Moment Infinite Divisibility of Weighted Shifts: Sequence Conditions

Abstract

We consider weighted shift operators having the property of moment infinite divisibility; that is, for any \(p > 0\), the shift is subnormal when every weight (equivalently, every moment) is raised to the p-th power. By reconsidering sequence conditions for the weights or moments of the shift, we obtain a new characterization for such shifts, and we prove that such shifts are, under mild conditions, robust under a variety of operations and also rigid in certain senses. In particular, a weighted shift whose weight sequence has a limit is moment infinitely divisible if and only if its Aluthge transform is. As a consequence, we prove that the Aluthge transform maps the class of moment infinitely divisible weighted shifts bijectively onto itself. We also consider back-step extensions, subshifts, and completions.