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.

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.

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.

On Additive Operators

1971 ◽  
Vol 23 (3) ◽  
pp. 468-480 ◽  
Author(s):  
N. A. Friedman ◽  
A. E. Tong
Keyword(s):  
Banach Space ◽  
Compact Hausdorff Space ◽  
Weak Topology ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Additive Functional ◽  
Representation Theorems ◽  
Image Position ◽  
Vector Valued ◽  
Kernel Representation

Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]):

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

1989 ◽  
Vol 31 (1) ◽  
pp. 59-64 ◽  
Author(s):  
Marek Nawrocki
Keyword(s):  
Hausdorff Space ◽  
Continuous Functions ◽  
Weighted Spaces ◽  
Weak Approximation ◽  
Completely Regular ◽  
Hausdorff Topological Vector Space ◽  
The Family ◽  
Image Position ◽  
Positive Functions ◽  
Vector Valued

Let X be a completely regular Hausdorff space. A Nachbin family of weights is a set V of upper-semicontinuous positive functions on X such that if u, υ ∈ V then there exists w ∈ V and t > 0 so that u, υ ≤ tw. For any Hausdorff topological vector space E, the weighted space CV0(X, E) is the space of all E-valued continuous functions f on X such that υf vanishes at infinity for all υ ∈ V. CV0(X, E) is equipped with the weighted topologywv = wv(X, E) which has as a base of neighbourhoods of zero the family of all sets of the formwhere υ ∈ Vand W is a neighbourhood of zero in E. If E is the scalar field, then the space CV0(X, E) is denoted by CV0(X). The reader is referred to [4, 6, 8] for information on weighted spaces.

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 Multiplication operators on weighted spaces of vector-valued continuous functions

1991 ◽  
Vol 50 (1) ◽  
pp. 98-107 ◽  
Author(s):  
R. K. Singh ◽  
Jasbir Singh Manhas
Keyword(s):  
Convex Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Weighted Spaces ◽  
Multiplication Operators ◽  
Supremum Norm ◽  
Locally Convex Spaces ◽  
Completely Regular ◽  
Vector Valued ◽  
Locally Convex

AbstractIf V is a system of weights on a completely regular Hausdorff space X and E is alocally convex space, then CV0(X, E) and CVb (X, E) are locally convex spaces of vector-valued continuous functions with topologies generated by seminorms which are weighted analogues of the supremum norm. In this paper we characterise multiplication operators on these spaces induced by scalar-valued and vector-valued mappings. Many examples are presented to illustrate the theory.

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.

scholarly journals Extension of vector-valued functions and weak–strong principles for differentiable functions of finite order

2021 ◽  
Vol 13 (1) ◽  
Author(s):  
Karsten Kruse
Keyword(s):  
Finite Order ◽  
Hausdorff Space ◽  
Linear Operators ◽  
Continuous Linear ◽  
Differentiable Functions ◽  
Weighted Banach Space ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Locally Convex ◽  
Continuous Linear Operators

AbstractIn this paper we study the problem of extending functions with values in a locally convex Hausdorff space E over a field $$\mathbb {K}$$ K , which has weak extensions in a weighted Banach space $${\mathcal {F}}\nu (\Omega ,\mathbb {K})$$ F ν ( Ω , K ) of scalar-valued functions on a set $$\Omega$$ Ω , to functions in a vector-valued counterpart $$\mathcal {F}\nu (\Omega ,E)$$ F ν ( Ω , E ) of $${\mathcal {F}}\nu (\Omega ,\mathbb {K})$$ F ν ( Ω , K ) . Our findings rely on a description of vector-valued functions as continuous linear operators and extend results of Frerick, Jordá and Wengenroth. As an application we derive weak-strong principles for continuously partially differentiable functions of finite order and vector-valued versions of Blaschke’s convergence theorem for several spaces.

scholarly journals Multiplication Operators and Dynamical Systems

1992 ◽  
Vol 53 (1) ◽  
pp. 92-102 ◽  
Author(s):  
R. K. Singh ◽  
Jasbir Singh Manhas
Keyword(s):  
Hausdorff Space ◽  
Continuous Functions ◽  
Multiplication Operators ◽  
Supremum Norm ◽  
Weighted Function ◽  
Completely Regular ◽  
Hausdorff Topological Vector Space ◽  
Weighted Function Space ◽  
Vector Valued ◽  
Locally Convex

AbstractLet X be a completely regular Hausdorff space, let V be a system of weights on X and let T be a locally convex Hausdorff topological vector space. Then CVb(X, T) is a locally convex space of vector-valued continuous functions with a topology generated by seminorms which are weighted analogues of the supremum norm. In the present paper we characterize multiplication operators on the space CVb(X, T) induced by operator-valued mappings and then obtain a (linear) dynamical system on this weighted function space.

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