Wiener–Hopf Factorization Through an Intermediate Space

Abstract

An operator factorization conception is investigated for a general Wiener–Hopf operator \({W = P_2 A |_{P_1 X}}\) where X, Y are Banach spaces, \({P_1 \in \mathcal{L}(X), P_2 \in \mathcal{L}(Y)}\) are projectors and \({A \in \mathcal{L}(X,Y)}\) is boundedly invertible. Namely we study a particular factorization of \({A = A_- C A_+ \,\,{\rm where}\,\, A_+ : X \rightarrow Z \,\,{\rm and} \,\,A_- : Z \rightarrow Y}\) have certain invariance properties and \({C : Z \rightarrow Z}\) splits the “intermediate space” Z into complemented subspaces closely related to the kernel and cokernel of W, such that W is equivalent to a “simpler” operator, \({W \sim P C|_{P X}}\). The main result shows equivalence between the generalized invertibility of the Wiener–Hopf operator and this kind of factorization (provided \({P_1 \sim P_2}\)) which implies a formula for a generalized inverse of W. It embraces I.B. Simonenko’s generalized factorization of matrix measurable functions in L p spaces, is significantly different from the cross factorization theorem and more useful in numerous applications. Various connected theoretical questions are answered such as: How to transform different kinds of factorization into each other? When is W itself the truncation of a cross factor?