Backward extensions of weighted shifts on directed trees

TL;DR

This paper investigates conditions under which weighted shifts on directed trees can be extended backward to larger trees while maintaining subnormal or power hyponormal properties, introducing a generalized framework for such shifts.

Contribution

It introduces a generalized framework for weighted shifts on directed forests and characterizes forests where all hyponormal shifts are power hyponormal.

Findings

Extension properties depend on individual forest extendability.

Characterization of forests with universally power hyponormal shifts.

Framework simplifies analysis of weighted shifts on directed structures.

Abstract

The weighted shifts are long known and important class of operators. One of known generalisation of this class are weighted shifts on directed trees, where we replace the linear order of coordinates in $\ell^2$ with a possibly more sophisticated graph structure. In this paper we focus on the question whether a weighted shift on a directed tree admits a subnormal or just power hyponormal (i.e. all powers of the operator are hyponormal) backward extension (a shift on larger directed tree). It comes out that in both cases the question whether we can obtain "joint extension" for a family of trees does not depend on any deep interrelations between the given trees but on their own "extendability" only. We introduce a generalised framework of weighted shifts on directed forests which seems to be slightly more convenient to work with. The characterisation of all the leafless directed forests on which all hyponormal weighted shifts are power hyponormal is also given.