Subnormal and completely hyperexpansive completion problem of weighted shifts on directed trees

TL;DR

This paper investigates the problem of completing weights on directed trees to form bounded weighted shifts that are subnormal or completely hyperexpansive, providing new characterizations and extension results.

Contribution

It introduces new results on backward extensions of moment sequences and characterizes the existence of such completions for weighted shifts on directed trees with one branching point.

Findings

Characterization of when weights can be completed to form subnormal shifts.

Development of new backward extension results for moment sequences.

Application to weighted shifts on directed trees with a single branching point.

Abstract

For a given directed tree and weights associated with vertices from a subtree the completion problem is to determine if these weights may be completed in a way to obtain a bounded weighted shift on the whole tree, which possibly satisfies also some more restrictive conditions. In this paper we consider subnormal and completely hyper-expansive completion problem for weighted shifts on directed trees with one branching point. We develop new results on backward extensions of truncated moment sequences and, exploiting these results, we obtain a characterization of existence of such a completion