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A probabilistic version of the Frequent Hypercyclicity Criterion

Sophie Grivaux (2006)

Studia Mathematica

For a bounded operator T on a separable infinite-dimensional Banach space X, we give a "random" criterion not involving ergodic theory which implies that T is frequently hypercyclic: there exists a vector x such that for every non-empty open subset U of X, the set of integers n such that Tⁿx belongs to U, has positive lower density. This gives a connection between two different methods for obtaining the frequent hypercyclicity of operators.

Bounded point evaluations for multicyclic operators

M. EL Guendafi, M. Mbekhta, E. H. Zerouali (2005)

Banach Center Publications

Let T be a multicyclic operator defined on some Banach space. Bounded point evaluations and analytic bounded point evaluations for T are defined to generalize the cyclic case. We extend some known results on cyclic operators to the more general setting of multicyclic operators on Banach spaces. In particular we show that if T satisfies Bishop’s property (β), then a = σ a p ( T ) . We introduce the concept of analytic structures and we link it to different spectral quantities. We apply this concept to retrieve...

Closed universal subspaces of spaces of infinitely differentiable functions

Stéphane Charpentier, Quentin Menet, Augustin Mouze (2014)

Annales de l’institut Fourier

We exhibit the first examples of Fréchet spaces which contain a closed infinite dimensional subspace of universal series, but no restricted universal series. We consider classical Fréchet spaces of infinitely differentiable functions which do not admit a continuous norm. Furthermore, this leads us to establish some more general results for sequences of operators acting on Fréchet spaces with or without a continuous norm. Additionally, we give a characterization of the existence of a closed subspace...

Co-analytic, right-invertible operators are supercyclic

Sameer Chavan (2010)

Colloquium Mathematicae

Let denote a complex, infinite-dimensional, separable Hilbert space, and for any such Hilbert space , let () denote the algebra of bounded linear operators on . We show that for any co-analytic, right-invertible T in (), αT is hypercyclic for every complex α with | α | > β - 1 , where β i n f | | x | | = 1 | | T * x | | > 0 . In particular, every co-analytic, right-invertible T in () is supercyclic.

Common Cesàro hypercyclic vectors

George Costakis (2010)

Studia Mathematica

In this work, which can be seen as a continuation of a paper by Hadjiloucas and the author [Studia Math. 175 (2006)], we establish the existence of common Cesàro hypercyclic vectors for the following classes of operators: (i) multiples of the backward shift, (ii) translation operators and (iii) weighted differential operators. In order to do so, we first prove a version of Ansari's theorem for operators that are hypercyclic and Cesàro hypercyclic simultaneously; then our argument essentially relies...

Dense range perturbations of hypercyclic operators

Luis Bernal-Gonzalez (2002)

Colloquium Mathematicae

We show that if (Tₙ) is a hypercyclic sequence of linear operators on a locally convex space and (Sₙ) is a sequence of linear operators such that the image of each orbit under every linear functional is non-dense then the sequence (Tₙ + Sₙ) has dense range. Furthermore, it is proved that if T,S are commuting linear operators in such a way that T is hypercyclic and all orbits under S satisfy the above non-denseness property then T - S has dense range. Corresponding statements for operators and sequences...

Disjoint hypercyclic operators

Luis Bernal-González (2007)

Studia Mathematica

We introduce the concept of disjoint hypercyclic operators. These are operators performing the approximation of any given vectors with a common subsequence of iterates applied on a common vector. The notion is extended to sequences of operators, and applied to composition operators and differential operators on spaces of analytic functions.

Disjoint hypercyclic powers of weighted translations on groups

Liang Zhang, Hui-Qiang Lu, Xiao-Mei Fu, Ze-Hua Zhou (2017)

Czechoslovak Mathematical Journal

Let G be a locally compact group and let 1 p < . Recently, Chen et al. characterized hypercyclic, supercyclic and chaotic weighted translations on locally compact groups and their homogeneous spaces. There has been an increasing interest in studying the disjoint hypercyclicity acting on various spaces of holomorphic functions. In this note, we will study disjoint hypercyclic and disjoint supercyclic powers of weighted translation operators on the Lebesgue space L p ( G ) in terms of the weights. Sufficient and...

Dynamics of differentiation and integration operators on weighted spaces of entire functions

María J. Beltrán (2014)

Studia Mathematica

We investigate the dynamical behavior of the operators of differentiation and integration and the Hardy operator on weighted Banach spaces of entire functions defined by integral norms. In particular we analyze when they are hypercyclic, chaotic, power bounded, and (uniformly) mean ergodic. Moreover, we estimate the norms of the operators and study their spectra. Special emphasis is put on exponential weights.

Dynamics of differentiation operators on generalized weighted Bergman spaces

Liang Zhang, Ze-Hua Zhou (2015)

Open Mathematics

The chaos of the differentiation operator on generalized weighted Bergman spaces of entire functions has been characterized recently by Bonet and Bonilla in [CAOT 2013], when the differentiation operator is continuous. Motivated by those, we investigate conditions to ensure that finite many powers of differentiation operators are disjoint hypercyclic on generalized weighted Bergman spaces of entire functions.

Frequently hypercyclic semigroups

Elisabetta M. Mangino, Alfredo Peris (2011)

Studia Mathematica

We study frequent hypercyclicity in the context of strongly continuous semigroups of operators. More precisely, we give a criterion (sufficient condition) for a semigroup to be frequently hypercyclic, whose formulation depends on the Pettis integral. This criterion can be verified in certain cases in terms of the infinitesimal generator of the semigroup. Applications are given for semigroups generated by Ornstein-Uhlenbeck operators, and especially for translation semigroups on weighted spaces of...

Frequently hypercyclic translation semigroups

Elisabetta M. Mangino, Marina Murillo-Arcila (2015)

Studia Mathematica

Frequent hypercyclicity for translation C₀-semigroups on weighted spaces of continuous functions is studied. The results are achieved by establishing an analogy between frequent hypercyclicity for translation semigroups and for weighted pseudo-shifts and by characterizing frequently hypercyclic weighted pseudo-shifts on spaces of vanishing sequences. Frequently hypercyclic translation semigroups on weighted L p -spaces are also characterized.

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