Differentiation of Banach-space-valued additive processes
Let X be a Banach space and ( ; ; ) be a -nite measure space. Let L be a Banach space of X-valued strongly measurable functions on ( ; ; ). We consider a strongly continuous d-dimensional semigroup T = fT (u) : u = (u1;:::;ud); ui > 0; 1 i dg of linear contractions on L. We assume that each T (u) has, in a sense, a contraction majorant and that the strong limitT (0) = strong-lim u!0 T (u) exists. Then we prove, under some suitable norm conditions on the Banach space L, that a dieren tiation theorem holds for d-dimensional bounded processes in L which are additive with respect to the semigroup T. This generalizes a dieren tiation theorem obtained previously by the author under the assumption that L is an X-valued Lp-space, with 1 p <1.