Factorizations of Matrix Functions

Abstract

We now begin the trek to a criterion for the Fredholmness or, equivalently, the invertibility of Wiener-Hopf operators with matrix-valued AP symbols. In one or another way, all invertibility criteria for matrix Wiener-Hopf operators are connected with certain factorizations of their symbols. In this chapter we introduce the notions of Wiener-Hopf factorization and almost periodic factorization. The first of these two types of factorizations has been used in Wiener-Hopf theory for many decades, while almost periodic factorization is less known. Both factorizations are special cases of factorizations of matrix functions into a product a_ua_±in which a+ are invertible analytic matrix functions and u is a unitary matrix function. We therefore conclude this chapter by collecting some results on the Fredholmness and invertibility of Wiener-Hopf operators with unitary-valued symbols.