Somewhere dense orbits are everywhere dense

Bourdon, Paul S.; Feldman, Nathan S.

doi:10.1512/iumj.2003.52.2303

Cited by 76 publications

(83 citation statements)

References 12 publications

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“…there is x ∈ X such that Orb( x ) = {αT n (x) : n ∈ N, α ∈ K} is dense ( x denotes the linear span of {x}). For locally convex spaces, Peris [8] proved that (in the obvious sense) multi-supercyclic operators are supercyclic and Bourdon and N. Feldman [4] even showed that Orb( x ) is either everywhere dense or nowhere dense for each vector individually. As for the hypercyclic case, local convexity was only used in the proof of the locally convex version of:…”

Section: : Q(y) = |Q(y)| Defines a Continuous And Open Mapmentioning

confidence: 99%

Hypercyclic operators on non-locally convex spaces

Wengenroth¹

2003

Proc. Amer. Math. Soc.

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Abstract. We transfer a number of fundamental results about hypercyclic operators on locally convex spaces (due to Ansari, Bès, Bourdon, Costakis, Feldman, and Peris) to the non-locally convex situation. This answers a problem posed by A. Peris [Multi-hypercyclic operators are hypercyclic, Math. Z. 236 (2001), 779-786].During the past years much research has been done about hypercyclic operators; the article [6] contains a rather complete survey of results until 1999. A (continuous linear) operator T : X → X on a topological vector space X is called hypercyclic if it admits a vector x ∈ X having dense orbit Orb(x) = {x, T x, T 2 x, . . .} (x is then called a hypercyclic vector). The following theorem collects some of the recent fundamental results:Theorem. Let X be a locally convex space and let T : X → X be an operator.( A. Peris asked in [8] whether in (3) local convexity is really needed and we now show that it is indeed not:All parts of the Theorem hold for topological vector spaces. The only place in the proof of the Theorem where local convexity plays a role is the following lemma which, for hypercyclic operators, is due to P. Bourdon [3] (the complex case) and J. Bès [2] (the real case). Our proof for the non-locally convex case is quite similar to their arguments.

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Section: : Q(y) = |Q(y)| Defines a Continuous And Open Mapmentioning

confidence: 99%

Hypercyclic operators on non-locally convex spaces

Wengenroth¹

2003

Proc. Amer. Math. Soc.

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“…However, through Example 3.10 (Example 3.11 and Example 3.12), we show that the adjoint (inverse) of diskcyclic operators need not be diskcyclic. In addition, the somewhere density of the orbit of an operator(the cone generated by orbit) implies the everywhere density of the orbit (cone generated by orbit) [3]. However, we show that the somewhere density of the disk orbit of an operator does not imply to the everywhere density of it, by giving the Counterexample 3.14.…”

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confidence: 69%

“…Also we have It follows from Corollary 2.15 that F is diskcyclic. Moreover, the inverse of F is the bilateral backward weighted shift Be n = (1/w n−1 ) e n−1 with weight sequence Bourdon and Feldman in [3] proved that if Orb(T, x) (or COrb (T, x)) is somewhere dense, then Orb(T, x) (or COrb (T, x) respectively) is everywhere dense set. However, If DOrb(T, x) is somewhere dense then the situation is different.…”

Section: Theorem 31 (Second Diskcyclicity Criterion) Let T ∈ B(h)mentioning

confidence: 99%

A Review of Some Works in the Theory of Diskcyclic Operators

Bamerni

Kılıçman

Noorani

2015

Bull. Malays. Math. Sci. Soc.

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In this paper, we give a brief review concerning diskcyclic operators and then we provide some further characterizations of diskcyclic operators on separable Hilbert spaces. In particular, we show that if x ∈ H has a disk orbit under T that is somewhere dense in H then the disk orbit of x under T need not be everywhere dense in H. We also show that the inverse and the adjoint of a diskcyclic operator need not be diskcyclic. Moreover, we establish another diskcyclicity criterion and use it to find a necessary and sufficient condition for unilateral backward shifts that are diskcyclic operators. We show that a diskcyclic operator exists on a Hilbert space H over the field of complex numbers if and only if dim(H) = 1 or dim(H) = ∞ . Finally we give a sufficient condition for the somewhere density disk orbit to be everywhere dense.

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“…In [8] G. Costakis and A. Manoussos have studied the dynamics of linear operators by making use of extended limit sets instead of their orbits. Especially a result due to P. Bourdon and N. Feldman ( [6]) has been generalized by them as follows.…”

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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Somewhere dense orbits are everywhere dense

Abstract: Abstract. Let T be a continuous linear operator on a locally convex topological vector space X. We show that if x ∈ X has orbit under T that is somewhere dense in X, then the orbit of x under T must be everywhere dense in X, answering a question raised by Alfredo Peris.

Cited by 76 publications

References 12 publications

Hypercyclic operators on non-locally convex spaces

Hypercyclic operators on non-locally convex spaces

A Review of Some Works in the Theory of Diskcyclic Operators

J-Class Sequences of Linear Operators

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