Matrix Wiener-Hopf Operators with APW Symbols

Abstract

With this chapter we reach the first big goal on our trek. If a matrix function a ∈ APW_N×N has a canonical right APW factorization a = a_a₊, then W(a) is obviously invertible on (R). We here prove that the converse is also true, that is, the symbol a necessarily has a canonical right APW factorization if W (a) is invertible. The proof of this result requires some heavy tools. We first establish an isomorphism theorem for C*-dynamical systems, which allows us to show that the invertibility of W(a) on (R) is equivalent to the invertibility of a certain operator on the vector-valued Besicovitch space. After that we employ the Bochner-Phillips theorem on the inverse closedness of certain matrix algebras in order to prove that the invertibility of the latter operator implies the canonical right APW factorability of a.