Invertibility and spectrum of the Riemann boundary value problem operator in a countably normed space of smooth functions on a circle

In a countably normed space of smooth functions on the unit circle, we consider the Riemann boundary value problem operator with smooth coefficients. The concept of smooth degenerate factorizations of types of plus and minus functions that are smooth on the unit circle is introduced. Criteria for the existence of such factorizations are given. An apparatus is given for calculating the indices of these factorizations in terms of coefficients. In terms of smooth degenerate factorizations, a criterion for the invertibility of the Riemann boundary value problem operator is obtained. This allows us to describe the spectrum of this operator. Relationships between the spectra of the Riemann operator in the spaces of smooth and summable functions with the same coefficients are indicated.