Weakly compact operators and u-additive measures

Let X be a completely regular space and E, F locally convex Hausdorff space with a directed family of semi-norms which generates their topologies. Denote by Cb(X, E~ the space of all E-valued, continuous and bounded function f from X into E. Cb(X, E) is endowing by locally convex topologies, so called strict topologies. We study the F-valued weakly compact operators T defined on Cb(X, E). . We characterize those operators which are continuous in the strict topologies, and wel find certain kind of vector measures associated with them. Necessary and sufficient conditions for a continuous and weakly compact operators to be represented by integrals with respect to £(E, F)-valued measures on the Baire algebra generated by zero sets are obtained. 1. DEFINITIONS AND PRELIMINARIES. Let X be a completely regular Hausdorff space. By B = B(X) we will denote the algebra of subsets of X generated by zero sets Z = f -1 (~0}} , where f is a real-valued continuous function defined on X and by ~a = Ba(X) we will denote the u-algebra of Baire sets, that is, ~ia is the smallest u-algebra of subsets of X which contains the zero sets. Let M(X ) denote the space of all bounded finitely-additive regular (with respect to the zero sets) measures on B. We recall that a measure ~c E M(X ) is said to be . T-additive if, for any net with U03B1 J, Ø, lima (U03B1) = 0. The space of all T-additives measures is denoted by Mr(X). . a-additive if, for any net with Un i 0, limn~~ (Un) = 0. The space of all ~-additives measures is denoted by Mu(X). . u-additive if, for any partition of unity positive functions in C(X) such that = 1 and is locally 1991 Mathematics Subject Classification. AMS subject classification(1991): Primary 46A03, 47A67, 28A25. Secundary 28A33, 46G10F.. Key wor~ds and phrases.

ABSTRACT. Let X be a completely regular space and E, F locally convex Hausdorff space with a directed family of semi-norms which generates their topologies. Denote by Cb(X, E~ the space of all E-valued, continuous and bounded function f from X into E. Cb(X, E) is endowing by locally convex topologies, so called strict topologies. We study the F-valued weakly compact operators T defined on Cb(X, E). . We characterize those operators which are continuous in the strict topologies, and wel find certain kind of vector measures associated with them. Necessary and sufficient conditions for a continuous and weakly compact operators to be represented by integrals with respect to £(E, F)-valued measures on the Baire algebra generated by zero sets are obtained.

DEFINITIONS AND PRELIMINARIES.
Let X be a completely regular Hausdorff space. By B = B(X) we will denote the algebra of subsets of X generated by zero sets Z = f -1 (~0}} , where f is a real-valued continuous function defined on X and by ~a = Ba(X) we will denote the u-algebra of Baire sets, that is, ia is the smallest u-algebra of subsets of X which contains the zero sets.
Let M(X ) denote the space of all bounded finitely-additive regular (with respect to the zero sets) measures on B. We recall that a measurẽ c E M(X ) is said to be . T-additive if, for any net with U03B1 J, Ø, lima ( U 0 3 B 1 ) The set M03C3,p(X, E') consists of those m in Mp(X, E') for which ms E M03C3(X) for all sEE. The spaces M,p(X, E') and M~,p(X, E') are defined similarly. It is known that if m is in any of the spaces Mp(X, E'), Mu,p(X, E'), Mr,p(X, E') and E'), then mp belongs to M (X ), Mu(X), Mr(X) and respectively (see (1~ and [8]). Let (p be a generating family of continuous semi-norms on E which is directed, i.e., given pi, p2 in I there exists p3 E I such that p3 > max {p1,p2}. We will denote by M(X,E') the space Up~IMp(X, E'). Analogous notation we will have for Mu(X, E'), M(X, E') and Moo (X, EI). respectively. S~~ will be the class of all compact subsets of X satisfying any of the equivalent assertions of the following lemma given in [4].. Lemma 1. For a compact K c 03B2X B X the following are equivalent: I. There is a co-zero cover of X which is (a) locaLly finite, (b) u-locally finite or (c) u-discrete such that cl03B2XU03B1 n K = 03A6 far all a E A for all x K aEA It can be proved that Hi C ~t~ C S~. If K belongs to some of these collections, then we define ,QK to be the locally convex topology on Cb (X E) generated by the family of the semi-norms The locally convex topologies /3, ,Q~ and /~1 defined on Cb(X, E) will be the inductive limits topologies of the topologies ,QK as K ranges over 03A9, 03A9~ and 03A91 respectively. For a fixed p ~ I, denotes the locally convex topology on Cb(X, E) generated by the semi-norms f --~ g E CK. The locally convex topologies /?p, 03B2u,p and will be the inductive limits topology on Cb (X E) of the topologies as K ranges over S~, S~ã nd S~1 respectively. The locally convex topologies ,Q', 03B2'u and 03B2'1 are the projective limits topologies on Cb(X, E) of the topologies ,Qp, 03B2u,p and respectively, as p ranges over I.
. the dual of provided by ,Ql and ~3~, is Mu(X, E' ) (see ~6~ ); . the dual of Crc, provided by ,Q and 03B2', is M(X, E') (see [6]). . Cb(X) ~ E is ,Qu--dense in Cb(X, E) (see ~1~). Definition 2. Let m E M(X, ~C(E, F)) and let f : X --~ E be a function. We will say that f is m -integrable over A in B if: : 1. For each x' E F' the integral exists 2. There exists a vector in F, denoted by Afdm, such that f or all x' E F' we have x'(Afdm) = f d(x'm). Remark 1. Since F is a locally convex Hausdorff space, the A fdm is unique whenever it exists. If f is m-integrable over all A E B, we will say that f is m-integrable. If f is m-integrable on X, ' X fdm will be denoted by simply I dm. The follow~ing Representation Theorem is found in ~5~ . Theorem 1. 1fT is a continuous weakly compact operator from (Crc, u) into F, then there exists a unique m E M(X, £(E, F)) such that: 1. Every f in Cr~ is m-integrable and T ( f ) = f dm 2. If p ~ I andq E J are such that ~T~p,q = sup q(T( f )) : IIf Ilp 1 oo, then ~m~p,q = ~T~p,q .
3. For every x' E F', we have T'x' = x'm.

