Weighted shifts on directed trees

TL;DR

This paper introduces and studies a new class of operators called weighted shifts on directed trees, generalizing classical weighted shifts and adjacency operators, with detailed analysis of their properties and specific models.

Contribution

It defines weighted shifts on directed trees, explores their fundamental properties, and characterizes their hyponormality, subnormality, and hyperexpansivity in terms of weights, including models and examples.

Findings

Characterization of hyponormality, subnormality, and hyperexpansivity in terms of weights

Construction of models for subnormal and hyperexpansive shifts on specific trees

Examples of weighted shifts with prescribed properties, simpler than previous models

Abstract

A new class of (not necessarily bounded) operators related to (mainly infinite) directed trees is introduced and investigated. Operators in question are to be considered as a generalization of classical weighted shifts, on the one hand, and of weighted adjacency operators, on the other; they are called weighted shifts on directed trees. The basic properties of such operators, including closedness, adjoints, polar decomposition and moduli are studied. Circularity and the Fredholmness of weighted shifts on directed trees are discussed. The relationships between domains of a weighted shift on a directed tree and its adjoint are described. Hyponormality, cohyponormality, subnormality and complete hyperexpansivity of such operators are entirely characterized in terms of their weights. Related questions that arose during the study of the topic are solved as well. Particular trees with one branching vertex are intensively studied mostly in the context of subnormality and complete hyperexpansivity of weighted shifts on them. A strict connection of the latter with $k$-step backward extendibility of subnormal as well as completely hyperexpansive unilateral classical weighted shifts is established. Models of subnormal and completely hyperexpansive weighted shifts on these particular trees are constructed. Various illustrative examples of weighted shifts on directed trees with the prescribed properties are furnished. Many of them are simpler than those previously found on occasion of investigating analogical properties of other classes of operators.