Limits of Weakly Hypercyclic and Supercyclic Operators

Prǎjiturǎ, Gabriel T.

doi:10.1017/s0017089505002466

Cited by 10 publications

(8 citation statements)

References 4 publications

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“…For instance, as a consequence of Theorem 2.1 we can show that a bounded operator T , on a separable Banach space X is weakly hypercyclic if and only if αT is weakly hypercyclic for every α of modulus 1. This result complements some spectral properties of weakly hypercyclic operators discovered in [15].…”

Section: Introduction and Main Resultssupporting

confidence: 84%

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On weak positive supercyclicity

León-Saavedra

Piqueras-Lerena

2008

Isr. J. Math.

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A bounded linear operator T on a separable complex Banach space X is called weakly supercyclic if there exists a vector x ∈ X such that the projective orbit {λT n x : n ∈ N λ ∈ C} is weakly dense in X. Among other results, it is proved that an operator T such that σp(T ) = ∅, is weakly supercyclic if and only if T is positive weakly supercyclic, that is, for every supercyclic vector x ∈ X, only considering the positive projective orbit: {rT n x : n ∈ N, r ∈ R + } we obtain a weakly dense subset in X. As a consequence it is established the existence of non-weakly supercyclic vectors (non-trivial) for positive operators defined on an infinite dimensional separable complex Banach space. The paper is closed with concluding remarks and further directions.

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Section: Introduction and Main Resultssupporting

confidence: 84%

“…As a consequence we obtain the following corollary which complements the spectral properties of weakly hypercyclic operators discovered in [15]. Corollary 2.3: Let T be a weakly hypercyclic operator defined on a complex Banach space X.…”

Section: Theorem 22supporting

confidence: 59%

On weak positive supercyclicity

León-Saavedra

Piqueras-Lerena

2008

Isr. J. Math.

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“…[12,14], in locally convex spaces [8,15,16]) or supercyclicity (e.g. [3], in weak topology [16]). Notice that there are slight differences between real and complex spaces in the characterizations of supercyclicity and positive supercyclicity.…”

Section: Theorem 1 Let S Be a Bounded Linear Operator On Xmentioning

confidence: 99%

“…[4,5,7,15]. The papers [6,9,11,16,17] are particularly devoted to studies of hypercyclicity and supercyclicity in weak topology.…”

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confidence: 99%

Spectral characterization of weak topological transitivity

Desch

Schappacher

2011

RACSAM

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“…We refer to the survey [18] for the details. Weak supercyclicity was introduced by Sanders [24] and studied in, for instance, [14,15,20,22,25,27]. Gallardo and Montes [8], answering a question raised by Salas, demonstrated that the Volterra operator…”

Section: C(t ) = {S ∈ L(x) : [T S] = 0}mentioning

confidence: 99%

Operators Commuting with the Volterra Operator are not Weakly Supercyclic

Shkarin

2010

Integr. Equ. Oper. Theory

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We prove that any bounded linear operator on L p [0, 1] for 1 p < ∞, commuting with the Volterra operator V , is not weakly supercyclic, which answers affirmatively a question raised by Léon-Saavedra and Piqueras-Lerena. It is achieved by providing an algebraic flavored condition on an operator which prevents it from being weakly supercyclic and is satisfied for any operator commuting with V . MSC: 47A16, 37A25

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Limits of Weakly Hypercyclic and Supercyclic Operators

Abstract: Abstract. We give a spectral characterization of the norm closure of the class of all weakly hypercyclic operators on a Hilbert space. Analogous results are obtained for weakly supercyclic operators.2000 Mathematics Subject Classification. Primary 47A16. Secondary 47B37.

Cited by 10 publications

References 4 publications

On weak positive supercyclicity

On weak positive supercyclicity

Spectral characterization of weak topological transitivity

Operators Commuting with the Volterra Operator are not Weakly Supercyclic

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