On local irreducibility of the spectrum

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 $x=0$.