Spaces of $\sigma (p)$-nuclear linear and multilinear operators and their duals

The theory of $\tau$-summing and $\sigma$-nuclear linear operators on Banach spaces was developed by Pietsch [12, 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 $\sigma (p)$-nuclear linear and multilinear operators. In order to develop the duality theory for the spaces of such operators, we introduce the concept of quasi-tau(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 $\sigma (p)$-nuclear $n$-linear operators are represented, via the Borel transform, by quasi-$\tau (p)$-summing $n$-linear operators. As far as we know, this result is new even in the linear case $n=1$.