Normal extensions escape from the class of weighted shifts on directed trees

TL;DR

This paper investigates the properties of normal extensions of weighted shifts on directed trees, establishing conditions under which such extensions are bounded and can be modeled as weighted shifts on possibly different directed trees.

Contribution

It proves that formally normal weighted shifts on directed trees are bounded normal operators and addresses whether their normal extensions can be represented as weighted shifts on other directed trees.

Findings

Formally normal weighted shifts on directed trees are bounded normal operators.

Normal extensions of subnormal weighted shifts can be modeled as weighted shifts on different directed trees.

Provides conditions for the existence and modeling of such extensions.

Abstract

A formally normal weighted shift on a directed tree is shown to be a bounded normal operator. The question of whether a normal extension of a subnormal weighted shift on a directed tree can be modeled as a weighted shift on some, possible different, directed tree is answered.