On reversibility and the spectrum of the Wiener–Hopf integral operator in a countably-normed space of functions with power behavior at infinity

We consider the Wiener–Hopf integral operator in a countable normed space of measurable functions on the real axis, decreasing faster then any power. It is shown that the class of bounded Wiener–Hopf operators contains with discontinuous symbols of a special form. The problems of boundedness, Noetherianity, and invertibility of such operators in the given countably normed space are studied. In particular, criteria for Noetherianity and invertibility in terms of a symbol are obtained. For this purpose, the concept of a canonical smooth degenerate factorization is introduced and it is established that the invertibility of the Wiener–Hopf operator is equivalent to the presence of a canonical smooth degenerate factorization of its symbol. The canonical smooth degenerate factorization is described using a functional called the singular index. As a corollary, the spectrum of the Wiener–Hopf operator in the considered topological space is described. Some relations are given that connect the spectra of the Wiener–Hopf integral operator with the same symbol in the countably normed spaces of measurable functions decreasing at infinity faster than any power.