Dense strong continuity of mappings and the Radon-Nikodym property.
In a former paper we describe the geometric properties of the space of continuous functions with values in the space of operators acting on a Hilbert space. In particular we show that dent B(L(H)) = ext B(L(H)) if dim H < 8 and card K < 8 and dent B(L(H)) = 0 if dim H < 8 or card K = 8, and x-ext C(K,L(H)) = ext C(K,L(H)).
We deal with the weak Radon-Nikodým property in connection with the dual space of (X,Y), the space of compact operators from a Banach space X to a Banach space Y. First, under the weak Radon-Nikodým property, we give a representation of that dual. Next, using this representation, we provide some applications to the dual spaces of (X,Y) and , the space of weak*-weakly continuous operators.
Several properties of weakly p-summable sequences and of the scale of p-converging operators (i.e., operators transforming weakly p-summable sequences into convergent sequences) in projective and natural tensor products with an lp space are considered. The last section studies the Dunford-Pettis property of order p (i.e., every weakly compact operator is p-convergent) in those spaces.