Transformation techniques towards the factorization of non-rational 2×2 matrix functions

Abstract

For the Wiener–Hopf factorization of 2×2 matrix functions G defined on a closed Carleson curve Γ, transformations G↦ UGV where U and V are invertible rational 2×2 matrix functions are important. In the first part of this paper we establish a classification scheme for 2×2 matrix functions, which is based on such transformations. We determine invariants under these transformations and describe those matrix functions which can be transformed to triangular or Daniele–Khrapkov form. In the second part we consider special rational transformations and study the same problem. For instance, we consider transformations where U and V are rational matrix functions that are analytic and invertible on an open neighborhood of Γ. In the more complicated, but for factorization theory important case where U and V are rational matrix functions that are analytic and invertible on an open neighborhood of the closure of the domain inside of Γ or outside of Γ, respectively, the answer is slightly different.