Moment Infinitely Divisible Weighted Shifts

Abstract

We say that a weighted shift $W_\alpha$ with (positive) weight sequence $\alpha : \alpha _0, \alpha _1, \ldots$ is moment infinitely divisible (MID) if, for every $t > 0$, the shift with weight sequence $\alpha ^t: \alpha _0^t, \alpha _1^t, \ldots$ is subnormal. Assume that $W_{\alpha }$ is a contraction, i.e., $0 < \alpha _i \le 1$ for all $i \ge 0$. We show that such a shift $W_\alpha$ is MID if and only if the sequence $\alpha$ 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.