Relations between the Different Concepts of Summability of Multilinear Mappings between Banach Spaces

The core of the theory of absolutely summing operators lie in the ideas of A. Grothendieck in the 1950s. Further work (after a decade) of A. Pietsch [21] and Lindenstrauss and Pe lczyński [11] clarified Grothendiecks insights and nowadays the ideal of absolutely summing operators is a central topic of investigation. A natural question is how to extend the concept of absolutely summing operators to multilinear mappings and polynomials. A first light in this direction is the work by Alencar-Matos [1], where several classes of multilinear mappings between Banach spaces were investigated. Since then, just concerning to the idea of lifting the ideal of absolutely summing operators to polynomials and multilinear mappings, there are several works in different directions (we mention Matos [12],[13],[14],[15], Dimant [10], Bombal-Pérez-Garćıa-Villanueva [2]). However, there seems to be no effort in the direction of comparing these different classes. The aim of this paper is to investigate the connections between these classes and their properties.