Functional Analysis and Applications Seminar 

25/07/2014, 14:30 — 15:30 — Room P3.10, Mathematics Building
Frank-Olme Speck, Instituto Superior Técnico, Universidade de Lisboa

Wiener-Hopf factorization through an intermediate space

An operator factorization conception is investigated for a general Wiener-Hopf operator $W = P_2 A|{P_1 X}$ in asymmetric Banach space setting. Namely, we study a particular factorization of the underlying operator $A = A_- C A_+$ where $A_+$ and $A_-$ are strong Wiener-Hopf factors and the cross factor $C$ maps an "intermediate space" $Z$ onto itself such that $Z$ is split into complemented subspaces closely related to the kernel and cokernel of $W$ and, moreover, such that $W$ is toplinear equivalent to a "simpler" symmetric Wiener-Hopf operator, $W \sim P C|_{PX}$.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 $W$. The conception embraces I.B. Simonenko's generalized factorization of matrix measurable functions in $L^p$ spaces and various other factorization approaches, particularly factorization of bounded into unbounded operators. It is quite different from the cross factorization approach and more useful in many applications. Some 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?