Quasi-reflexive Fréchet spaces and mean ergodicity

Piszczek, Krzysztof

doi:10.1016/j.jmaa.2009.08.060

Cited by 13 publications

(16 citation statements)

References 17 publications

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“…Theorem 4.2). For recent results on mean ergodic operators in lcHs' we refer to [4], [5], [6], [7], [28], [29], for example, and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On the continuous Cesàro operator in certain function spaces

2015

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Abstract. Various properties of the (continuous) Cesàro operator C, acting on Banach and Fréchet spaces of continuous functions and L p -spaces, are investigated. For instance, the spectrum and point spectrum of C are completely determined and a study of certain dynamics of C is undertaken (eg. hyperand supercyclicity, chaotic behaviour). In addition, the mean (and uniform mean) ergodic nature of C acting in the various spaces is identied.

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“…Theorem 4.2). For recent results on mean ergodic operators in lcHs' we refer to [4], [5], [6], [7], [28], [29], for example, and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On the continuous Cesàro operator in certain function spaces

2015

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“…So, Fix (S(·)) separates Fix (S(·) ), i.e., Ker(A−I) separates Ker(A − I). An examination of the proof of Sine's Theorem for an individual operator in a Banach space, as given in [21, p.74], shows it is based purely on a duality argument and so carries over to lcHs', [27,Theorem 13], thereby allowing us to conclude that X = Ker(A − I) ⊕ Im(A − I). As A is power bounded, we have Im(A − I) = {x ∈ X : lim n→∞ A [n] x = 0}, [33, p.213].…”

Section: Proof (I)⇒(ii) As a Reflexive Space Is Semi-reflexive Coromentioning

confidence: 99%

“…As D ∈ B(X β ) and (T (s) ) s≥0 is a C 0 -semigroup in X β = (X β ) β , it follows lim n→∞ sup u∈D | u, (T (s n ) −I)x | = 0, i.e., (27) holds.…”

Section: Semigroups and Mean Ergodicity In Gdp-spacesmentioning

confidence: 99%

“…So, there is an interest in extending the Banach space results for mean ergodicity to this setting. For individual operators (especially Banach space results related to the theory of bases, [13]) this has been carried out in recent years, [1], [3], [4], [6], [7], [27], [28]. For certain aspects of the theory of mean ergodic semigroups of operators in the non-normable setting we refer to [9], [21,Ch.2], [31,Ch.III,§7], and the references therein.…”

Section: Introductionmentioning

confidence: 99%

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Mean ergodic semigroups of operators

Albanese

Bonet

Ricker

2011

RACSAM

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“…By Proposition 2.3 we may assume that T = I + R with R reflexive. Since by [16,Theorem 4.8] T or T is mean ergodic, we may assume that T is mean ergodic. If F (T ) = {0} then X = (I − T )X (use Hahn-Banach) and by [1, Theorem 2.4] T is mean ergodic.…”

Section: Lemma 43 Let U Be a Z-selection On X And Fixmentioning

confidence: 99%

Quasi-reflexive Fréchet spaces and contractively power bounded operators

Piszczek

2010

Arch. Math.

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Abstract. In this paper we introduce a new class of operators acting on a locally convex space. We show that for some Fréchet spaces all these operators are mean ergodic. This leads to the conclusion that the classes of reflexive and non-reflexive Fréchet spaces are, in a sense, close to each other. Mathematics Subject Classification (2010). Primary 46A04, 47A35.Keywords. Fréchet space, Quasi-reflexive space, Mean ergodic operator, Power bounded operator. Introduction. In [3] the following definition is introduced: a Banach spaceThis definition arose from the very well known paper of R. C. James who constructed in [9] an example of a non-reflexive Banach space X of codimension 1 in its bidual. Properties of such spaces have been studied in [4,8,10,17,18]. One of the outstanding profits of these spaces is that they have served as examples to settle several conjectures. Recently Fonf, Lin, and Wojtaszczyk have constructed in [7] a quasi-reflexive Banach space of order 1 with all contractions on the space itself and on its dual mean ergodic. Thus answering negatively a question of Sucheston posed in [19].The aim of this paper is to consider (after introducing a reasonable notation) an analogous problem in the Fréchet space setting. Here the notion of quasi-reflexivity is also available since, whenever we start with an arbitrary Fréchet space X, by [15, 25.3 and 25.10] all the spaces π(X), X and X /π(X) are Fréchet. Examples of such spaces without infinite-dimensional Banach subspaces may be found in [14]. Let us recall them for the convenience of the

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Quasi-reflexive Fréchet spaces and mean ergodicity

Cited by 13 publications

References 17 publications

On the continuous Cesàro operator in certain function spaces

On the continuous Cesàro operator in certain function spaces

Mean ergodic semigroups of operators

Quasi-reflexive Fréchet spaces and contractively power bounded operators

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