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Weighted shifts on directed trees
About this Title
Zenon Jan Jabłoński, Instytut Matematyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, PL-30348 Kraków, Il Bong Jung, Department of Mathematics, Kyungpook National University, Daegu 702-701 South Korea and Jan Stochel, Instytut Matematyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, PL-30348 Kraków
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 216, Number 1017
ISBNs: 978-0-8218-6868-3 (print); 978-0-8218-8527-7 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00644-1
Published electronically: May 25, 2011
Keywords: Directed tree,
weighted shift,
adjoint operator,
polar decomposition,
circular operator,
inclusion of domains,
Fredholm operator,
semi-Fredholm operator,
hyponormal operator,
cohyponormal operator,
subnormal operator,
completely hyperexpansive operator.
MSC: Primary 47B37, 47B20; Secondary 47A05, 44A60
Table of Contents
Chapters
- 1. Introduction
- 2. Prerequisites
- 3. Fundamental Properties
- 4. Inclusions of Domains
- 5. Hyponormality and Cohyponormality
- 6. Subnormality
- 7. Complete Hyperexpansivity
- 8. Miscellanea
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 extendability 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.- Jim Agler and Mark Stankus, $m$-isometric transformations of Hilbert space. I, Integral Equations Operator Theory 21 (1995), no. 4, 383–429. MR 1321694, DOI 10.1007/BF01222016
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