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The theory of $\tau$-summing and $\sigma$-nuclear linear operators on Banachspaces was developed by Pietsch [12, Chapter 23]. Extending the linear case tothe range p > 1 and generalizing all cases to the multilinear setting, in thispaper we introduce the concept of $\sigma(p)$-nuclear linear and multilinearoperators. In order to develop the duality theory for the spaces of suchoperators, we introduce the concept of quasi-tau(p)-summing linear/multilinearoperators and prove Pietsch-type domination theorems for such operators. Themain result of the paper shows that, under usual conditions, linear functionalson the space of $\sigma(p)$-nuclear $n$-linear operators are represented, viathe Borel transform, by quasi-$\tau(p)$-summing $n$-linear operators. As far aswe know, this result is new even in the linear case $n = 1$.

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