A note on J-sets of linear operators

Azimi, Mohammad Reza; Müller, Veronika

doi:10.1007/s13398-011-0042-6

Cited by 8 publications

(17 citation statements)

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“…Proof. We prove at first that (1) implies (2). Let x ∈ U and y ∈ V and U, V be nonempty relatively open subsets of M .…”

Section: For Every Non-empty Openmentioning

confidence: 97%

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$J$-Class Semigroup Operators

Tajmouati¹,

Berrag²

2016

Int. J. of Pure and Appl. Math.

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A C0-semigroup T = (Tt) t≥0 on an infinite-dimensional separable complex Banach space X is called subspace-hypercyclic for a subspace M, if Orb(T , x) M is dense in M for a vector x ∈ M . In this paper, we localize the notion of M-extended semigroup(resp.M-extended semigroup mixing) limit set of x under T and We give sufficient conditions of being M -hypercyclic for this semigroup. Then by this result, we prove that (T −1 t ) t≥0 is a M -hypercyclic. This result is an answer of the question of B. F. Madore and R. A. Martnez-Avendano for C0-semigroup.

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“…Proof. We prove at first that (1) implies (2). Let x ∈ U and y ∈ V and U, V be nonempty relatively open subsets of M .…”

Section: For Every Non-empty Openmentioning

confidence: 97%

“…It is not difficult to show that T is topologically transitive if and only if J(x) = X for every x ∈ X and that T is topologically mixing if, and only if J mix (x) = X for every x ∈ X, see [6]. For more information on the J-class set, see [2], [5], [6].…”

Section: Introductionmentioning

confidence: 99%

$J$-Class Semigroup Operators

Tajmouati¹,

Berrag²

2016

Int. J. of Pure and Appl. Math.

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“…They have also posed several open problems which one of them was answered in [2]. In particular they have studied J-class weighted shifts in [7].…”

Section: Introductionmentioning

confidence: 99%

J-Class Sequences of Linear Operators

Azimi

2017

Complex Anal. Oper. Theory

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In this paper we first introduce the extended limit set J {T n } (x) for a sequence of bounded linear operators {T n } ∞ n=1 on a separable Banach space X . Then we study the dynamics of sequence of linear operators by using the extended limit set. It is shown that the extended limit set is strongly related to the topologically transitive of a sequence of linear operators. Finally we show that a sequence of operators {T n } ∞ n=1 ⊆ B(X) is hypercyclic if and only if there exists a cyclic vector x ∈ X such that J {T n } (x) = X.

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“…Recall that T is called a J-class operator if for every non-zero vector x in X and for every open neighborhood U ⊂ X of x and every non-empty open set V ⊂ X, there exists a positive integer n such that T n (U) ∩V = / 0. Recently, there are much research that study or use J-class operators, We mention for instance, the series of papers by Azimi and Müller [3], Nasseri [10], Chan and Seceleanu [5]. Among properties, there are locally hypercyclic, non hypercyclic operators and that finite dimensional Banach spaces do not admit locally hypercyclic operators (see [7]).…”

Section: Introductionmentioning

confidence: 99%

J-class abelian semigroups of matrices on ℝⁿ

Marzougui

2017

Applied Mathematics and Nonlinear Sciences

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We establish, for finitely generated abelian semigroups G of matrices on ℝn, and by using the extended limit sets (the J-sets), the following equivalence analogous to the complex case: (i) G is hypercyclic, (ii) JG(vη) = ℝn for some vector vη given by the structure of G, (iii) G(vη) = ℝn. This answer a question raised by the author. Moreover we construct for any n = 2 an abelian semigroup G of GL(n, ℝ) generated by n + 1 diagonal matrices which is locally hypercyclic (or J-class) but not hypercyclic and such that JG(ek) = ℝn for every k = 1,…, n, where (e1,…, en) is the canonical basis of ℝn. This gives a negative answer to a question raised by Costakis and Manoussos

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A note on J-sets of linear operators

Abstract: We construct a Banach space operator T ∈ B(X ) such that the set J T (0) has a nonempty interior but J T (0) = X . This gives a negative answer to a problem raised by Costakis and Manoussos (J. Oper. Theory [in press], 2011).

Cited by 8 publications

References 1 publication

$J$-Class Semigroup Operators

$J$-Class Semigroup Operators

J-Class Sequences of Linear Operators

J-class abelian semigroups of matrices on ℝⁿ

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A note on J-sets of linear operators

Abstract: We construct a Banach space operator T ∈ B(X ) such that the set J T (0) has a nonempty interior but J T (0) = X . This gives a negative answer to a problem raised by Costakis and Manoussos (J. Oper. Theory [in press], 2011).

Cited by 8 publications

References 1 publication

$J$-Class Semigroup Operators

$J$-Class Semigroup Operators

J-Class Sequences of Linear Operators

J-class abelian semigroups of matrices on ℝ n

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Product

Resources

About

J-class abelian semigroups of matrices on ℝⁿ