scholarly journals Multiplication operators on weighted spaces in the non-locally convex framework

1997 ◽  
Vol 20 (1) ◽  
pp. 75-79 ◽  
Author(s):  
L. A. Khan ◽  
A. B. Thaheem
Keyword(s):  
Topological Space ◽  
Hausdorff Space ◽  
Weighted Space ◽  
Topological Algebra ◽  
Sufficient Conditions ◽  
Multiplication Operators ◽  
Completely Regular ◽  
General Topological Space ◽  
Necessary And Sufficient ◽  
Locally Convex

LetXbe a completely regular Hausdorff space,Ea topological vector space,Va Nachbin family of weights onX, andCV0(X,E)the weighted space of continuousE-valued functions onX. Letθ:X→Cbe a mapping,f∈CV0(X,E)and defineMθ(f)=θf(pointwise). In caseEis a topological algebra,ψ:X→Eis a mapping then defineMψ(f)=ψf(pointwise). The main purpose of this paper is to give necessary and sufficient conditions forMθandMψto be the multiplication operators onCV0(X,E)whereEis a general topological space (or a suitable topological algebra) which is not necessarily locally convex. These results generalize recent work of Singh and Manhas based on the assumption thatEis locally convex.

scholarly journals WEIGHTED COMPOSITION OPERATORS AND DYNAMICAL SYSTEMS

1998 ◽  
Vol 29 (2) ◽  
pp. 101-107
Author(s):  
R. K. SINGH ◽  
BHOPINDER SINGH
Keyword(s):  
Hausdorff Space ◽  
Weighted Space ◽  
Composition Operators ◽  
Continuous Functions ◽  
Weighted Composition Operators ◽  
Linear Dynamical System ◽  
Completely Regular ◽  
Weighted Composition ◽  
Locally Convex

Let $X$ be a completely regular Hausdorff space, $E$ a Hausdorff locally convex topo­logical vector space, and $V$ a system of weights on $X$. Denote by $CV_b(X, E)$ ($CV_o(X, E)$) the weighted space of all continuous functions $f : X \to E$ such that $vf (X)$ is bounded in $E$ (respectively, $vf$ vanishes at infinity on $X$) for all $v \in V$. In this paper, we give an improved characterization of weighted composition operators on $CV_b(X, E)$ and present a linear dynamical system induced by composition operators on the metrizable weighted space $CV_o(\mathbb{R}, E)$.

scholarly journals Completion in the bidual

1968 ◽  
Vol 8 (2) ◽  
pp. 238-241 ◽  
Author(s):  
R. Nielsen
Keyword(s):  
Hausdorff Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
Locally Convex

Let E, Ê, and E′ denote a locally convex linear Hausdorff space, completion of E and the dual of E, respectively. It has been observed that Ê is a subspace of E″ under certain conditions on E. It is the primary goal of this paper to give necessary and sufficient conditions for the Ê ⊂ E″ to be valid. Such conditions are found and are given Theorem 4. With a variation of the technique used, several equivalent characterizations of semi-reflexive spaces are given in Theorem 5. The nationa throughtout will follow that in [2].

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.

Monadic binary relations and the monad systems at near-standard points

1987 ◽  
Vol 52 (3) ◽  
pp. 689-697
Author(s):  
Nader Vakil
Keyword(s):  
Topological Space ◽  
Binary Relation ◽  
Hausdorff Space ◽  
Equivalence Relations ◽  
Binary Relations ◽  
Completely Regular ◽  
General Topological Space ◽  
Image Position

AbstractLet (*X, *T) be the nonstandard extension of a Hausdorff space (X, T). After Wattenberg [6], the monad m(x) of a near-standard point x in *X is defined as m{x) = μT(st(x)). Consider the relationFrank Wattenberg in [6] and [7] investigated the possibilities of extending the domain of Rns to the whole of *X. Wattenberg's extensions of Rns were required to be equivalence relations, among other things. Because the nontrivial ways of constructing such extensions usually produce monadic relations, the said condition practically limits (to completely regular spaces) the class of spaces for which such extensions are possible. Since symmetry and transitivity are not, after all, characteristics of the kind of nearness that is obtained in a general topological space, it may be expected that if these two requirements are relaxed, then a monadic extension of Rns to *X should be possible in any topological space. A study of such extensions of Rns is the purpose of the present paper. We call a binary relation W ⊆ *X × *X an infinitesimal on *X if it is monadic and reflexive on *X. We prove, among other things, that the existence of an infinitesimal on *X that extends Rns is equivalent to the condition that the space (X, T) be regular.

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.

Eberlein compacts and spaces of continuous functions

1979 ◽  
Vol 85 (2) ◽  
pp. 305-313
Author(s):  
Richard J. Hunter ◽  
J. W. Lloyd
Keyword(s):  
Topological Space ◽  
Weak Topology ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Locally Convex Spaces ◽  
Weakly Compact ◽  
Spaces Of Continuous Functions ◽  
Eberlein Compact ◽  
Necessary And Sufficient ◽  
Locally Convex

AbstractLet X be a Hausdorff topological space. We consider various locally convex spaces of continuous real valued functions on X and give necessary and sufficient conditions in order that (i) they contain an absolutely convex weakly compact total subset and (ii) they contain an absolutely convex total subset which is an Eberlein compact, when given the weak topology.

scholarly journals Note on pointwise contractive projections

1993 ◽  
Vol 16 (4) ◽  
pp. 817-818
Author(s):  
L. M. Sanchez Ruiz ◽  
J. R. Ferrer Villanueva
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Compact Open Topology ◽  
Completely Regular ◽  
Contractive Projection ◽  
Necessary And Sufficient ◽  
Contractive Projections

LetC(X)be the space of real-valued continuous functions on a Hausdorff completely regular topological spaceX. endowed with the compact-open topology. In this paper necessary and sufficient conditions are given for a subspace ofC(X)to be the range of a pointwise contractive projection inC(X).

scholarly journals On a topology between Tα and Tγα

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3209-3221
Author(s):  
Dimitrije Andrijevic
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Closure Operators ◽  
New Class ◽  
Necessary And Sufficient

Using the topology T in a topological space (X,T), a new class of generalized open sets called ?-preopen sets, is introduced and studied. This class generates a new topology Tg which is larger than T? and smaller than T??. By means of the corresponding interior and closure operators, among other results, necessary and sufficient conditions are given for Tg to coincide with T? , T? or T??.

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.

Interpolation in Separable Frechet Spaces with Applications to Spaces of Analytic Functions

1975 ◽  
Vol 27 (5) ◽  
pp. 1110-1113 ◽  
Author(s):  
Paul M. Gauthier ◽  
Lee A. Rubel
Keyword(s):  
Sufficient Conditions ◽  
Fréchet Space ◽  
Complex Numbers ◽  
Frechet Space ◽  
Fréchet Spaces ◽  
Continuous Linear Functional ◽  
Interpolating Sequence ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Spaces Of Analytic Functions

Let E be a separable Fréchet space, and let E* be its topological dual space. We recall that a Fréchet space is, by definition, a complete metrizable locally convex topological vector space. A sequence {Ln} of continuous linear functional is said to be interpolating if for every sequence {An} of complex numbers, there exists an ƒ ∈ E such that Ln(ƒ) = An for n = 1, 2, 3, … . In this paper, we give necessary and sufficient conditions that {Ln} be an interpolating sequence. They are different from the conditions in [2] and don't seem to be easily interderivable with them.

Sign in / Sign up

Export Citation Format

Share Document