Spaces of σ(p)-nuclear linear and multilinear operators on Banach spaces and their duals

The theory of τ-summing and σ-nuclear linear operators on Banach spaces was developed by Pietsch [20, Chapter 23]. Extending the linear case to the range p>1 and generalizing all cases to the multilinear setting, in this paper we introduce the concept of σ(p)-nuclear linear and multilinear operators. In order to develop the duality theory for the spaces of such operators, we introduce the concept of quasi-τ(p)-summing linear/multilinear operators and prove Pietsch-type domination theorems for such operators. The main result of the paper shows that, under usual conditions, linear functionals on the space of σ(p)-nuclear n-linear operators are represented, via the Borel transform, by quasi-τ(p)-summing n-linear operators. As far as we know, this result is new even in the linear case n=1.