Geometrically regular weighted shifts

TL;DR

This paper introduces a new class of weighted shifts with geometrically regular weights, exploring their diverse properties such as subnormality and hyperexpansiveness, and their relation to well-known functions.

Contribution

It defines a novel family of weighted shifts with geometric weight patterns, expanding the set of examples for studying properties like subnormality and moment divisibility.

Findings

The shifts exhibit a wide range of properties depending on parameters.

They include subshifts of the Bergman shift with geometric spacing.

Some shifts are moment infinitely divisible and hyperexpansive.

Abstract

We study a general class of weighted shifts whose weights $\alpha$ are given by $\alpha_n = \sqrt{\frac{p^n + N}{p^n + D}}$, where $p > 1$ and $N$ and $D$ are parameters so that $(N,D) \in (-1, 1)\times (-1, 1)$. Some few examples of these shifts have appeared previously, usually as examples in connection with some property related to subnormality. In sectors nicely arranged in the unit square in $(N,D)$, we prove that these geometrically regular weighted shifts exhibit a wide variety of properties: moment infinitely divisible, subnormal, $k$- but not $(k+1)$-hyponormal, or completely hyperexpansive, and with a variety of well-known functions (such as Bernstein functions) interpolating their weights squared or their moment sequences. They provide subshifts of the Bergman shift with geometric, not linear, spacing in the weights which are moment infinitely divisible. This new family of weighted shifts provides a useful addition to the library of shifts with which to explore new definitions and properties.