Moment infinitely divisible weighted shifts

Abstract

We say that a weighted shift $W_{\alpha }$ with (positive) weight sequence $\alpha :{\alpha }_0,{\alpha }_1,\dots$ is {\it moment infinitely divisible} (MID) if, for every $t>0$, the shift with weight sequence ${\alpha }^t:{\alpha }_0^t,{\alpha }_1^t,\dots$ 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.

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