Let S be a locally compact space and let X be a Banach space. Let us consider the function space ... more Let S be a locally compact space and let X be a Banach space. Let us consider the function space C 0 (S, X) of all continuous func-tions f : S → X vanishing at infinity, endowed with the uniform topology. We shall be concerned with integral representations of linear bounded operators T : C 0 (S, X) → X. The main result is a complete characterization of those operators which enjoy an integral form with respect to a scalar measure μ on S. Furthermore we show that such operators also have an integral representation with respect to an operator valued measure G on S with values in L (X, X) , the space of bounded operators on X. Finally, relationships between the dif-ferent measures are established and this allows to characterize the operators under consideration by their representing measures.
Let (S, F , μ) be a finite measure space and let X be a Banach space. As usual L 1 (μ, X) is the ... more Let (S, F , μ) be a finite measure space and let X be a Banach space. As usual L 1 (μ, X) is the Banach space of all Bochner μ−integrable functions f : S → X, with L 1 (μ, X) = L 1 (μ) if X = R. This work is intended for the study of a class of linear bounded operators T : L 1 (μ, X) → X, whose integral structure is much similar to that of bounded functionals on L 1 (μ). We give two complete characterizations of this class. The first one, which may be considered as a Riesz type theorem, is obtained via integrals by functions in L ∞ (μ). Actually the identified class is isometrically isomorphic to L ∞ (μ). The second characterization is more specific. It pertains to an operator valued measure, that will be attached to each operator of the class. This operator valued measure will be absolutely continuous with respect to μ and this property will be used to get another interesting characterization of the class under consideration.
ABSTRACT ABSTRACT Let X be a topological vector space and let S be a locally compact space. Let u... more ABSTRACT ABSTRACT Let X be a topological vector space and let S be a locally compact space. Let us consider the function space of all continuous functions , vanishing outside a compact set of S, equipped with an appropriate topology. In this work we will be concerned with the relationship between bounded operators , and X-valued integrals on . When X is a Banach space, such relation has been completely achieved via Bochner integral in [1]. In this paper we investigate the context of locally convex spaces and we will focus attention on weak integrals, namely the Pettis integrals. Some results in this direction have been obtained, under some special conditions on the structure of X and its topological dual X*. In this work we consider the case of a semi reflexive locally convex space and prove that each Pettis integral with respect to a signed measure μ, on S gives rise to a unique bounded operator , which has the given Pettis integral form.
Let X; Y be Banach spaces (or either topological vector spaces) and let us consider the function ... more Let X; Y be Banach spaces (or either topological vector spaces) and let us consider the function space C (S;X) of all continuous functions f : S ! X; from the compact (locally compact) space S into X; equipped with some appropriate topology. Put C (S;X) = C (S) if X = R: In this work we will mainly be concerned with the problem of representing linear bounded operators T : C (S;X) ! Y in an integral form: f 2 C (S;X) ; Tf = R S f d¹, for some integration process with respect to a measure ¹ on the Borel ¾¡field BS of S: The prototype of such representation is the theorem of F. Riesz according to which every continuous functional T : C (S) ! R has the Lebesgue integral form Tf = R S f d¹: This paper is intended to present various extensions of this theorem to the Banach spaces setting alluded to above, and to the context of locally convex spaces.
Let S be a locally compact space and let X be a Banach space. Let us consider the function space ... more Let S be a locally compact space and let X be a Banach space. Let us consider the function space C 0 (S, X) of all continuous func-tions f : S → X vanishing at infinity, endowed with the uniform topology. We shall be concerned with integral representations of linear bounded operators T : C 0 (S, X) → X. The main result is a complete characterization of those operators which enjoy an integral form with respect to a scalar measure μ on S. Furthermore we show that such operators also have an integral representation with respect to an operator valued measure G on S with values in L (X, X) , the space of bounded operators on X. Finally, relationships between the dif-ferent measures are established and this allows to characterize the operators under consideration by their representing measures.
Let (S, F , μ) be a finite measure space and let X be a Banach space. As usual L 1 (μ, X) is the ... more Let (S, F , μ) be a finite measure space and let X be a Banach space. As usual L 1 (μ, X) is the Banach space of all Bochner μ−integrable functions f : S → X, with L 1 (μ, X) = L 1 (μ) if X = R. This work is intended for the study of a class of linear bounded operators T : L 1 (μ, X) → X, whose integral structure is much similar to that of bounded functionals on L 1 (μ). We give two complete characterizations of this class. The first one, which may be considered as a Riesz type theorem, is obtained via integrals by functions in L ∞ (μ). Actually the identified class is isometrically isomorphic to L ∞ (μ). The second characterization is more specific. It pertains to an operator valued measure, that will be attached to each operator of the class. This operator valued measure will be absolutely continuous with respect to μ and this property will be used to get another interesting characterization of the class under consideration.
ABSTRACT ABSTRACT Let X be a topological vector space and let S be a locally compact space. Let u... more ABSTRACT ABSTRACT Let X be a topological vector space and let S be a locally compact space. Let us consider the function space of all continuous functions , vanishing outside a compact set of S, equipped with an appropriate topology. In this work we will be concerned with the relationship between bounded operators , and X-valued integrals on . When X is a Banach space, such relation has been completely achieved via Bochner integral in [1]. In this paper we investigate the context of locally convex spaces and we will focus attention on weak integrals, namely the Pettis integrals. Some results in this direction have been obtained, under some special conditions on the structure of X and its topological dual X*. In this work we consider the case of a semi reflexive locally convex space and prove that each Pettis integral with respect to a signed measure μ, on S gives rise to a unique bounded operator , which has the given Pettis integral form.
Let X; Y be Banach spaces (or either topological vector spaces) and let us consider the function ... more Let X; Y be Banach spaces (or either topological vector spaces) and let us consider the function space C (S;X) of all continuous functions f : S ! X; from the compact (locally compact) space S into X; equipped with some appropriate topology. Put C (S;X) = C (S) if X = R: In this work we will mainly be concerned with the problem of representing linear bounded operators T : C (S;X) ! Y in an integral form: f 2 C (S;X) ; Tf = R S f d¹, for some integration process with respect to a measure ¹ on the Borel ¾¡field BS of S: The prototype of such representation is the theorem of F. Riesz according to which every continuous functional T : C (S) ! R has the Lebesgue integral form Tf = R S f d¹: This paper is intended to present various extensions of this theorem to the Banach spaces setting alluded to above, and to the context of locally convex spaces.
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