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

TL;DR

This paper extends the theory of $ au$-summing and $\sigma$-nuclear operators to the multilinear setting for p > 1, establishing duality results and introducing quasi-$ au(p)$-summing operators.

Contribution

It introduces the concept of $\sigma(p)$-nuclear multilinear operators and develops their duality theory, including Pietsch-type domination theorems, generalizing previous linear results.

Findings

Duality between $\sigma(p)$-nuclear and quasi-$ au(p)$-summing operators

Representation of linear functionals via Borel transform in the multilinear setting

New results even in the linear case $n=1$

Abstract

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 $n = 1$.