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We correct a logic mistake in our paper “On statistical convergence and strong Cesàro convergence by moduli” (León-Saavedra et al. in J. Inequal. Appl. 23:298, 2019).
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A Banach space operator $$T$$T is said to be weakly super convex-cyclic if there exists $$x \in X$$x∈X such that $$\{\lambda p(T )x : p\, \mathrm{convex \,polynomial}, \lambda \in \mathbb {C}\}$${λp(T)x:pconvexpolynomial,λ∈C} is weakly... more
A Banach space operator $$T$$T is said to be weakly super convex-cyclic if there exists $$x \in X$$x∈X such that $$\{\lambda p(T )x : p\, \mathrm{convex \,polynomial}, \lambda \in \mathbb {C}\}$${λp(T)x:pconvexpolynomial,λ∈C} is weakly dense in $$X$$X. The notion of convex-cyclicity was introduced recently by Rezaei in Linear Algebra Appl 438(11):4190–4203, (2013). We provide a simple argument, to show that many elements in the commutant of the Volterra operator acting on $$L^p_\mathbb {C}[0,1]$$LCp[0,1] spaces are not weakly super convex-cyclic.
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Пусть $\alpha$ - комплексный скаляр, а $A$ - ограниченный линейный оператор в гильбертовом пространстве $H$. Говорят, что $\alpha$ является расширенным собственным значением оператора $A$, если существует ненулевой ограниченный линейный... more
Пусть $\alpha$ - комплексный скаляр, а $A$ - ограниченный линейный оператор в гильбертовом пространстве $H$. Говорят, что $\alpha$ является расширенным собственным значением оператора $A$, если существует ненулевой ограниченный линейный оператор $X$, такой, что выполняется равенство $AX=\alpha XA$. В весовых пространствах Харди, инвариантных относительно автоморфизмов, мы полностью вычисляем расширенные собственные значения операторов композиции, индуцированных дробно-линейными отображениями единичного круга $\mathbb{D}$ в себя с внутренней неподвижной точкой в $\mathbb{D}$ и еще одной неподвижной точкой вне $\overline{\mathbb{D}}$. К таким классам преобразований относятся эллиптическое и локсодромное отображения, а также гиперболическое неавтоморфное отображение.
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A continuous linear operator on a Fréchet space $$\mathcal {X}$$ X is frequently hypercyclic if there exists a vector x such that for any nonempty open subset $$U\subset \mathcal {X}$$ U ⊂ X the set of $$n\in \mathbb {N}\cup \{0\}$$ n ∈ N... more
A continuous linear operator on a Fréchet space $$\mathcal {X}$$ X is frequently hypercyclic if there exists a vector x such that for any nonempty open subset $$U\subset \mathcal {X}$$ U ⊂ X the set of $$n\in \mathbb {N}\cup \{0\}$$ n ∈ N ∪ { 0 } for which $$T^nx\in U$$ T n x ∈ U has a positive lower density. Here we determine when an operator that commutes up to a factor with the differentiation operator D, defined on the space of entire functions, is frequently hypercyclic.
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In this paper we will characterize the completeness and barrelledness of a normed space through the strong p-Ces?ro summability of series. A new characterization of weakly unconditionally Cauchy series and unconditionally convergent... more
In this paper we will characterize the completeness and barrelledness of a normed space through the strong p-Ces?ro summability of series. A new characterization of weakly unconditionally Cauchy series and unconditionally convergent series through the strong p-Ces?ro summability is obtained.
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In this paper we study the cyclic properties of the infinite continuous Cesàro operator defined on L 2 ( 0 , ∞ ) L^2(0,\infty ) by ( C ∞ f ) ( x ) = 1 x ∫ 0 x f ( s ) d s (C_\infty f)(x)=\frac {1}{x}\int _0^x f(s) ds . Despite this... more
In this paper we study the cyclic properties of the infinite continuous Cesàro operator defined on L 2 ( 0 , ∞ ) L^2(0,\infty ) by ( C ∞ f ) ( x ) = 1 x ∫ 0 x f ( s ) d s (C_\infty f)(x)=\frac {1}{x}\int _0^x f(s) ds . Despite this operator being cyclic, we show that it is not supercyclic; even more, it is not weakly supercyclic. These results complement some recent ones on the cyclic behavior of Cesàro operators.
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Fernando León-Saavedra Notes about the hypercyclicity criterion ... Persistent URL: http://dml.cz/dmlcz/136887 ... © Mathematical Institute of the Slovak Academy of Sciences, 2003 ... Institute of Mathematics of the Academy of Sciences of... more
Fernando León-Saavedra Notes about the hypercyclicity criterion ... Persistent URL: http://dml.cz/dmlcz/136887 ... © Mathematical Institute of the Slovak Academy of Sciences, 2003 ... Institute of Mathematics of the Academy of Sciences of the Czech Republic provides ...
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We present some recent results related with supercyclic operators, also some of its consequences. We will finalize with new related questions.
