scholarly journals The Dunford-Pettis property on vector-valued continuous and bounded functions

1993 ◽  
Vol 48 (2) ◽  
pp. 303-311 ◽  
Author(s):  
Jose Aguayo ◽  
Jose Sanchez
Keyword(s):  
Banach Space ◽  
Regular Space ◽  
Weakly Compact ◽  
Completely Regular ◽  
Bounded Functions ◽  
Continuous Operators ◽  
Vector Valued

Let X be a completely regular space, E a Banach space, Cb(X, E) the space of all continuous, bounded and E-valued functions defined on X, M(X, L(E, F)) the space of all L(E, F)-valued measures defined on the algebra generated by zero subsets of X. Weakly compact and β0-continuous operators defined from Cb(X, E) into a Banach space F are represented by integrals with respect to L(E, F)-valued measures. The strict Dunford-Pettis and the Dunford-Pettis properties are established on (Cb(X, E), βi), where βi denotes one of the strict topologies β0, β or β1, when E is a Schur space; the same properties are established on (Cb(X, E), β0), when E is an AM-space or an AL-space.

scholarly journals Weakly compact operators and the strict topologies

1989 ◽  
Vol 39 (3) ◽  
pp. 353-359 ◽  
Author(s):  
José Aguayo ◽  
José Sánchez
Keyword(s):  
Banach Space ◽  
Compact Operators ◽  
Continuous Functions ◽  
Supremum Norm ◽  
Regular Space ◽  
Weakly Compact ◽  
Completely Regular ◽  
Weakly Compact Operators ◽  
Bounded Continuous Functions

Let X be a completely regular space. We denote by Cb(X) the Banach space of all real-valued bounded continuous functions on X endowed with the supremum-norm.In this paper we prove some characterisations of weakly compact operators defined from Cb(X) into a Banach space E which are continuous with respect to fit, βt, βr and βσ introduced by Sentilles.We also prove that (Cb,(X), βi), i = t, τσ , has the Dunford-Pettis property.

scholarly journals Operators on Spaces of Bounded Vector-Valued Continuous Functions with Strict Topologies

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Marian Nowak
Keyword(s):  
Hausdorff Space ◽  
Continuous Functions ◽  
Linear Operators ◽  
Representation Theorems ◽  
Completely Regular ◽  
General Integral ◽  
Continuous Operators ◽  
Bounded Continuous Functions ◽  
Vector Valued ◽  
Continuous Linear Operators

LetXbe a completely regular Hausdorff space, and let(E,‖·‖E)and(F,‖·‖F)be Banach spaces. LetCb(X,E)be the space of allE-valued bounded, continuous functions defined onX, equipped with the strict topologiesβz, where  z=σ,∞,p,τ,t. General integral representation theorems of(βz,‖·‖F)-continuous linear operators  T:Cb(X,E)→F  with respect to the corresponding operator-valued measures are established. Strongly bounded and(βz,‖·‖F)-continuous operatorsT:Cb(X,E)→Fare studied. We extend to “the completely regular setting” some classical results concerning operators on the spacesC(X,E)andCo(X,E), whereX  is a compact or a locally compact space.

scholarly journals Some Classes of Continuous Operators on Spaces of Bounded Vector-Valued Continuous Functions with the Strict Topology

2015 ◽  
Vol 2015 ◽  
pp. 1-13
Author(s):  
Marian Nowak
Keyword(s):  
Continuous Functions ◽  
Linear Operators ◽  
Strict Topology ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Completely Regular ◽  
Strictly Singular ◽  
Continuous Operators ◽  
Bounded Continuous Functions ◽  
The Relationship

LetXbe a completely regular Hausdorff space and letE,·Eand(F,·F)be Banach spaces. LetCb(X,E)be the space of allE-valued bounded, continuous functions onX, equipped with the strict topologyβσ. We study the relationship between important classes of(βσ,·F)-continuous linear operatorsT:Cb(X,E)→F(strongly bounded, unconditionally converging, weakly completely continuous, completely continuous, weakly compact, nuclear, and strictly singular) and the corresponding operator measures given by Riesz representing theorems. Some applications concerning the coincidence among these classes of operators are derived.

scholarly journals On approximation in weighted spaces of continuous vector-valued functions

1987 ◽  
Vol 29 (1) ◽  
pp. 65-68 ◽  
Author(s):  
Liaqat Ali Khan
Keyword(s):  
Continuous Functions ◽  
Weighted Spaces ◽  
Regular Space ◽  
Local Convexity ◽  
Range Space ◽  
Completely Regular ◽  
Spaces Of Continuous Functions ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex

The fundamental work on approximation in weighted spaces of continuous functions on a completely regular space has been done mainly by Nachbin ([5], [6]). Further investigations have been made by Summers [10], Prolla ([7], [8]), and other authors (see the monograph [8] for more references). These authors considered functions with range contained in the scalar field or a locally convex topological vector space. In the present paper we prove some approximation results without local convexity of the range space.

