WEIGHTED SHIFTS ON DIRECTED TREES WITH ONE BRANCHING VERTEX: BETWEEN QUASINORMALITY AND PARANORMALITY

LetT & eta;,& kappa; be a directed tree consisting of one branching vertex, & eta; branches and a trunk of length & kappa; and let S & lambda; be the associated weighted shift on T & eta;,& kappa; with positive weight se-quence & lambda; . Intouorecently was a collection of classical weighted shifts, "the i-th branching weighted shifts" W (i) for 0 5 i 5 & eta; , whose weights are derived from those of S & lambda; by slicing the branches of the treeT & eta;,& kappa; ([9]). As a contrast contrasting to "slicing" we consider "collapsing the branches of a tree" and define "the k-step collapsed weighted shift" S & lambda;�(k) on T & eta;-k,& kappa; for 1 5 k 5 & eta; -1 so that S & lambda;(& eta; -1) may become the basic branching shift W(0) . In this paper we discuss the relationships between operator properties of S & lambda; such as quasinormality, subnormal-ity, & INFIN;-hyponormality, p-hyponormality, and p-paranormality, and these properties for the W (i) and S & lambda;--(k) .