Weighted shifts on directed forests and hyponormality

TL;DR

This paper explores weighted shifts on directed forests, extending previous work on directed trees, and demonstrates how graph structures influence operator properties, including a full characterization of hyponormal shifts.

Contribution

It introduces weighted shifts on directed forests, revealing deeper connections between graph theory and operator theory, and characterizes forests where all hyponormal shifts are power hyponormal.

Findings

Weighted shifts on directed forests generalize those on directed trees.

Graph structure critically affects operator hyponormality.

Complete characterization of forests with universally hyponormal shifts.

Abstract

In a paper from 2012 Jab{\l}o\'nski, Jung and Stochel introduced the weighted shifts on directed trees, a generalisation of well known weighted shift operators on $\ell^2$. In the last decade this class has proven itself handy for finding counterexamples in operator theory. Properties of underlying graph structure had essential influence on the operator. It appears that a slight generalisation of the class, namely weighted shifts on directed forests, shows even deeper relations between graph theory and functional analysis. Several operations on directed forests have their natural operator-theoretic counterparts. This paper is meant to present advantages of the directed forest approach. As an application of the interrelation between graphs and operators we provide full characterisation of directed forests on which every hyponormal bounded weighted shift is power hyponormal.