# Proper holomorphic mappings of the spectral unit ball

## TL;DR

This paper proves that the spectral unit ball admits only trivial proper holomorphic mappings, extending classical results to a matrix spectral setting and showing rigidity in complex analysis.

## Contribution

It establishes an Alexander type theorem for the spectral unit ball, demonstrating the absence of non-trivial proper holomorphic mappings for dimensions n ≥ 2.

## Key findings

- No non-trivial proper holomorphic mappings in spectral unit ball for n ≥ 2
- Extension of classical Alexander theorem to spectral matrix domains
- Rigidity result in complex analysis of spectral domains

## Abstract

We prove an Alexander type theorem for the spectral unit ball $\Omega_n$ showing that there are no non-trivial proper holomorphic mappings in $\Omega_n$, $n\geq 2$.