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

Geraldo Botelho, Ximena Mujica

Published 2016-08-24Version 1

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.