Chaos for Cowen-Douglas operators

Hou, Bingzhe; Cui, Puyu; Cao, Yang

doi:10.1090/s0002-9939-09-10046-1

Cited by 30 publications

(16 citation statements)

References 19 publications

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“…The following criterion for distributional chaos was introduced in [17]. Since, for several reasons that will be clarified soon, this criterion is somehow very restrictive, we will call it the "strong" criterion.…”

Section: The Strong Criterion For Distributional Chaos and Spectral Pmentioning

confidence: 99%

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Li–Yorke and distributionally chaotic operators

Bermúdez

Bonilla

Martínez-Giménez

et al. 2011

Journal of Mathematical Analysis and Applications

107

110

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We study Li-Yorke chaos and distributional chaos for operators on Banach spaces. More precisely, we characterize Li-Yorke chaos in terms of the existence of irregular vectors. Sufficient "computable" criteria for distributional and Li-Yorke chaos are given, together with the existence of dense scrambled sets under some additional conditions. We also obtain certain spectral properties. Finally, we show that every infinite dimensional separable Banach space admits a distributionally chaotic operator which is also hypercyclic.

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Section: The Strong Criterion For Distributional Chaos and Spectral Pmentioning

confidence: 99%

“…This concept was widely studied by several authors and can be formulated in any metric space. In particular, it is considered for linear operators defined on Banach or Fréchet spaces in [25,22,[17][18][19].…”

mentioning

confidence: 99%

Li–Yorke and distributionally chaotic operators

Bermúdez

Bonilla

Martínez-Giménez

et al. 2011

Journal of Mathematical Analysis and Applications

107

110

View full text Add to dashboard Cite

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“…For each pair x, y ∈ X and each n ∈ N, the distributional function F The following notions were introduced in [23] and [21], respectively. They were considered for linear operators on Banach or Fréchet spaces in [6,12,13,14,16,17,20,24]. Definition 1.…”

Section: Introductionmentioning

confidence: 99%

Distributional chaos for linear operators

Bernardes

Bonilla

Müller

et al. 2013

Journal of Functional Analysis

135

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ElsevierPeris Manguillot, A.; Bernardes, NC.; Müller, V. (2013) AbstractWe characterize distributional chaos for linear operators on Fréchet spaces in terms of a computable condition (DCC), and also as the existence of distributionally irregular vectors. A sufficient condition for the existence of dense uniformly distributionally irregular manifolds is presented, which is very general and can be applied to many classes of operators. Distributional chaos is also analyzed in connection with frequent hypercyclicity, and the particular cases of weighted shifts and composition operators are given as an illustration of the previous results.

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“…Schweizer and Smítal introduced the concept of distributional chaos in [26] as a natural extension of the notion of chaos given by Li and Yorke. There are several recent studies of distributional chaos for linear operators on Banach or Fréchet spaces (see, e.g., [21,[27][28][29]). …”

Section: Distributional Chaos For Polynomials On Banach Spacesmentioning

confidence: 99%

Inverse functions of polynomials and its applications to initialize the search of solutions of polynomials and polynomial systems

Moreno

Sáiz

2011

Numer Algor

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We establish a general result on the existence of hypercyclic (resp., transitive, weakly mixing, mixing, frequently hypercyclic) polynomials on locally convex spaces. As a consequence we prove that every (real or complex) infinite-dimensional separable Frèchet space admits mixing (hence hypercyclic) polynomials of arbitrary positive degree. Moreover, every complex infinitedimensional separable Banach space with an unconditional Schauder decomposition and every complex Frèchet space with an unconditional basis support chaotic and frequently hypercyclic polynomials of arbitrary positive degree. We also study distributional chaos for polynomials and show that every infinite-dimensional separable Banach space supports polynomials of arbitrary positive degree that have a dense distributionally scrambled linear manifold.

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Chaos for Cowen-Douglas operators

Abstract: Abstract. In this article, we provide a sufficient condition which gives Devaney chaos and distributional chaos for Cowen-Douglas operators. In fact, we obtain a distributionally chaotic criterion for bounded linear operators on Banach spaces.

Cited by 30 publications

References 19 publications

Li–Yorke and distributionally chaotic operators

Li–Yorke and distributionally chaotic operators

Distributional chaos for linear operators

Inverse functions of polynomials and its applications to initialize the search of solutions of polynomials and polynomial systems

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