Wiener-Hopf factorization, inverse Fourier transforms and exponentially dichotomous operators

Abstract

For a certain class of matrix functions that are analytic on the real line but not at infinity an explicit method of factorization and of constructing inverse Fourier transforms is developed. This method is applied to invert Wiener-Hopf integral equations on the half line and the full line. The results obtained extend analogous results of the authors for rational matrix functions and for functions that are analytic on the real line and at infinity. The analysis is based on an infinite dimensional realization theorem which involves operators that are a direct sum of two infinitesimal generators of C 0 -semigroups of negative exponential type, one of which has support on the negative half line and the other on the positive half line. The latter operators are called exponentially dichotomous and the study of their properties forms an essential part of the paper.