Propagation phenomena for hyponormal 2-variable weighted shifts

TL;DR

This paper investigates hyponormal 2-variable weighted shifts, revealing how equal weights induce flatness and exploring the conditions under which these shifts are hyponormal or subnormal, including a complete characterization for symmetric cases.

Contribution

It demonstrates that equal weights cause flatness in 2-variable shifts and provides a comprehensive analysis of hyponormality and subnormality, including new examples and characterizations.

Findings

Equal weights lead to flatness in 2-variable shifts.

Many flat 2-variable shifts are hyponormal but not subnormal.

Complete characterization of hyponormality and subnormality for symmetric flat shifts.

Abstract

We study the class of hyponormal 2-variable weighted shifts with two consecutive equal weights in the weight sequence of one of the coordinate operators. We show that under natural assumptions on the coordinate operators, the presence of consecutive equal weights leads to horizontal or vertical flatness, in a way that resembles the situation for 1-variable weighted shifts. In 1-variable, it is well known that flat weighted shifts are necessarily subnormal (with finitely atomic Berger measures). By contrast, we exhibit a large collection of flat (i.e., horizontally and vertically flat) 2-variable weighted shifts which are hyponormal but not subnormal. Moreover we completely characterize the hyponormality and subnormality of symmetrically flat contractive 2-variable weighted shifts.