On holomorphic matrices on bordered Riemann surfaces

Abstract

Let double-struckD be the unit disk. Kutzschebauch and Studer (Bull. Lond. Math. Soc. 51 (2019) 995–1004) recently proved that, for each continuous map A:double-struckD¯→SLfalse(2,double-struckCfalse), which is holomorphic in double-struckD, there exist continuous maps E,F:double-struckD¯→slfalse(2,double-struckCfalse), which are holomorphic in double-struckD, such that A=eEeF. Also they asked if this extends to arbitrary compact bordered Riemann surfaces. We prove that this is possible.