# A Subnormal Completion Problem for Weighted Shifts on Directed Trees, II

## TL;DR

This paper investigates the subnormal completion problem for weighted shifts on a specific class of directed trees with one branching point, providing necessary and sufficient conditions for completion with 2-atomic measures and explicit solutions for certain cases.

## Contribution

It offers a characterization of subnormal completions on directed trees with a single branching point, including explicit solutions when the number of branches is two.

## Key findings

- Necessary and sufficient conditions for subnormal completion with 2-atomic measures.
- Solution of the completion problem for trees with finite branches.
- Explicit solution when the number of branches is two.

## Abstract

The subnormal completion problem on a directed tree is to determine, given a collection of weights on a subtree, whether the weights may be completed to the weights of a subnormal weighted shift on the directed tree. We study this problem on a directed tree with a single branching point, $\eta$ branches and the trunk of length $1$ and its subtree which is the "truncation" of the full tree to vertices of generation not exceeding $2$. We provide necessary and sufficient conditions written in terms of two parameter sequences for the existence of a subnormal completion in which the resulting measures are $2$-atomic. As a consequence, we obtain a solution of the subnormal completion problem for this pair of directed trees when $\eta < \infty$. If $\eta=2$, we present a solution written explicitly in terms of initial data.