An operator factorization conception is investigated for a general Wiener–Hopf operator W=P2A|P1X\documentclass[12pt]{minimal}
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\begin{document}$${W = P_2 A |_{P_1 X}}$$\end{document} where X, Y are Banach spaces, P1∈L(X),P2∈L(Y)\documentclass[12pt]{minimal}
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\begin{document}$${P_1 \in \mathcal{L}(X), P_2 \in \mathcal{L}(Y)}$$\end{document} are projectors and A∈L(X,Y)\documentclass[12pt]{minimal}
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\begin{document}$${A \in \mathcal{L}(X,Y)}$$\end{document} is boundedly invertible. Namely we study a particular factorization of A=A-CA+whereA+:X→ZandA-:Z→Y\documentclass[12pt]{minimal}
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\begin{document}$${A = A_- C A_+ \,\,{\rm where}\,\, A_+ : X \rightarrow Z \,\,{\rm and} \,\,A_- : Z \rightarrow Y}$$\end{document} have certain invariance properties and C:Z→Z\documentclass[12pt]{minimal}
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\begin{document}$${C : Z \rightarrow Z}$$\end{document} 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∼PC|PX\documentclass[12pt]{minimal}
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\begin{document}$${W \sim P C|_{P X}}$$\end{document}. The main result shows equivalence between the generalized invertibility of the Wiener–Hopf operator and this kind of factorization (provided P1∼P2\documentclass[12pt]{minimal}
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\begin{document}$${P_1 \sim P_2}$$\end{document}) which implies a formula for a generalized inverse of W. It embraces I.B. Simonenko’s generalized factorization of matrix measurable functions in Lp 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?
