A local form for the automorphisms of the spectral unit ball

TL;DR

This paper demonstrates that automorphisms of the spectral unit ball near cyclic matrices can be expressed as conjugations by holomorphic matrices, offering a shorter proof and a stronger result than previous work.

Contribution

It provides a new, simplified proof of Rostand's theorem, showing local conjugation representations for automorphisms of the spectral unit ball near cyclic matrices.

Findings

Automorphisms near cyclic matrices are conjugations by holomorphic matrices.

The proof is shorter and more direct than previous methods.

The result extends Rostand's theorem with a stronger local characterization.

Abstract

If F is an automorphism of the spectral unit ball, we show that, in a neighborhood of any cyclic (i.e. non-derogatory) matrix of the ball, the map F can be written as conjugation by a holomorphically varying non singular matrix. This provides a shorter proof of a theorem of J. Rostand, with a slightly stronger result.