A hyponormal weighted shift on a directed tree whose square has trivial domain

TL;DR

This paper classifies directed trees that support hyponormal weighted shifts with nonzero weights, showing only two types have the property that their square has a trivial domain, based on their structure.

Contribution

It proves that only two specific types of enumerable directed trees admit hyponormal weighted shifts with nonzero weights whose square has trivial domain.

Findings

Only two isomorphism classes of directed trees support such shifts.

These trees are enumerable with each vertex having an enumerable set of successors.

The result characterizes the structure of trees supporting these operators.

Abstract

It is proved that, up to isomorphism, there are only two directed trees that admit a hyponormal weighted shift with nonzero weights whose square has trivial domain. These are precisely those enumerable directed trees, one with root, the other without, whose every vertex has enumerable set of successors.