Ergodic Characterizations of Reflexivity of Banach Spaces

Fonf, V. P.; Lin, Michael; Wojtaszczyk, Przemysław

doi:10.1006/jfan.2001.3806

Cited by 32 publications

(55 citation statements)

References 18 publications

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“…The last Section contains the main result of the paper which gives examples of non-reflexive Fréchet spaces so that all contractively power bounded operators are mean ergodic. Combining these results with [6] we see that the classes of reflexive and non-reflexive Fréchet spaces are, in a sense, close to each other.…”

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confidence: 57%

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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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confidence: 57%

“…In 1939 Lorch proved that all reflexive Banach spaces are mean ergodic and raised the question of whether the converse holds. The affirmative answer for spaces with bases is given in [6]. For more informations on that topic see [5, Chapter VIII, Section 4] and [13,Chapter 2].…”

Section: Preliminariesmentioning

confidence: 99%

Quasi-reflexive Fréchet spaces and contractively power bounded operators

Piszczek

2010

Arch. Math.

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“…Observe now that formula (2.5) in [FLW01] shows that the power-bounded operator T is in general not a contraction. Define a norm on X by…”

Section: Lemma 35 a Power-bounded Operator Is Mean Ergodic If And Omentioning

confidence: 99%

“…To this purpose, we introduce in Section 2 a semigroup that in turn permits us to prove an ergodic characterization of finite-dimensional Banach spaces that is the semigroup analogue of [FLW01,Cor. 3].…”

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A semigroup analogue of the Fonf–Lin–Wojtaszczyk ergodic characterization of reflexive Banach spaces with a basis

Mugnolo¹

2004

Studia Math.

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Abstract. In analogy to a recent result by V. Fonf, M. Lin, and P. Wojtaszczyk, we prove the following characterizations of a Banach space X with a basis.(i) X is finite-dimensional if and only if every bounded, uniformly continuous, mean ergodic semigroup on X is uniformly mean ergodic. (ii) X is reflexive if and only if every bounded strongly continuous semigroup is mean ergodic if and only if every bounded uniformly continuous semigroup on X is mean ergodic.

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“…There is a classical theory of mean ergodic operators which goes back to fundamental papers of Yosida and Hille especially in the Banach case; cf. [20] and [29]. For more details on the locally convex theory see [2,3,41] and the references therein.…”

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confidence: 99%

Power bounded composition operators on spaces of analytic functions

Bonet

Domański

2010

Collect. Math.

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We study the dynamical behaviour of composition operators defined on spaces of real analytic functions. We characterize when such operators are power bounded, i.e. when the orbits of all the elements are bounded. In this case this condition is equivalent to the composition operator being mean ergodic. In particular, we show that the composition operator is power bounded on the space of real analytic functions on if and only if there is a basis of complex neighbourhoods U of such that the operator is an endomorphism on the space of holomorphic functions on each U .

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Ergodic Characterizations of Reflexivity of Banach Spaces

Cited by 32 publications

References 18 publications

Quasi-reflexive Fréchet spaces and contractively power bounded operators

Quasi-reflexive Fréchet spaces and contractively power bounded operators

A semigroup analogue of the Fonf–Lin–Wojtaszczyk ergodic characterization of reflexive Banach spaces with a basis

Power bounded composition operators on spaces of analytic functions

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