  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 $n = 1$.
