Generalized multipliers for left-invertible analytic operators will be
introduced. It will be shown that they form a Banach algebra and
characterize the commutant of such operators in its terms. In the special case, we describe the commutant of balanced weighted shift on directed tree only in terms of its weights. The reflexivity results for weighted shifts on directed tree will presented.
Joint work with Piotr Dymek, Artur Planeta.