A Note on Supercyclic Operators in Locally Convex Spaces

Albanese, Angela A.; Jornet, David

doi:10.1007/s00009-019-1386-y

Cited by 3 publications

(4 citation statements)

References 13 publications

(16 reference statements)

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“…From this result it follows that isometries in Banach spaces are never supercyclic. Albanese and Jornet [1] have recently extended this result for operators in locally convex spaces. Concretely, they show that if E is a locally convex space and T : E → E is a supercyclic operator such that (T n ) n is an equicontinuous sequence in L(E), then (T n (e)) n converges to 0 for any e ∈ E. In particular, this property applies to barrelled spaces, where the condition of equicontinuity of (T n ) n is equivalent to the boundedness of the sequence (T n (e)) n in E for any e ∈ E. As an application of Theorem 8 we get below Ansari-Bourdon type results for weakly supercyclic operators.…”

Section: Weak Supercyclicity On Fréchet Spacesmentioning

confidence: 87%

“…The Ansari-Bourdon theorem mentioned above about the non supercyclicity of isometries on Banach spaces is a consequence of a theorem which asserts that for a supercyclic power bounded operator T defined on a Banach space, the orbits of the adjoint T * are everywhere ω * convergent to 0. For recent research extending these classic results we refer to [1,2,13]. In the last section of the paper we use our results about weighted composition operators to get results for arbitrary operators defined on locally convex spaces, obtaining conditions in the dynamics of the adjoint T ′ which are necessary for T being weakly supercyclic.…”

Section: Introductionmentioning

confidence: 99%

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Supercyclicity of weighted composition operators on spaces of continuous functions

2019

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Our study is focused on the dynamics of weighted composition operators defined on a locally convex space E ֒→ (C(X), τ p ) with X being a topological Hausdorff space containing at least two different points and such that the evaluations {δ x : x ∈ X} are linearly independent in E ′ . We prove, when X is compact and E is a Banach space containing a nowhere vanishing function, that a weighted composition operator C w,ϕ is never weakly supercyclic on E. We also prove that if the symbol ϕ lies in the unit ball of A(D), then every weighted composition operator can never be τ psupercyclic neither on C(D) nor on the disc algebra A(D). Finally, we obtain Ansari-Bourdon type results and conditions on the spectrum for arbitrary weakly supercyclic operators, and we provide necessary conditions for a composition operator to be weakly supercyclic on the space of holomorphic functions defined in non necessarily simply connected planar domains. As a consequence, we show that no composition operator can be weakly supercyclic neither on the space of holomorphic functions on the punctured disc nor in the punctured plane.

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Section: Weak Supercyclicity On Fréchet Spacesmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Supercyclicity of weighted composition operators on spaces of continuous functions

2019

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“…Ergodic dynamical systems seem to be of interest for a few decades, with an increasing number of papers appearing lately (see, for example, [1,2,5,9,14]), a large number of them concerning convex-cyclic operators.…”

Section: Introductionmentioning

confidence: 99%

On Subspace Convex-Cyclic Operators

Woźniak¹,

Ahmed²,

Hama³

et al. 2020

Z. mat. fiz. anal. geom.

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Let H be an infinite dimensional real or complex separable Hilbert space. We introduce a special type of a bounded linear operator T and study its important relation with the invariant subspace problem on H: the operator T is said to be subspace convex-cyclic for a subspace M if there exists a vector whose orbit under T intersects the subspace M in a relatively dense set. We give the sufficient condition for a subspace convex-cyclic transitive operator T to be subspace convex-cyclic. We also give a special type of the Kitai criterion related to invariant subspaces which implies subspace convex-cyclicity. Finally we show a counterexample of a subspace convexcyclic operator which is not subspace convex-cyclic transitive.

show abstract

“…Ergodic dynamical systems seem to be of interest for a few decades, with an increasing number of papers appearing lately (see, for example, [1][2][3][4][5]), a large number of them concerning convex-cyclic operators.…”

Section: Introductionmentioning

confidence: 99%

On subspace convex-cyclic operators

Ahmed,

Hama,

Woźniak

et al. 2019

Preprint

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Let H be an infinite dimensional real or complex separable Hilbert space. We introduce a special type of a bounded linear operator T and its important relation with invariant subspace problem on H: operator T is said to be is subspace convexcyclic for a subspace M, if there exists a vector whose orbit under T intersects the subspace M in a relatively dense set. We give the sufficient condition for a subspace convex-cyclic transitive operator T to be subspace convex-cyclic. We also give a special type of Kitai criterion related to invariant subspaces which implies subspace convex-cyclicity. We conclude showing a counterexample of a subspace convex-cyclic operator which is not subspace convex-cyclic transitive.

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A Note on Supercyclic Operators in Locally Convex Spaces

Cited by 3 publications

References 13 publications

Supercyclicity of weighted composition operators on spaces of continuous functions

Supercyclicity of weighted composition operators on spaces of continuous functions

On Subspace Convex-Cyclic Operators

On subspace convex-cyclic operators

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