Weak Integrals and Bounded Operators in Topological Vector Spaces

Home , Mackey topology

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)

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 integrals 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

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 compact 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 ds holds for all  in X and all x  X . So we have  f sMp  fs if f CSX0  ,  . Moreover for each seminorm p on X f CS ,,KX , and Uo iK  fs   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 UfSup  fs MSup p fs .  tK tK   TpTffSup  : Bp p    Since by Formula (*), the right side of this inequality is with Mpf   , we deduce that U is continuous. Now  ,K  Bp  fCSX,: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 WfSXTfP,,C  fsd s UfsUhsx hs, 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 continuous functional  in X * , we have: 3.1. Operators via Pettis Integrals f CSX,,, Tf , fs ds 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 topology 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-

LC0 SX,, X of all bounded operators from CSX0 ,  tails).

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 . Therefore,  ,dfs s 0, because the whose dual is a Mackey space. It is natural to ask if * * measure μ is of bounded variation. Hence f X  . similar representations can be established if the dual is * * endowed with another topology, e.g. the strong topology. Since X is semireflexive, f X ; by Proposition 3.5(a), there is a unique  f  X such that * 3.4. Definition f   , f ,  X . Now let us define the op-

* * erator TC:,0  SXX  , by Tf   f , f CSX0  ,  . The strong topology   X , X  of X is the topology It is easily checked that T is linear, and satisfies the con- generated by the family of the seminorms: dition of the theorem by construction. We have to show * **X ,PxB  SupB  that T is bounded. Let p be a seminorm on X, and let K be a compact subset of X. For f CS ,,KX , we where B is running over all the bounded sets of X. have: It is the topology of uniform convergence on the ppTfoSup Tf bounded sets of X. When we restrict ** to the ﬁnite  f   Ba  pa sets B of X we get the so called weak * topology * Sup ,fs d s  X , X , which is the topology of simple convergence B      p ** * on X. We shall denote by X X the space X * *  Sup Sup ,fs . B sK  equipped with the   X , X  -topology (the   X , X  - p topology). Then we have:  Sup Sup ,fs . sK B  p 3.5. Proposition  pf ,K .     1) For each xX there exists a unique x  X which proves the continuity of T.   Now to compute T , observe from the integral such that: x  xX,   . p   form of oTf that oTfSup , f s : s S . . 2) XX , that is, every weak * continuous   Taking the supremum in both sides over   B0 , the functional on X is strongly continuous . p polar set of the unit ball BxX,px1 of X, p      3.6. Definition we get:

We say that the space X is semireﬂexive if Sup0 oTf  p  Tf Bp   XX .   Sup Sup ,fs . B0 sS  Now we are in a position to state the main results of p  Sup Sup ,fs ( ) . this paper. sS B0 p Sup pfsfor fB 3.7. Theorem   p Let X be a locally convex Hausdorff semireﬂexive space. So we deduce that T   . To see the reverse ine- p If  is a bounded signed measure on BS , then there is quality, let us consider a function f CSX0  ,  of the a unique bounded operator TC:,0  SX X in the form f  gx , with g CS0   satisfying g  1

and x fixed in X such that px    1 . With this choice, bounded signed measure  on BS. This correspondence the function f belongs to the unit ball B . Then we is given by the relation W : p   have f CSX0  ,,Tf P fsd  s , f sgs x and 4. Acknowledgements ,,Tf f s ds   s gs d;  s This work has been done under the Project No. 121/130/ 1432. The authors are grateful to the Deanship of Scien- so that tific Research of the King Abdulaziz University, Jeddah, for their financial support. Sup oTf  p Tf p x g s d s B0      p  gsd,  s REFERENCES [1] L. Meziani, “Integral Representation for a Class of Vector since px 1 . So we get   Valued Operators,” Proceedings of the American Mathe- matical Society, Vol. 130, No. 7, 2002, pp. 2067-2077. pTf  gs d s T  p doi:10.1090/S0002-9939-02-06336-0 because f  B . [2] S. Kakutani, “Concrete Representation of Abstract (M). p Therefore Spaces,” Annals of Mathematics, Vol. 42, No. 4, 1941, pp. 994-1024. doi:10.2307/1968778

Supg ssgCSg d  ,0  , 1  T ■ [3] J. Diestel and J. J. Uhl, “Vector Measures,” Mathematical  p Surveys, No. 15, 1977. By appealing to theorem 2.3, we get the following [4] L. Meziani, “A Theorem of Riesz Type with Pettis Inte- rather precise theorem: grals in TVS,” Journal of Mathematical Analysis and Ap- plications, Vol. 340, No. 2, 2008, pp. 817-824. 3.8. Theorem doi:10.1016/j.jmaa.2007.09.005 Let X be a locally convex Hausdorff semireflexive space. [5] A. Wilanski, “Modern Methodsin Topological Vector Spaces,” McGraw-Hill, New York, 1978. Then there is a one to one correspondence between the bounded operators TC:,0  SX X of the class P and the X-valued Pettis integrals with respect to some