A relationship: Subnormal, polynomially hyponormal and semi-weakly hyponormal weighted shifts

Li, Chunji; Lee, Mi Ryeong; Baek, Seung‐Hwan

doi:10.1016/j.jmaa.2019.06.046

Cited by 3 publications

(2 citation statements)

References 30 publications

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“…Obviously, 1-hyponormal [or weakly 1-hyponormal] operator T ∈ B(H) is hyponormal. It holds that every subnormal operator is polynomially hyponormal and every k-hyponormal operator is weakly k-hyponormal for each k ∈ N. Also it is well known that subnormal ⇒ nhyponormal ⇒ weakly n-hyponormal ⇒ hyponormal for every n ≥ 2; many operator theorists have studied the converse implications; for example, see [1], [4], [6][7][8][9][10], [21], [23], etc. In [14, Theorem 2.1], Curto-Putinar proved theoretically that there exists a polynomially hyponormal operator which is not 2-hyponormal.…”

Section: Introductionmentioning

confidence: 99%

A distinction of k-hyponormal and weakly k-hyponormal weighted shifts

Lee

Xiao

2021

Filomat

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Let ?(x) : ?x/2, ?2/3, ?3/4, ?4/5, ... be a sequence with a real variable x > 0 and let W?(x) be the associated weighted shift with weight sequence ?(x). In [17], Exner-Jung-Park provided an algorithm to distinguish weak k-hyponormality and k-hyponormality of weighted shift W?(x), and obtained sn > 0 for some low numbers n = 4,..., 10, such that W?(sn) is weakly n-hyponormal but not n-hyponormal. In this paper, we obtain a formula of sn (for all positive integer n) such that W?(sn) is weakly n-hyponormal but not n-hyponormal, which improves Exner-Jung-Park?s result above.

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Section: Introductionmentioning

confidence: 99%

A distinction of k-hyponormal and weakly k-hyponormal weighted shifts

Lee

Xiao

2021

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“…Although the existence of a weighted shift, which is polynomially hyponormal, but not subnormal, was established in [12,13], a concrete example of such weighted shifts has not been found yet. Recently, the authors in [14] proved that the subnormality is equivalent to the polynomial hyponormality for recursively-weighted shift W α with α :…”

Section: Introductionmentioning

confidence: 99%

On Positive Quadratic Hyponormality of a Unilateral Weighted Shift with Recursively Generated by Five Weights

Ryoo

2019

Mathematics

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Weakly n-hyponormal weighted shifts: A sufficient condition and their examples

Baek,

Lee,

2023

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The n-hyponormal and weakly n-hyponormal weighted shifts were developed to study bridges of operators between the subnormal and hyponormal operators on an infinite dimensional complex Hilbert space about 30 years ago. In this paper we discuss the distinction between the classes of n-hyponormal and weakly n-hyponormal weighted shifts. For such a purpose we consider an arbitrary contractive hyponormal weighted shift W? and find a sufficient condition for the weak n-hyponormality of W?. We provide a general technique for distinction between the n-hyponormality and the weak n-hyponormality of W?, and investigate the distinction between the classes of n-hyponormal and weakly n-hyponormal weighted shifts with Bergman shift and some other examples.

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A relationship: Subnormal, polynomially hyponormal and semi-weakly hyponormal weighted shifts

Cited by 3 publications

References 30 publications

A distinction of k-hyponormal and weakly k-hyponormal weighted shifts

A distinction of k-hyponormal and weakly k-hyponormal weighted shifts

On Positive Quadratic Hyponormality of a Unilateral Weighted Shift with Recursively Generated by Five Weights

Weakly n-hyponormal weighted shifts: A sufficient condition and their examples

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