On the similarity of holomorphic matrices

TL;DR

This paper extends Guralnick's theorem from Riemann surfaces to Stein spaces, showing that local holomorphic similarity implies global holomorphic similarity, and explores conditions under which smooth or continuous similarities imply holomorphic similarity.

Contribution

The authors generalize Guralnick's result to Stein spaces of arbitrary dimension and establish that smooth similarity implies holomorphic similarity, unlike continuous similarity.

Findings

Holomorphic similarity on Stein spaces extends Guralnick's theorem.

Smooth similarity implies holomorphic similarity in Stein spaces.

Continuous similarity does not guarantee holomorphic similarity.

Abstract

R. Guralnick (Linear Algebra Appl. 99, 85-96, 1988) proved that two holomorphic matrices on a noncompact connected Riemann surface, which are locally holomorphically similar, are globally holomorphically similar. We generalize this to (possibly, non-smooth) one-dimensional Stein spaces. For Stein spaces of arbitrary dimension, we prove that global $\mathcal C^\infty$ similarity implies global holomorphic similarity, whereas global continuous similarity is not sufficient.