Local and global similarity of holomorphic matrices

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.

arXiv commentarXiv admin note: substantial text overlap with arXiv:1703.09530