On multiplicative inversion for Wolff-Denjoy series

Let a function $f$ with real poles that form a monotone bounded sequence be expanded in a Wolff–Denjoy series with positive coefficients. The main result of the paper states that, if we subtract the “linear part” from the function $1/f$, then the remaining “fractional part” is also expanded in a Wolff–Denjoy series (its poles are also real and the coefficients of the series are negative). An application of the result to operator theory is given.