Convergent Martingales of Operators and the Radon Nikodým Property in Banach Spaces

We extend Troitsky’s ideas on measure-free martingales on Banach lattices to martingales of operators acting between a Banach lattice and a Banach space. We prove that each norm bounded martingale of cone absolutely summing (c.a.s.) operators (also known as 1-concave operators), from a Banach lattice E to a Banach space Y , can be generated by a single c.a.s. operator. As a consequence, we obtain a characterization of Banach spaces with the Radon Nikodým property in terms of convergence of norm bounded martingales defined on the Chaney-Schaefer l-tensor product E⊗̃lY . This extends a classical martingale characterization of the Radon Nikodým property, formulated in the Lebesgue-Bochner spaces Lp(μ, Y ) (1 < p < ∞).