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In this note, we prove a Schur-type lemma for bounded multiplier series. This result allows us to obtain a unified vision of several previous results, focusing on the underlying structure and the properties that a summability method must... more
In this note, we prove a Schur-type lemma for bounded multiplier series. This result allows us to obtain a unified vision of several previous results, focusing on the underlying structure and the properties that a summability method must satisfy in order to establish a result of Schur’s lemma type.
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In this manuscript we characterize the completeness of a normed space through the strong lacunary ( N θ ) and lacunary statistical convergence ( S θ ) of series. A new characterization of weakly unconditionally Cauchy series through N θ... more
In this manuscript we characterize the completeness of a normed space through the strong lacunary ( N θ ) and lacunary statistical convergence ( S θ ) of series. A new characterization of weakly unconditionally Cauchy series through N θ and S θ is obtained. We also relate the summability spaces associated with these summabilities with the strong p-Cesàro convergence summability space.
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In this paper we will establish a result by Connor, Khan and Orhan (Analysis 8:47–63, 1988; Publ. Math. (Debr.) 76:77–88, 2010) in the framework of the statistical convergence and the strong Cesàro convergence defined by a modulus... more
In this paper we will establish a result by Connor, Khan and Orhan (Analysis 8:47–63, 1988; Publ. Math. (Debr.) 76:77–88, 2010) in the framework of the statistical convergence and the strong Cesàro convergence defined by a modulus function f. Namely, for every modulus function f, we will prove that a f-strongly Cesàro convergent sequence is always f-statistically convergent and uniformly integrable. The converse of this result is not true even for bounded sequences. We will characterize analytically the modulus functions f for which the converse is true. We will prove that these modulus functions are those for which the statistically convergent sequences are f-statistically convergent, that is, we show that Connor–Khan–Orhan’s result is sharp in this sense.
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We aim to unify several results which characterize when a series is weakly unconditionally Cauchy (wuc) in terms of the completeness of a convergence space associated to the wuc series. If, additionally, the space is completed for each... more
We aim to unify several results which characterize when a series is weakly unconditionally Cauchy (wuc) in terms of the completeness of a convergence space associated to the wuc series. If, additionally, the space is completed for each wuc series, then the underlying space is complete. In the process the existing proofs are simplified and some unanswered questions are solved. This research line was originated in the PhD thesis of the second author. Since then, it has been possible to characterize the completeness of a normed spaces through different convergence subspaces (which are be defined using different kinds of convergence) associated to an unconditionally Cauchy sequence.
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Abstract. A vector x in a Hilbert space H is called hypercyclic for a bounded operator T : H → H if the orbit {Tnx : n ≥ 1} is dense in H. Our main result states that if T satisfies the Hypercyclicity Criterion and the essential spectrum... more
Abstract. A vector x in a Hilbert space H is called hypercyclic for a bounded operator T : H → H if the orbit {Tnx : n ≥ 1} is dense in H. Our main result states that if T satisfies the Hypercyclicity Criterion and the essential spectrum intersects the closed unit disk, then there is an ...
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This paper unifies several versions of the Orlicz–Pettis theorem that incorporate summability methods. We show that a series is unconditionally convergent if and only if the series is weakly subseries convergent with respect to a regular... more
This paper unifies several versions of the Orlicz–Pettis theorem that incorporate summability methods. We show that a series is unconditionally convergent if and only if the series is weakly subseries convergent with respect to a regular linear summability method. This includes results using matrix summability, statistical convergence with respect to an ideal, and other variations of summability methods.
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A bounded operatorTon a Banach spaceXis convex cyclic if there exists a vectorxsuch that the convex hull generated by the orbitTnxn≥0is dense inX. In this note we study some questions concerned with convex-cyclic operators. We provide an... more
A bounded operatorTon a Banach spaceXis convex cyclic if there exists a vectorxsuch that the convex hull generated by the orbitTnxn≥0is dense inX. In this note we study some questions concerned with convex-cyclic operators. We provide an example of a convex-cyclic operatorTsuch that the powerTnfails to be convex cyclic. Using this result we solve three questions posed by Rezaei (2013).
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Page 1. SEMI-FREDHOLM THEORY: HYPERCYCLIC AND SUPERCYCLIC SUBSPACES MANUEL GONZA Â LEZ, FERNANDO LEO Â N-SAAVEDRA and ALFONSO MONTES-RODRI Â GUEZ [Received 27 April 1998; revised 28 June 1999] 1. Introduction ...
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1 Departamento de Matemáticas, Universidad de Cádiz, Avda. de la Universidad s/n. 11405, Jerez de la Frontera, Cádiz, Spain 2 Departamento de Matemáticas, Universidad de Cádiz, Sacramento, 82, 11002 Cádiz, Spain 3 Departamento de... more
1 Departamento de Matemáticas, Universidad de Cádiz, Avda. de la Universidad s/n. 11405, Jerez de la Frontera, Cádiz, Spain 2 Departamento de Matemáticas, Universidad de Cádiz, Sacramento, 82, 11002 Cádiz, Spain 3 Departamento de Análisis Matemático, Facultad ...