (x)$ for $x\in D$; (2) if $\rho (F(x))=\rho (x)$ for $x\in D$, then again $\sigma (F(x))=\sigma (x)$ for $x\in D$. Both results are special cases of theorems expressing the irreducibility of the spectrum $\sigma (x)$ near $x=0$.