Joint backward extension property for weighted shifts on directed trees

TL;DR

This paper introduces the joint backward extension property (JBEP) for weighted shifts on directed trees, showing which classes satisfy it, and explores its implications for various operator classes.

Contribution

It defines JBEP for weighted shifts on directed trees and characterizes which classes satisfy this property, including positive results for subnormal and power hyponormal shifts.

Findings

Subnormal and power hyponormal shifts satisfy JBEP.

Completely hyperexpansive and quasinormal shifts do not satisfy JBEP.

Positive results for joint backward extensions of completely hyperexpansive shifts.

Abstract

Weighted shifts on directed trees are a decade old generalisation of classical shift operators in the sequence space $\ell^2$. In this paper we introduce the joint backward extension property (JBEP) for classes of weighted shifts on directed trees. If a class satisfies JBEP, the existence of a common backward extension within the class for a family of weighted shifts on rooted directed trees does not depend on the additional structure of the big tree (of fixed depth). We decide whether several classes of operators have JBEP. For subnormal or power hyponormal weighted shifts the property is satisfied, while it fails for completely hyperexpansive or quasinormal. Nevertheless some positive results on joint backward extensions of completely hyperexpansive weighted shifts are proven.