Moment infinitely divisible weighted shifts

TL;DR

This paper characterizes moment infinitely divisible weighted shifts as those with log completely alternating weights, providing new conditions for subnormality and revealing connections to infinitely divisible Hankel matrices.

Contribution

It establishes a precise characterization of MID weighted shifts via log complete alternation, improving understanding and conditions for subnormality.

Findings

Characterization of MID shifts as having log completely alternating weights

New sufficient conditions for subnormality of weighted shifts

Identification of new infinitely divisible Hankel matrices

Abstract

We say that a weighted shift $W_\alpha$ with (positive) weight sequence $\alpha: \alpha_0, \alpha_1, \ldots$ is {\it 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.