Moment Infinite Divisibility of Weighted Shifts: Sequence Conditions

TL;DR

This paper characterizes weighted shift operators that remain subnormal when all weights are raised to any positive power, revealing their robustness and rigidity under various transformations.

Contribution

It provides a new sequence-based characterization of moment infinite divisibility for weighted shifts and explores their stability under transformations like the Aluthge transform.

Findings

Weighted shifts with limit weights are moment infinitely divisible iff their Aluthge transform is.

Such shifts are robust under operations like back-step extensions and subshifts.

The paper establishes new sequence conditions for moment infinite divisibility.

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. We also consider back-step extensions, subshifts, and completions.