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 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 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 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.

Continuous linear operators on Orlicz-Bochner spaces

2019 ◽  
Vol 17 (1) ◽  
pp. 1147-1155 ◽  
Author(s):  
Marian Nowak
Keyword(s):  
Finite Measure ◽  
Compact Operators ◽  
Linear Operators ◽  
Continuous Linear ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Completely Continuous Operators ◽  
Weakly Compact Operators ◽  
Continuous Operators ◽  
Continuous Linear Operators

Abstract Let (Ω, Σ, μ) be a complete σ-finite measure space, φ a Young function and X and Y be Banach spaces. Let Lφ(X) denote the corresponding Orlicz-Bochner space and $\begin{array}{} \displaystyle \mathcal T^\wedge_\varphi \end{array}$ denote the finest Lebesgue topology on Lφ(X). We examine different classes of ( $\begin{array}{} \displaystyle \mathcal T^\wedge_\varphi \end{array}$, ∥ ⋅ ∥Y)-continuous linear operators T : Lφ(X) → Y: weakly compact operators, order-weakly compact operators, weakly completely continuous operators, completely continuous operators and compact operators. The relationships among these classes of operators are established.

The Strict Topology in a Completely Regular Setting: Relations to Topological Measure Theory

1972 ◽  
Vol 24 (5) ◽  
pp. 873-890 ◽  
Author(s):  
Steven E. Mosiman ◽  
Robert F. Wheeler
Keyword(s):  
Measure Theory ◽  
Continuous Functions ◽  
Topological Spaces ◽  
Strict Topology ◽  
Completely Regular ◽  
Borel Measures ◽  
Topological Measure Theory ◽  
Locally Convex Topologies ◽  
Bounded Continuous Functions ◽  
Locally Convex

Let X be a locally compact Hausdorff space, and let C*(X) denote the space of real-valued bounded continuous functions on X. An interesting and important property of the strict topology β on C*(X) was proved by Buck [2]: the dual space of (C*(X), β) has a natural representation as the space of bounded regular Borel measures on X.Now suppose that X is completely regular (all topological spaces are assumed to be Hausdorff in this paper). Again it seems natural to seek locally convex topologies on the space C*(X) whose dual spaces are (via the integration pairing) significant classes of measures.

scholarly journals Characterizations of continuous operators on $$C_b(X)$$ with the strict topology

2021 ◽  
Vol 12 (2) ◽  
Author(s):  
Marian Nowak ◽  
Juliusz Stochmal
Keyword(s):  
Banach Space ◽  
Hausdorff Space ◽  
Compact Operators ◽  
Continuous Functions ◽  
Linear Operators ◽  
Strict Topology ◽  
Completely Regular ◽  
Absolutely Summing Operators ◽  
Nuclear Operators ◽  
Nikodym Property

AbstractLet X be a completely regular Hausdorff space and $$C_b(X)$$ C b ( X ) be the space of all bounded continuous functions on X, equipped with the strict topology $$\beta $$ β . We study some important classes of $$(\beta ,\Vert \cdot \Vert _E)$$ ( β , ‖ · ‖ E ) -continuous linear operators from $$C_b(X)$$ C b ( X ) to a Banach space $$(E,\Vert \cdot \Vert _E)$$ ( E , ‖ · ‖ E ) : $$\beta $$ β -absolutely summing operators, compact operators and $$\beta $$ β -nuclear operators. We characterize compact operators and $$\beta $$ β -nuclear operators in terms of their representing measures. It is shown that dominated operators and $$\beta $$ β -absolutely summing operators $$T:C_b(X)\rightarrow E$$ T : C b ( X ) → E coincide and if, in particular, E has the Radon–Nikodym property, then $$\beta $$ β -absolutely summing operators and $$\beta $$ β -nuclear operators coincide. We generalize the classical theorems of Pietsch, Tong and Uhl concerning the relationships between absolutely summing, dominated, nuclear and compact operators on the Banach space C(X), where X is a compact Hausdorff space.

Nuclear operators on Cb(X, E) and the strict topology

2018 ◽  
Vol 68 (1) ◽  
pp. 135-146
Author(s):  
Marian Nowak ◽  
Juliusz Stochmal
Keyword(s):  
Banach Spaces ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Strict Topology ◽  
Completely Regular ◽  
Borel Measures ◽  
Nuclear Operators ◽  
Bounded Continuous Functions

AbstractLetXbe a completely regular Hausdorff space,EandFbe Banach spaces. LetCb(X,E) be the space of allE-valued bounded, continuous functions onX, equipped with the natural strict topologyβ. We study nuclear operatorsT:Cb(X,E) →Fin terms of their representing operator-valued Borel measures.

Semigroups of Operators in C(S)

1970 ◽  
Vol 22 (1) ◽  
pp. 47-54 ◽  
Author(s):  
F. Dennis Sentilles
Keyword(s):  
Transition Function ◽  
Continuous Functions ◽  
Linear Operators ◽  
Semigroups Of Operators ◽  
Supremum Norm ◽  
Strict Topology ◽  
Borel Measures ◽  
Transition Functions ◽  
Semigroup Of Linear Operators ◽  
Bounded Continuous Functions

Our study in this paper is two-fold: One is that of a semigroup of linear operators on the space C(S) of bounded continuous functions on a locally compact Hausdorff space S, while the other is that of a transition function of measures in the Banach space M(S) of bounded regular Borel measures on S. It will be seen that an informative and essentially non-restrictive theory of the former can be obtained when C(S) is given the strict topology rather than the usual supremum norm topology and that, in this setting, the natural relationship between semigroups and transition functions obtained when S is compact is maintained, essentially because the dual of C(S) with the strict topology is M(S).

Strict Topology on Spaces of Continuous Vector-Valued Functions

1979 ◽  
Vol 31 (4) ◽  
pp. 890-896 ◽  
Author(s):  
Seki A. Choo
Keyword(s):  
Tensor Product ◽  
Convex Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Locally Convex Space ◽  
Completely Regular ◽  
Vector Valued Functions ◽  
Bounded Continuous Functions ◽  
Vector Valued ◽  
Locally Convex

In this paper, X denotes a completely regular Hausdorff space, Cb(X) all real-valued bounded continuous functions on X, E a Hausforff locally convex space over reals R, Cb(X, E) all bounded continuous functions from X into E, Cb(X) ⴲ E the tensor product of Cb(X) and E. For locally convex spaces E and F, E ⴲ, F denotes the tensor product with the topology of uniform convergence on sets of the form S X T where S and T are equicontinuous subsets of E′, F′ the topological duals of E, F respectively ([11], p. 96). For a locally convex space G , G ′ will denote its topological dual.

The Uncomplemented Subspace K(X,Y)

2013 ◽  
Vol 56 (1) ◽  
pp. 65-69
Author(s):  
Ioana Ghenciu
Keyword(s):  
Vector Measure ◽  
Compact Operators ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Spaces Of Operators ◽  
Completely Continuous Operators ◽  
Weakly Compact Operators ◽  
Continuous Operators

AbstractA vector measure result is used to study the complementation of the space K(X,Y) of compact operators in the spaces W(X,Y) of weakly compact operators, CC(X,Y) of completely continuous operators, and U(X,Y) of unconditionally converging operators. Results of Kalton and Emmanuele concerning the complementation of K(X,Y) in L(X,Y) and in W(X,Y) are generalized. The containment of c0 and ℓ∞ in spaces of operators is also studied.

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