Canonical Wiener-Hopf Factorization via Corona Problems

Abstract

Abstract Explicit Wiener-Hopf factorization of matrix functions is a white whale of the theory of convolution type equations. Enormous effort has been spent on it but the cases when it has been found are still scarce. The Portuguese transformation and its applications (Chapters 13 14) can be thought of as results in this direction for triangular AP matrix functions based on the solution of certain corona problems. As it happens there is an alternative approach to explicit factorization also using corona problems which allows us to consider more general 2 × 2 matrix functions (not necessarily triangular and not necessarily AP). Assuming the existence of a non-trivial solution of the corresponding homogeneous Riemann-Hilbert problem in \({\left[ {H_\pm ^\infty } \right]_2}\) we get criteria for the existence of a canonical left WH factorization G = G + G_ in L P (R w) (1 < p <∞w∈ Ap(R)) and a bounded canonical left WH factorization in case \(G \in L_{2 \times 2}^\infty \left( R \right)\) anddetG = 1,as well as explicit formulas for G ± in terms of the solutions of two associated corona problems (in the half-planes C + and C_). A separation principle for constructing corona solutions from simpler ones is developed and corona solutions for a number of corona problems in \(H_ + ^\infty \) are obtained. Making use of these results we explicitly construct canonical left WH factorizations for triangular2 x 2matrix functions with diagonal entries e ± λ and certain classes of L ∞ off diagonal entries. KeywordsHardy SpaceMatrix FunctionDiagonal EntrySeparation PrincipleExplicit FactorizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.