WEIGHTED SHIFTS ON DIRECTED TREES WITH ONE BRANCHING VERTEX: n-CONTRACTIVITY AND HYPONORMALITY

Let S-lambda be a weighted shift on a rooted directed tree with one branching vertex (u) over tilde, eta branches (2 <= eta < infinity) and positive weight sequence lambda. We define a collection of (classical) weighted shifts, the so-called "the i-th branching weighted shifts" W-(i) for 0 <= i <= eta, whose weights are derived from those of S-lambda. In this note we discuss the relationships between n-contractivity, n-hypercontractivity and hyponormality of S-lambda and these properties for the W-(i) (0 <= i <= eta).