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

Abstract

In a countable normed space of smooth functions on the unit circle, we consider the Toeplitz operator with a smooth symbol. Boundedness, Notherianity and invtrbillity of such operators are studied. The concepts of a smooth canonical degenerate factorization on the minus type of smooth functions and the associated local degenerate canonical factorization of the minus type are introduced. Criteria are obtained in terms of the symbol for existence of a canonical degenerate factorization of type minus. As in the classical case of the Toeplitz operator in spaces of summable functions with Wiener symbols, the Toeplitz operator being Noterian turned out to be equivalent to the presence of a canonical factorization its symbol. The equivalence of the degenerate canonical factorizability is established, which makes it possible to use the localization of the symbol of certain characteristic arcs of the circle when studying Invertibility questions. The equivalence of the degenerate canonical factorizability is established, which makes it possible to use the localization of the symbol on certain characteristic arcs of the circle when studying Invertibility questions. Relations are obtained that relate the spectra of some Toeplitz operators in the spaces smooth and summable functions. A description is given of the resolvent set of the Toeplitz operator with smooth symbol.