On the Toeplitz operator on a torus in some topological spaces

In countably-normed spaces of functions on a torus that are smooth in one or both variables, we study Toeplitz operators with symbols that ensure that the operators are bounded in these spaces. To study operators of bisingular type in the spaces of summable and Hölder functions, the method of partial regularization was used. Using this method, we obtain the construction of a regularizer and the condition for the Toeplitz operator to be Noetherian in a countably-normed space of functions that are smooth in one of the variables. Despite the fact that the set of Noetherian operators in this space, unlike in the case of Banach spaces, contains operators with symbols vanishing on the torus, these results are quite analogous to the case of Banach spaces. The situation is different in the space of functions that are smooth in both variables. It is shown that the partial regularization method is inapplicable in this space.