Local and global similarity of holomorphic matrices

TL;DR

This paper extends Guralnick's result on holomorphic similarity of matrices from Riemann surfaces to Stein spaces, showing that smooth similarity implies holomorphic similarity in higher dimensions.

Contribution

It generalizes the concept of local and global similarity of holomorphic matrices from Riemann surfaces to Stein spaces of arbitrary dimension.

Findings

Global $ ext{C}^ ext{infty}$ similarity implies holomorphic similarity.

Global continuous similarity does not guarantee holomorphic similarity.

Extension of Guralnick's theorem to higher-dimensional Stein spaces.

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.