Matrix Wiener-Hopf Operators with SAPW Symbols

Abstract

In this chapter we prove a Fredholm criterion and an index formula for Wiener-Hopf operators with matrix symbols in SAPW, where SAPW denotes the algebra of all semi-almost periodic functions whose almost periodic representatives a _l and a _r belong to APW. By using an argument familiar from the theory of so-called limit operators, we first show that W(a_l)and W(a_r)are invertible if W (a) is Fredholm. Then we construct a factorization a = f_ b f ₊ with the outer factors in and the middle factor in PC. In this way we can employ what we know about Wiener-Hopf operators with matrix-valued symbols in and PC in order to dispose of the Fredholm theory of operators with matrix symbols in SAPW.