Generalized AP Factorization

Abstract

Simonenko’s theorem tells us that the invertibility of the Wiener-Hopf operator W (a) on $L_N^2\left( {{R_ + }} \right)$ is equivalent to the existence of a right Wiener-Hopf factorization of a. If a is continuous, then the factors of the Wiener-Hopf factorization need not be continuous. AP factorization is a factorization with “nice” factors, and we know from Section 9.4 that for a ∈ APW_N×Nthe existence of a canonical right AP factorization is equivalent to the invertibility of W(a). In this chapter, we prove the AP analogue of Simonenko’s theorem, that is we introduce the notion of generalized AP factorization and then show that if a ∈ APW_N×N, then W(a) is invertible on $L_N^2\left( {{R_ + }} \right)$ if and only if a possesses a canonical generalized right AP factorization.