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Weak Integrals and Bounded Operators in Topological Vector Spaces

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Weak Integrals and Bounded Operators in Topological Vector Spaces Advances in Pure Mathematics, 2013, 3, 475-478 http://dx.doi.org/10.4236/apm.2013.35068 Published Online August 2013 (http://www.scirp.org/journal/apm) Weak Integrals and Bounded Operators in Topological Vector Spaces Lakhdar Meziani, Saud M. Alsulami Mathematics Department, King Abdulaziz University, Jeddah, KSA Email: [email protected], [email protected] Received May 4, 2013; revised June 8, 2013; accepted July 11, 2013 Copyright © 2013 Lakhdar Meziani, Saud M. Alsulami. This is an open access article distributed under the Creative Commons At- tribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop- erly cited. ABSTRACT Let X be a topological vector space and let S be a locally compact space. Let us consider the function space CSX0 , of all continuous functions f : SX , 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 TC:,0 SX X, and X-valued in- tegrals on CSX0 , . 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 , on S gives rise to a unique bounded operator TC:,0 SXX , which has the given Pettis integral form. Keywords: Bounded Operators; Integral Representation; Pettis Integral 1. Topological Preliminaries union being performed over all the compact subsets K of S. On the other hand if K1 is a subset of K2, then the Suppose that S is a locally compact space and let X be a natural embedding iCSKXCSKX:,, ,, is locally convex TVS. We denote by the set of KK12 12 CSX0 , continuous. This allows one to provide the space all continuous functions f : S X vanishing outside a CSX , with the inductive topology induced by compact set of S, put CSX, CS if X = R. We 0 00 the subspaces CSKX ,, , . The facts we need are interested in representing linear bounded operators K about the space, CSX0 , is well known: TC:,0 SX X, by means of weak integrals against scalar measures on the Borel -field BS of S. Before 1.1. Proposition handling more closely this problem, we need some topo- 1) The space CSX, , is locally convex Haus- logical facts about the space CS0 , X. 0 If K is a compact set in S, let CSKX,, be the set dorff and for each compact K, the relative topology of of all continuous functions f : SX , vanishing out- on CS ,,KX is K , this means that the canonical side K. It is clear that CS ,,KX is a linear subspace embedding iCk :,,SKX CSX0 , is continuous. 2) Let TC:,SX V be a linear operator of of CSX0 , . We equip CS ,,KX with the topology 0 CSX, into the locally convex Hausdorff space V, K generated by the family of seminorms: 0 then T is continuous if and only if the restriction ToiK *,,,Supf CSKX p p ft ,tKK of T to the subspace CSKX ,, is continuous for each compact K. where p is the family of seminorms generating the locally convex topology of X. The topology is the K 1.2. Definition topology of uniform convergence on K. * Next let us observe that CSX0 ,, K CSKX , , the For each in the topological dual X of X and for Copyright © 2013 SciRes. APM 476 L. MEZIANI, S. M. ALSULAMI each function f CSX0 , , define the function U f to X. Also one can prove that P is closed in the weak on S by Uf s fs , fs. Then U sends operator topology of LC 0 SX,, X . Note also that CSX0 , into CS0 . Recall that CS0 is equipped for a given bounded TC:,0 SX X, Definition 1.4 with the uniform norm. implies condition (I) i.e. TP . The crucial point is that condition (I) implies the Pettis integral form of Defini- 1.3. Lemma tion 1.4, for some bounded scalar measure on BS. This is the content of the following theorem proved in The operator U is linear and bounded. Moreover for [4]. each 0 , U is onto. Proof: First it is clear that Uf C S. Now by 0 2.2. Theorem Proposition 1.1(b), we have to show that for each com- pact set K of S the operator Uo iK :,CSK, X CS0 Let TC:,0 SX X be in the class P. Then there is a is bounded. Since is bounded, there is a seminorm p unique bounded signed measure on BS such that * on X and a constant M such that x Mp x for ,,Tf f s ds holds for all in X and all x X . So we have f sMp fs if f CSX0 , . Moreover for each seminorm p on X f CS ,,KX , and Uo iK fs f s , s S ; we have T , where is the total variation of p it follows that and T is the p -norm of T defined by p UfSup fs MSup p fs . tK tK TpTffSup : Bp p Since by Formula (*), the right side of this inequality is with Mpf , we deduce that U is continuous. Now ,K Bp fCSX,:Sup fs1. suppose 0 . Then there exists x X such that pS 0 x 0 and . It is clear that we can assume x 0 By this theorem we may denote each operator T in the x 1 . Now let hC S and define f : SX 0 class P by the conventional symbol by f tht .x, then f CS0 , X and we have WfSXTfPC,, fsd s UfsUhsx hs, because x 1 . It 0 follows that U is onto. ■ Now we consider the relationship between bounded where the letter P stands for Pettis integral. operators TC:,0 SX X, and weak integrals in the sense of the following definition. Such relationship is 3. Operators Associated to Scalar Measures reminiscent to the classical Riesz theorem [2]. via Pettis Integrals In this section we start with a bounded scalar measure 1.4. Definition on BS and we seek for a linear bounded TC:,0 SX X such that the correspondence be- We say that a bounded operator TC:,0 SXX has a Pettis integral form if there exists a scalar measure of tween and T would be given by formula (W). First let us make some observations. bounded variation on BS such that, for every con- tinuous functional in X * , we have: 3.1. Operators via Pettis Integrals f CSX,,, Tf , fs ds 0 A little inspection of (W) suggests the following quite See Reference [3] for details on Pettis integral. plausible observations: First the integral ,df s s , as a linear functional of on X * , 2. Integral Representation by Pettis Integral should beat least continuous for some convenient topol- ogy on X * Also the existence of the corresponding Tf In what follows, we introduce a class of bounded opera- in (W) will require that such topology on X should be tors TC:, SX X, which is, in this context, similar 0 compatible for the dual pair X * , X . Finally, to get the to the class CXX used in [1]. continuity of the functional S ,f sd s , one can seek conditions such that if 0 in an appropri- 2.1. Definition ate manner, then , f s goes to 0 uniformly for Let P be the class of all bounded operators s S . Since is bounded this will give TC:,0 SX X satisfying the following condition: ,dfs s 0. * (I) For , X and f ,,gCSX 0 , if Uf Ug Such a program has been realized in [4], for a locally then Tf Tg. convex space having the convex compactness property It is easy to check that P is a subspace of the space [5], according to the following theorems (see [4] for de- LC0 SX,, X of all bounded operators from CSX0 , tails). Copyright © 2013 SciRes. APM L. MEZIANI, S. M. ALSULAMI 477 3.2. Theorem class P satisfying: Let X be a locally convex space with the convex com- f CSX0 ,,, Tf , fs d s, pactness property, and whose dual X * is equipped with * T . the Mackey topology X ,X . If is a bounded p scalar measure on B , then there is a unique bounded S where is the variation of . operator TC:,0 SXX in the class P satisfying Proof: Fix f in CSX0 , and define the functional (W), with T for each seminorm p on X. p * f : X R , by f ,df s s . It is clear 3.3. Theorem * * that f is linear. Moreover f X . Indeed it is Let X be a locally convex Hausdorff space whose dual * * enough to prove that lim f 0 . If 0 , in X , X is a barrelled space. If is a bounded signed 0 then for each bounded subset B of X, x 0 uni- measure on B , then there is a unique bounded operator S formly for x B . But since f CSX, , the set TC:,SX X in the class P satisfying (W) with 0 0 f ss: S is bounded, so ,fs 0 uniformly respect to and such that T . p Most of these results have been obtained for a space in s S .
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