| A complex Banach space X has the analytic Radon-Nikodym property if every X-valued measure, (mu), of bounded variation defined on the Borel subsets of the unit circle, (PI), with.;for all n (ELEM) ZZ with n < 0 has a Radon-Nikodym derivative.;Theorem. Let X be a complex Banach space. The following condi- tions are equivalent. (1) X has the analytic Radon-Nikodym property.;(2) Tq: L(,1) (--->) X is representable for all bounded linear operators T: L(,1)/H(,0)('1) (--->) X and where q: L(,1) (--->) L(,1)/H(,0)('1) is the natural quotient opera- tor. (3) Every bounded linear operator from L(,1) into X which factors through L(,1)/H(,0)('1) is representable. (4) Every completely continuous.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;operator from L(,1) into X which factors through L(,1)/H(,0)('1) is represent- able. (5) For each L(,1)/H(,0)('1)-valued uniformly bounded martingale (f(,n))(,n=0)('(INFIN)) and each bounded linear operator T: L(,1)/H(,0)('1) (--->) X the mar- tingale (T(f(,n)))(,n=0)('(INFIN)) is almost everywhere convergent. (6) For each.;L(,1)/H(,0)('1)-valued L(,1)-bounded martingale (f(,n))(,n=0)('(INFIN)) and each bounded linear operator T: L(,1)/H(,0)('1) (--->) X the martingale (T(f(,n)))(,n=0)('(INFIN)) is almost everywhere convergent. (7) Every bounded subset of L(,1)/H(,0)('1) is T-(sigma)-dentable for each bounded linear operator T: L(,1)/H(,0)('1) (--->) X. (8) For.;each bounded subset B of L(,1)/H(,0)('1) and each bounded linear operator T: L(,1)H(,0)(' )(--->) X, B has slices such that the image of these slices under T are of arbitrarily small diameter.;Theorem. (a) If a complex Banach space X has the analytic Radon-Nikodym property then every bounded linear operator from L(,1)/H(,0)('1).;into X is completely continuous. (b) If X is a complemented sub- space of L(,1)/H(,0)('1) having the analytic Radon-Nikodym property then X has the Schur property. |
