Analog for the Wiener Lemma for Wolff-Denjoy Series

Abstract

Let a function f with real poles be expanded in a Wolff-Denjoy series with positive coefficients. The main result of the note states that if we subtract its "linear part" from the function 1/f , then the remaining "fractional part" of this function will also expand into Wolff-Denjoy series (its poles are also real, and the coefficients of the series are negative). In other words, for Wolff-Denjoy series of the indicated form, an analogue of the well-known Wiener lemma in the theory of Fourier series is true up to a linear term. Applications of the result to operator theory are given.