On the Invertibility and Spectrum of the Wiener–Hopf Integral Operator in a Countably-Normed Space of Functions of Rapid Decay at Infinity

We consider the Wiener–Hopfintegral operator in a countably-normed space of measurable functions of rapid decayon the real axis and show that the class of bounded Wiener–Hopfoperators contains those with discontinuous symbol of a special form.Under study are the problems ofboundedness, noetherianity, and invertibility of such operators.In particular, we obtain some criteria for invertibility interms of a symbol by introducing the concept of canonical smooth degenerate factorizationand establishing that the invertibility of theWiener–Hopf operator is equivalent to the presence of a canonical smooth degeneratefactorization of the symbol of the operator.We describe the canonical smooth degenerate factorizationby some functional called the singular index as well as the spectrum of theWiener–Hopf operator and some relationsconnecting the spectra of the Wiener–Hopf integral operators.