Weighted shifts on directed trees

Operators with absolute continuity properties: an application to quasinormality

“…The following characterization of quasinormality of weighted shifts on directed trees generalizes that of [5,Proposition 8.1.7] to the case of unbounded operators. The present proof is quite different from that for bounded operators.…”

“…Since, by [5,Proposition 3.4.3], E V ⊆ D(|S λ | α ) and |S λ | α e u = S λ e u α e u for every u ∈ V , we deduce that a function f ∈ 2 (V ) belongs to D(|S λ | α ) if and only if…”

“…The basic facts on directed trees and weighted shifts on directed trees can be found in [5]. We refer the reader to [6] for the recent applications of this idea to general operator theory.…”

“…Hence the operator S λ is quasinormal. It is easily seen, by using [5,Proposition 3.1.8], that for every u ∈ V , the operator S λ | Lin{ev : v∈Des(u)} (which acts in 2 (Des(u))) is unbounded. The injectivity of S λ follows from [5,Proposition 3.1.7].…”

“…Theorem 7.2 (the concept of a weighted shift on a directed tree has been developed in [5]). This enables us to illustrate the theme of this article by various examples and to show that there is no relationship between the hyponormality class and the classes of operators being considered in this paper (cf.…”