Minimal Normalization of Wiener–Hopf Operators in Spaces of Bessel Potentials

Abstract

A class of operators is investigated which results from certain boundary and transmission problems, the so-called Sommerfeld diffraction problems. In various cases these are of normal type but not normally solvable, and the problem is how to normalize the operators in a physically relevant way, i.e., not loosing the Hilbert space structure of function spaces defined by a locally finite energy norm. The present approach solves this question rigorously for the case where the lifted Fourier symbol matrix function is Hölder continuous on the real line with a jump at infinity. It incorporates the intuitive concept of compatibility conditions which is known from some canonical problems. Further it presents explicit analytical formulas for generalized inverses of the normalized operators in terms of matrix factorization.