Weighted shifts on directed trees

By Zenon Jan Jablo'nski, I. Jung, Jan Stochel. 2009

90 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b835d47ecef356e354d83440c841dc2860a52a16

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In Algebra and Number Theory

1. Zenon Jan Jablo'nski
2. Sameer Chavan
3. Jan Stochel
4. Akash Anand

In Applied Mathematics

1. Jan Stochel
2. Zenon Jan Jablo'nski
3. Sameer Chavan
4. Akash Anand

In Mathematical Physics

1. Jan Stochel
2. Zenon Jan Jablo'nski
3. Sameer Chavan
4. Akash Anand

Citations in the corpus

• A solution to the Cauchy dual subnormality problem for 2-isometries (2017, background): “ted trees the moduli of whose weights λvdepend only on the parents of v. This class of 2-isometries contains the adjacency operators of directed trees that play an important role in graph theory (see [21] for more details). We prove that the Cauchy dual of a 2-isometric adjacency operator of a direc”