Similarity of holomorphic matrices on 1-dimensional Stein spaces

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. In the preprints [arXiv:1703.09524] and [arXiv:1703.09530], a generalization of this to arbitrary (possibly, nonsmooth) 1-dimensional Stein spaces was obtained. The present paper contains a revised version of the proof from [arXiv:1703.09524]. The method of this revised proof can be used also in the higher dimensional case, which will be the subject of a forthcoming paper.Comment: This is a revised version of a part of a preprint from Sep 2016, later (Mar 2017) posted as arXiv:1703.09524. Version 2 is a mistake (the wrong file was sent). The differences of version 3 to version 1 are: a "Note added in proof" and a reference are added, typos are corrected. The difference of version 4 to version 3 is: some small corrections are carried out, rusults unchange

On holomorphic matrices on bordered Riemann surfaces

Let \D be the unit disk. Kutzschebauch and Studer \cite{KS} recently proved that, for each continuous map A:\overline D\to \mathrm{SL}(2,\C), which is holomorphic in \D, there exist continuous maps E,F:\overline \D\to \mathfrak{sl}(2,\C), which are holomorphic in \D, such that $A=e^Ee^F$. Also they asked if this extends to arbitrary compact bordered Riemann surfaces. We prove that this is possible.Comment: 11 page

Smoothness of generalized inverses

The paper is largely expository. It is shown that if a (x) is a smooth unital Banach algebra valued function of a parameter x, and if a(x) has a locally bounded generalized inverse in the algebra, then a generalized inverse of a(x) exists which is as smooth as a(x) is. Smoothness is understood in the sense of having a certain number of continuous derivatives, being real-analytic, or complex holomorphic. In the complex holomorphic case, the space of parameters is required to be a Stein manifold. Local formulas for the generalized inverses are given. In particular, the Moore-Penrose and the generalized Drazin inverses are studied in this context. (C) 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved