Disjoint mixing operators

Bès, Juan; Martin, Özgür; Peris, Alfredo; Shkarin, Stanislav

doi:10.1016/j.jfa.2012.05.018

Cited by 65 publications

(53 citation statements)

References 25 publications

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“…We refer the readers to section 4 in [7] and chapter 4 in [12], respectively. The papers by Bès et al [3,4,5,7,6], Salas [16,17], and Shkarin [18] further explored different aspects of disjoint hypercyclicity. In our paper, we mainly investigate the weaker property-disjoint supercyclic of weighted shifts acting on the weighed sequence spaces l 2 (N, w), l 2 (Z, w) , c 0 (N, w), and c 0 (Z, w).…”

Section: Remark 12 the Supercyclicity Criterion Is A Sufficient Condmentioning

confidence: 99%

Disjoint supercyclic powers of weighted shifts on weighted sequence spaces

Liang¹,

Zhou²

2014

Turk J Math

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Section: Remark 12 the Supercyclicity Criterion Is A Sufficient Condmentioning

confidence: 99%

Disjoint supercyclic powers of weighted shifts on weighted sequence spaces

Liang¹,

Zhou²

2014

Turk J Math

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“…3.7]. Interesting constructions and counterexamples have been given in the framework of certain subspaces of analytic functions, see for instance [10,40,14]. Godefroy & Shapiro also considered Hardy and Bergman spaces for studying the dynamics of shift operators, see [28].…”

Section: Introductionmentioning

confidence: 99%

Chaotic semigroups from second order partial differential equations

Conejero

Lizama

Murillo‐Arcila

2017

Journal of Mathematical Analysis and Applications

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Abstract. We give general conditions on given parameters to ensure Devaney and distributional chaos for the solution C 0 -semigroup corresponding to a class of second-order partial differential equations. We also provide a critical parameter that led us to distinguish between stability and chaos for these semigroups. In the case of chaos, we prove that the C 0 -semigroup admits a strongly mixing measure with full support. We also give concrete examples of partial differential equations, such as the telegraph equation, whose solutions satisfy these properties.

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“…Now, let B w given by proposition 59, then B w is mixing on l p (Z + ) and does not satisfy property P BD . Finally take into account the following fact proved in (14): B w is mixing if and only if B w ⊕ B 2 w ⊕ ... ⊕ B r w is mixing, for any r ∈ N, thus the converse of proposition 57 does not hold.…”

Section: Recurrence Properties Defined Via Essential Idempotents Of βNmentioning

confidence: 99%

“…The case when F is the family of cofinite sets, i.e., d-mixing tuples of operators, has been studied in (14) . Let X = c 0 or l p , 1 ≤ p < ∞ and let B w be a weighted backward shift on X.…”

Section: Disjoint Hypercyclicity Along Filtersmentioning

confidence: 99%

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Recurrence in Linear Dynamics

Dios¹

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This thesis is divided into four chapters. In the first one, we give some preliminaries by mentioning some definitions and known results that will be of great help later.In chapter 2, we introduce a refinement of the notion of hypercyclicity, relative to the set N (U, V ) = {n ∈ N : T −n U ∩ V = ∅} when belonging to a certain collection F of subsets of N, namely a bounded and linear operator T is called F -operator if N (U, V ) ∈ F , for any pair of non-empty open sets U, V in X. First, we do an analysis of the hierarchy established between F -operators, whenever F covers those families mostly studied in Ramsey theory. Second, we investigate which kind of properties of density can the sets N (x, U ) = {n ∈ N : T n x ∈ U } and N (U, V ) have for a given hypercyclic operator, and classify the hypercyclic operators accordingly to these properties.In chapter three, we introduce the following notion: an operator T on X satisfies property P F if for any U non-empty open set in X, there exists x ∈ X such that N (x, U ) ∈ F . Let BD the collection of sets in N with positive upper Banach density. We generalize the main result of (19) using a strong result of Bergelson and Mccutcheon (10) in the vein of Szemerédi's theorem, leading us to a characterization of those operators satisfying property P BD . It turns out that operators having property P BD satisfy a kind of recurrence described in terms of essential idempotents of βN (the Stone-Čech compactification of N). We will discuss the case of weighted backward shifts satisfying property P BD . On the other hand, as a consequence we obtain a characterization of reiteratively hypercyclic operators, i.e. operators for which there exists x ∈ X such that for any U non-empty open set in X, the set N (x, U ) ∈ BD. We also, investigate the relationship between reiteratively hypercyclic operators and d-F tuples, for filters F contained in the family of syndetic sets. Finally, we examine conditions to impose in order to get reiterative hypercyclicity from syndeticity in the weighted shift frame. ResumenUn operador lineal y acotado se dice hipercíclico si existe un vector cuya órbita es densa. El primer ejemplo de operador hipercíclico sobre un espacio de Banach fue dado por Rolewicz en 1969, quien prueba que B es hipercíclico si y sólo si |λ| > 1, para B operador desplazamiento unilateral en l 2 . Entre los primeros resultados vinculados a la hiperciclicidad podríamos mencionar el hecho que ningún espacio finito dimensional puede soportar un operador hipercíclico y que en el contexto de los espacios de Hilbert, todo operador hipercíclico tiene un conjunto G δ -denso de vectores hipercíclicos.La tesis está dividida en cuatro capítulos. En el primero, se dan algunos preliminares, repasando aquellas definiciones y resultados ya existentes en la literatura que nos serán necesarios más adelante.En el capítulo dos, introducimos un refinamiento del conceptoT −n U ∩ V = ∅}, cuando éste pertenece a una cierta colección de subconjuntos de N. En otras palabras, un operador lineal y continuo ...

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Disjoint mixing operators

Cited by 65 publications

References 25 publications

Disjoint supercyclic powers of weighted shifts on weighted sequence spaces

Disjoint supercyclic powers of weighted shifts on weighted sequence spaces

Chaotic semigroups from second order partial differential equations

Recurrence in Linear Dynamics

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