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.