Weighted shifts on directed trees (Q2880224)

This memoir is divided into eight chapters, and is an introduction to and investigation of a new class of (not necessarily bounded) operators related to (mainly infinite) directed trees, which the authors call weighted shifts on directed trees. These operators generalize the notion of classical weighted shift operators and of weighted adjacency operators.NEWLINENEWLINEIn Chapter 2, the authors provide some basic notions and concepts from the theory of graphs and discuss some properties of directed trees. In Chapter 3, they introduce weighted shifts on directed trees and derive several basic properties of such operators, including closedness, adjoints, polar decomposition. In particular, they show that any weighted shift on a directed tree is a closed operator, and describe the domain and graph norm of such a weighted shift. Furthermore, for a densely defined weighted shift on a directed tree, they show that the spectra are circular about the origin, and characterize when such a shift is a Fredholm or semi-Fredholm operator. In Chapter 4, they characterize the circumstances under which the domain of a densely defined weighted shift on a directed tree is contained in the domain of its adjoint, and discuss when the reverse inclusion holds. In Chapter 5 (resp., 6 and 7), they characterize, in particular, which bounded weighted shifts on directed trees are hyponormal/cohyponormal (resp., subnormal and completely hyperexpansive) in terms of their weights, and discuss \(k\)-step backward extendability of subnormal as well as completely hyperexpansive unilateral classical weighted shifts. In Chapter 8, they describe directed trees admitting weighted shifts with dense range, bounded hyponormal (subnormal, normal, etc.), and characterize bounded \(p\)-hyponormal weighted shifts on directed trees.NEWLINENEWLINEThe exposition is clear, the arguments seem well worth understanding, and many examples and remarks are given which nicely illustrate the results considered.