scholarly journals Vector-valued holomorphic and harmonic functions

2016 ◽  
Vol 3 (1) ◽  
pp. 68-76
Author(s):  
Wolfgang Arendt
Keyword(s):  
Banach Space ◽  
Convergence Theorem ◽  
Harmonic Functions ◽  
Dual Space ◽  
Holomorphic Functions ◽  
Bounded Functions ◽  
Vector Valued ◽  
Separating Subspace

AbstractHolomorphic and harmonic functions with values in a Banach space are investigated. Following an approach given in a joint article with Nikolski [4] it is shown that for bounded functions with values in a Banach space it suffices that the composition with functionals in a separating subspace of the dual space be holomorphic to deduce holomorphy. Another result is Vitali’s convergence theorem for holomorphic functions. The main novelty in the article is to prove analogous results for harmonic functions with values in a Banach space.

scholarly journals A Riesz representation theory for completely regular Hausdorff spaces and its applications

2016 ◽  
Vol 14 (1) ◽  
pp. 474-496 ◽  
Author(s):  
Marian Nowak
Keyword(s):  
Representation Theory ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Weakly Compact ◽  
Completely Regular ◽  
Riesz Representation ◽  
Continuous Operators ◽  
Bounded Continuous Functions ◽  
The Relationship ◽  
Stieltjes Type

AbstractLet X be a completely regular Hausdorff space, E and F be Banach spaces. Let Cb(X, E) be the space of all E-valued bounded, continuous functions on X, equipped with the strict topology β. We develop the Riemman-Stieltjes-type Integral representation theory of (β, || · ||F) -continuous operators T : Cb(X, E) → F with respect to the representing Borel operator measures. For X being a k-space, we characterize strongly bounded (β, || · ||F)-continuous operators T : Cb(X, E) → F. As an application, we study (β, || · ||F)-continuous weakly compact and unconditionally converging operators T : Cb(X, E) → F. In particular, we establish the relationship between these operators and the corresponding Borel operator measures given by the Riesz representation theorem. We obtain that if X is a k-spaceand E is reflexive, then (Cb(X, E), β) has the V property of Pełczynski.

scholarly journals Uniform Convergence and Spectra of Operators in a Class of Fréchet Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Angela A. Albanese ◽  
José Bonet ◽  
Werner J. Ricker
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Bounded Operator ◽  
Operator Norm ◽  
Ergodic Theorems ◽  
Fréchet Spaces ◽  
Norm Convergence ◽  
Operator Ideals ◽  
Weakly Compact ◽  
Continuous Operators

Well-known Banach space results (e.g., due to J. Koliha and Y. Katznelson/L. Tzafriri), which relate conditions on the spectrum of a bounded operatorTto the operator norm convergence of certain sequences of operators generated byT, are extended to the class of quojection Fréchet spaces. These results are then applied to establish various mean ergodic theorems for continuous operators acting in such Fréchet spaces and which belong to certain operator ideals, for example, compact, weakly compact, and Montel.

scholarly journals Convergence theorem of Pettis integrable multivalued pramart

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
M'Hamed El-Louh ◽  
Mohammed El Allali ◽  
Fatima Ezzaki
Keyword(s):  
Banach Space ◽  
Separable Banach Space ◽  
Almost Sure Convergence ◽  
Stopping Times ◽  
Weakly Compact ◽  
Content Type ◽  
Convergence Results ◽  
The Family ◽  
Vector Valued ◽  
Generally True

PurposeIn this work, the authors are interested in the notion of vector valued and set valued Pettis integrable pramarts. The notion of pramart is more general than that of martingale. Every martingale is a pramart, but the converse is not generally true.Design/methodology/approachIn this work, the authors present several properties and convergence theorems for Pettis integrable pramarts with convex weakly compact values in a separable Banach space.FindingsThe existence of the conditional expectation of Pettis integrable mutifunctions indexed by bounded stopping times is provided. The authors prove the almost sure convergence in Mosco and linear topologies of Pettis integrable pramarts with values in (cwk(E)) the family of convex weakly compact subsets of a separable Banach space.Originality/valueThe purpose of the present paper is to present new properties and various new convergence results for convex weakly compact valued Pettis integrable pramarts in Banach space.

scholarly journals Separable measures and the Dunford-Pettis property

1991 ◽  
Vol 43 (3) ◽  
pp. 423-428 ◽  
Author(s):  
José Aguayo ◽  
José Sánchez
Keyword(s):  
Banach Space ◽  
Compact Space ◽  
Compact Operators ◽  
Continuous Functions ◽  
Regular Space ◽  
Weakly Compact ◽  
Weakly Compact Operators ◽  
Bounded Continuous Functions ◽  
Locally Convex Topology ◽  
Locally Convex

Let X be a complete regular space. We denote by Cb(X) the Banach space of all real-valued bounded continuous functions on X endowed with the supremumnorm.In this paper we give a characterisation of weakly compact operators defined from Cb(X) into a Banach space E which are β∞-continuous, where β∞ is a locally convex topology on Cb(X) introduced by Wheeler. We also prove that (Cb(X), β∞) has the strict Dunford-Pettis property and, if X is a σ-compact space, (Cb(X), β∞), has the Dunford-Pettis property.

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