### A Completely Hyperexpansive Completion Problem for Weighted Shifts on Directed Trees with one branching vertex

#### Abstract

Let $\boldsymbol{\alpha }=\left\{ \alpha _{k}\right\} _{k=0}^{n}$ be given a finite sequence of positive real numbers. The completely hyperexpansive completion problem seeks equivalence conditions for the existence of a completely hyperexpansive weighted shift $W_{\hat{\boldsymbol{\alpha }}}$ such that $\boldsymbol{\alpha}\subseteq \hat{\boldsymbol{\alpha }}$. Let $\mathscr{T}_{\eta ,\kappa }$ be a directed tree consisting of one branching vertex, $\eta $ branches and a trunk of length $\kappa ,$ and let $\mathscr{T}_{\eta ,\kappa ,p}$ be a subtree of $\mathscr{T}_{\eta,\kappa }$ whose members consist of the $p$-generation family from branching vertex. Suppose $S_{\boldsymbol{\lambda }}$ is the weighted shift acting on the tree $\mathscr{T}_{\eta ,\kappa }$. This object $S_{\boldsymbol{\lambda }}$ on the tree $\mathscr{T}_{\eta ,\kappa }$ has been applied to the several topics. Recently, Exner-Jung-Stochel-Yun studied the subnormal completion problem for weighted shifts on $\mathscr{T}_{\eta ,\kappa }$ in 2018. In this paper we discuss the completely hyperexpansive completion problem for weighted shifts on $\mathscr{T}_{\eta ,\kappa }$ as a counterpart of the subnormal completion problem for $S_{\boldsymbol{\lambda }}$.

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