Weighted shifts on directed trees with one branching vertex: $n$-contractivity and hyponormality

Abstract

Let $S_{\boldsymbol{\lambda} }$ be a weighted shift on a rooted directed tree with one branching vertex $\widetilde{u}$, $\eta $ branches ($2\leq \eta \lt\infty $) and positive weight sequence $\boldsymbol{\lambda }$. We define a collection of (classical) weighted shifts, the so-called ``the $i$-th branching weighted shifts'' $W^{(i)}$ for $0\leq i\leq \eta $, whose weights are derived from those of $S_{\boldsymbol{\lambda}}$. In this note we discuss the relationships between $n$-contractivity, $n$-hypercontractivity and hyponormality of $S_{\boldsymbol{\lambda} }$ and these properties for the $W^{(i)}$ $(0\leq i\leq \eta )$.