The complete hyperexpansivity analog of the Embry conditions

Athavale, Ameer

doi:10.4064/sm154-3-4

Cited by 7 publications

(6 citation statements)

References 12 publications

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“…The concept of complete hyperexpansitivity was introduced by Athavale in [1]. Let us recall that this type of operators have been studied in [1,3,20,6,7,2,8,13]. Let X be an arbitrary nonempty set.…”

Section: Abstract Given a Familymentioning

confidence: 99%

A moment theorem for completely hyperexpansive operators

Jabłoński

2007

Acta Math Hung

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Given a family {A x m } m∈Z d + x∈X (X is a non-empty set) of bounded linear operators between the complex inner product space D and the complex Hilbert space H we characterize the existence of completely hyperexpansive dtuples T = (T1, . . . , T d ) on H such that A x m = T m A x 0 for all m ∈ Z d + and x ∈ X. 1. From now on D stands for a complex inner product space and H for a complex Hilbert space. Let us denote by L(D, H) (resp. B(D, H)) the set of all linear (resp. bounded linear) operators from D to H. For simplicity we write L(H), B(H) instead of B(H, H); I D stands for the identity operator on D. For T ∈ L(D, H), we set R(T ) = T (D). Denote by Z the additive group of all integers. Let Z + stand for the set of all nonnegative integers and let Z d + = Z + × · · · × Z + (d-times) for the Abelian semigroup under coordinatewise addition with the identity element 0 = (0, . . . , 0). For m = (m 1 , . . . , m d ) and n = (n 1 , . . . , n d ) in Z d + we write m n if m i n i for all i ∈ {1, . . . , d} and use |m| to denote m 1 + · · · + m d . Let e d j = (δ j,1 , . . . , δ j,d ), where δ k,l stands for the Kronecker symbol * This work was supported by the MNiSzW grant N201 026 32/1350.

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Section: Abstract Given a Familymentioning

confidence: 99%

A moment theorem for completely hyperexpansive operators

Jabłoński

2007

Acta Math Hung

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“…1.5) Completely hyperexpansive operators are antithetical to contractive subnormal operators in the sense that their defining properties and behavior are related to the theory of completely alternating functions on abelian semigroups (subnormality is connected with positive definiteness). This is their great advantage and one of the reasons why they attract attention of researchers (see e.g., [4,1,2,3,7,11,72,9,10,46,8,47,27]).…”

Section: )mentioning

confidence: 99%

Weighted shifts on directed trees

Jabłoński¹,

Jung²,

Stochel³

2009

Preprint

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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.

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“…Completely hyperexpansive operators were introduced in [4] and studied extensively by Athavale and co-workers (see [5][6][7][8]25]) as well as by Jabloński and Stochel (see [14][15][16][17][18]). The class of completely hyperexpansive operators is closely related to the theory of negative definite functions on abelian semigroups, and it is in some sense antithetical to the class of subnormal contractions.…”

Section: Preliminariesmentioning

confidence: 99%

“…5) where U is unitary on H u and A is an analytic 2-hyperexpansion on H a . Since [T * , T ] = 0 ⊕ [A * , A], it suffices to check that [A * , A] is a trace class operator.…”

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confidence: 99%

On Operators Cauchy Dual to 2-Hyperexpansive Operators

Chavan¹

2007

Proceedings of the Edinburgh Mathematical Society

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The operator Cauchy dual to a $2$-hyperexpansive operator $T$, given by $T'\equiv T(T^*T)^{-1}$, turns out to be a hyponormal contraction. This simple observation leads to a structure theorem for the $C^*$-algebra generated by a $2$-hyperexpansion, and a version of the Berger–Shaw theorem for $2$-hyperexpansions.As an application of the hyperexpansivity version of the Berger–Shaw theorem, we show that every analytic $2$-hyperexpansive operator with finite-dimensional cokernel is unitarily equivalent to a compact perturbation of a unilateral shift.

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The complete hyperexpansivity analog of the Embry conditions

Cited by 7 publications

References 12 publications

A moment theorem for completely hyperexpansive operators

A moment theorem for completely hyperexpansive operators

Weighted shifts on directed trees

On Operators Cauchy Dual to 2-Hyperexpansive Operators

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