Abstract. A famous result due to Grothendieck asserts that every continuous linear operator from ℓ 1 to ℓ 2 is absolutely (1, 1)-summing. If n ≥ 2, however, it is very simple to prove that every continuous n-linear operator from ℓ 1 × · · · × ℓ 1 to ℓ 2 is absolutely (1; 1, ..., 1)-summing, and even absolutely 2 n ; 1, ..., 1 -summing. In this note we deal with the following problem:Given a positive integer n ≥ 2, what is the best constant g n > 0 so that every n-linear operator from ℓ 1 × · · · × ℓ 1 to ℓ 2 is absolutely (g n ; 1, ..., 1)-summing?We prove that g n ≤ 2 n+1 and also obtain an optimal improvement of previous recent results (due to Heinz Juenk et al , Geraldo Botelho et al and Dumitru Popa) on inclusion theorems for absolutely summing multilinear operators.

Abstract. In the last decades many authors have become interested in the study of multilinear and polynomial generalizations of families of operator ideals (such as, for instance, the ideal of absolutely summing operators). However, these generalizations must keep the essence of the given operator ideal and there seems not to be a universal method to achieve this. The main task of this paper is to discuss, study, and introduce multilinear and polynomial extensions of the aforementioned operator ideals taking into account the already existing methods of evaluating the adequacy of such generalizations. Besides this subject's intrinsic mathematical interest, the main motivation is our belief (based on facts that shall be presented) that some of the already existing approaches are not adequate.

No abstract

No abstract

In this short note we present some new results concerning cotype and absolutely summing multilinear operators, extending recent results from different authors. RESUMENEn esta nota presentamos nuevos resultados sobre cotipo y suma absoluta de operadores multilineales, extendiendo resultados recientes de diferentes autores.