The set Vm,S = { 0 3 A 3 m ( A i ) s i :
{Ai} is a finite B-partition of X, si E S} is weakly relatively compact in F, for every bounded set ,5~ in E.
Conversely, if m E M(X, £(E, F)) is such that (4) holds, then every f E Cr~ is m-integrable and the operatorT(J) = fdm is u-continuous and weakly compact. Remark 2. Since T can be extended to C(03B2X, E) and if T denotes its extension, we have T ( f ) = T ( f ) which is ucontinuous, where id enotes the uniform topology on C(03B2X, E). The dual of (C(03B2X, E),u) is MT (,QX, E'). Using the above theorem, we can get a vector measure representation of T, say, T( f ) = d.

REPRESENTATION
In this section we will study the F-valued weakly compact operators T defined on Cb(X, E), where F is another locally convex Hausdorff space with a directed family of semi-norms J which generates its topology. We shall characterize those operators which are 03B2'u -continuous and we shall find certain kind of vector measures associated with them.
In section 1, we announced the Katsaras' Theorem which gives us an integral representation of weakly compact operators. The following theorem characterizes the 03B2'u-continuous operators defined on Crc(X, F). First we need the following very well known Grothendieck's Lemma ~3~. . Lemma 2. Let T be an operator from a topological vector space V into another topological vector space Wand let T' and T" denote, respectively, the transpose and the second transpose of T. The following statements are equivalent: 1   onversely, we already know that T is weakly compact under the duality Crc, M (X , E' ) y ; since 03B2' and the uniform topology u have the same bounded sets ( [6] , Th. 5.4, p. 225) and since ~Q' ,Qu u, we have T is weakly compact under the duality (Crc, , E')) . The 03B2'u-continuity of T is coming from the fact that = 0, for all K E 03A9~, and Th. 7. Theorem 5. Let T be a F-valued, weakly compact and continuous operator on Cb(X, E), where F is, in addition, complete. Then, there exists a unique m E M~(X, £(E, F)) such that: 1. . Every f E Gb(X, E) is m-integrable and T( f ) = fdm.  (4) holds, then every f E Cb(X, E) is m-integrable and the operator T ( f ) = fdm is 03B2'u-continuous and weakly compact.
Proof. Suppose that T is weakly compact and ~D~--continuous operator on Cb(X, E). If we put T = T|Crc, then it is clear that T is also weakly compact and 03B2'u -continuous operator on Crc-So, there exists a unique m E M~(X, (F, F)) which satisfies (1), (2), (3) and (4). Now, we shall prove that f E Cb(X, E) is m-integrable. Fix x' E F' and G E B, we define La : Cr~ -+ I~, ' by La(g) = .
By the 03B2'u-denseness of Crc in Cb(X, E) (see [1]), LG can be uniquely extended to Cb(X, E) and then exists. Conversely, we will suppose that m ~ M~(X, £(E, F)) and we will prove that there exists a 03B2'u-continuous and weakly compact operator T such that T( f ) = fdm. By Th. 7, we have a 13'.. -continuous and weakly compact operator T defined on Cr~ such that f(f) = J fdm. Now, the 03B2'u-denseness of Crc gives us a unique 03B2'u-continuous operator T defined on Cb(X, E) such that T ( f ) = T ( f ), for all f E Crc. Using the same arguments given before, we prove that each f E Cb(X, E) is m-integrable and then T( f ) = J fdm. The remainder of the proof, that is, T is weakly compact, follows easily from the fact that T' = T' which implies that T"((Cb(X, E ) , 0 3 B 2 ' u ) " ) = 03B2'u)" C F. Remark 3. Since ~i' we can rewrite Th. 5, p. 55.~, given by Katsaras ~6J , , as follows : Theorem 6. Let T be a F-valued, weakly compact and ,0'continuous operator on Cb(X, E), where F is, in addition, complete. Then, there exists a unique m E M(X, £(E, F)) such that: l. every f E Cb(X, E) is m-integrable and T( f) = fdm.
2. Given q E J, there exists p ~ I such that = ~m~p,q .
3. For every x' E F', we have T' x' = x'rrz. Conversely, if m E Mr(X,£(E,F)) is such that (4) holds, then every f E Cb(X, E) is m-integrable and the operator T(f) = fdm is 03B2'-continuous and weakly compact.