On the spectrum of a Toeplitz operator in a countable normed space of smooth functions

In a countable normed space of smooth functions on the unit circle, we consider the Toeplitz operator with a symbol that is the ratio of smooth functions. The questions of boundedness, Notherianness and invertibillity of such operators are studied. The notions of a smooth degenerate factorization of the minus type for smooth functions and the related appropriate degenerate factorization of the minus type are introduced. A criterion is obtained in terms of the symbol for existence of a suitable degenerate factorization of type minus. As in the classical case of Toeplitz operators in spaces of summable functions with Wiener symbols, a Toeplitz operator is Notherian if and only if its symbol admits a suitable factorization of the type minus. Moreover, the index of this factorization, which determines the index of the Toeplitz operator, can be expressed in terms of some functional defined by the operator symbol. In particular, a criterion for the invertibility of this operator in terms of the operator symbol is obtained. This criterion is formulated in the form of a relationship between the number of zeros, the number of poles and the singular index of a symbol. The latter enables one to obtain an effective description of the spectrum of the Toeplitz operator in a countable normed space of smooth functions on the unit circle. Relations are obtained that connect the spectra of some special Toeplitz operators in spaces of smooth and summable functions. Examples are given showing that the spectrum of a Toeplitz operator in a countably normed space, generally speaking, differs significantly from the spectrum of the Toeplitz operator in spaces of summable functions. In particular, the spectrum of a bounded Toeplitz operator in a countably normed space may turned out be open and (or) unbounded subset of the complex plane.