A CHAIN CONDITION FOR OPERATORS FROM C(K)-SPACES

We introduce a chain condition <inline-graphic xlink:href="HAT006IM1" xlink:type="simple"/>, defined for operators acting on C(K)-spaces, which is intermediate between weak compactness and having weakly compactly generated range. It is motivated by Pe<remove>czyA"ski's characterization of weakly compact operators on C(K)-spaces. We prove that if K is extremally disconnected and X is a Banach space, then, for an operator T : C(K)-> X, T is weakly compact if and only if T satisfies <inline-graphic xlink:href="HAT006IM2" xlink:type="simple"/> if and only if the representing vector measure of T satisfies an analogous chain condition. As a tool for proving the above-mentioned result, we derive a topological counterpart of Rosenthal's lemma. We exhibit several compact Hausdorff spaces K for which the identity operator on C(K) satisfies <inline-graphic xlink:href="HAT006IM3" xlink:type="simple"/>, for example, both locally connected compact spaces having countable cellularity and ladder system spaces have this property. Using a Ramsey-type theorem, due to Dushnik and Miller, we prove that the collection of operators on a C(K)-space satisfying <inline-graphic xlink:href="HAT006IM4" xlink:type="simple"/> forms a closed left ideal of a"not sign(C(K)).