## Operators on Continuous Function Spaces and Weak Precompactness

Description:
If T:C(H,X)-->Y is a bounded linear operator then there exists a unique weakly regular finitely additive set function m:-->L(X,Y**) so that T(f) = ∫Hfdm. In this paper, bounded linear operators on C(H,X) are studied in terms the measure given by this representation theorem. The first chapter provides a brief history of representation theorems of these classes of operators. In the second chapter the represenation theorem used in the remainder of the paper is presented. If T is a weakly compact operator on C(H,X) with representing measure m, then m(A) is a weakly compact operator for every Borel set A. Furthermore, m is strongly bounded. Analogous statements may be made for many interesting classes of operators. In chapter III, two classes of operators, weakly precompact and QSP, are studied. Examples are provided to show that if T is weakly precompact (QSP) then m(A) need not be weakly precompact (QSP), for every Borel set A. In addition, it will be shown that weakly precompact and GSP operators need not have strongly bounded representing measures. Sufficient conditions are provided which guarantee that a weakly precompact (QSP) operator has weakly precompact (QSP) values. A sufficient condition for a weakly precomact operator to be strongly bounded is given. In chapter IV, weakly precompact subsets of L1(μ,X) are examined. For a Banach space X whose dual has the Radon-Nikodym property, it is shown that the weakly precompact subsets of L1(μ,X) are exactly the uniformly integrable subsets of L1(μ,X). Furthermore, it is shown that this characterization does not hold in Banach spaces X for which X* does not have the weak Radon-Nikodym property.

Date:
August 1988

Creator:
Abbott, Catherine Ann