On the Structure of the Spectrum and the Resolvent Set of a Toeplitz Operator in a Countably Normed Space of Smooth Functions

In the countably normed space of smooth functions on the unit circle, we consider a Toeplitz operator with a smooth symbol. Questions about the boundedness, Fredholm property, and invertibility of such operators are studied. The concepts of smooth canonical degenerate factorization of minus type of smooth functions and the associated local degenerate canonical factorization of minus type are introduced. Criteria are obtained in terms of the symbol for the existence of a canonical degenerate factorization of minus type. Just as in the classical case of the Toeplitz operator in spaces of integrable functions with Wiener symbols, the Fredholm property of the Toeplitz operator turned out to be equivalent to the existence of a smooth degenerate canonical factorization of the minus type of its symbol. The equivalence of degenerate canonical factorizability and similar local factorizability is established, which permits one to use the localization of the symbol on certain characteristic arcs of a circle when studying invertibility issues. Relations between the spectra of some Toeplitz operators in spaces of smooth and integrable functions are obtained. A description of the resolvent set of the Toeplitz operator with a smooth symbol is given.