Moment Infinitely Divisible Weighted Shifts

Abstract

We say that a weighted shift Wα with (positive) weight sequence α: α, α1, … is moment infinitely divisible (MID) if, for every t> 0 , the shift with weight sequence αt:α0t,α1t,… is subnormal. Assume that Wα is a contraction, i.e., 0 < αi≤ 1 for all i≥ 0. We show that such a shift Wα is MID if and only if the sequence α is log completely alternating. This enables the recapture or improvement of some previous results proved rather differently. We derive in particular new conditions sufficient for subnormality of a weighted shift, and each example contains implicitly an example or family of infinitely divisible Hankel matrices, many of which appear to be new.