Nonlinear absolutely summing operators revisited

Abstract

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.

Generalized Multiple Summing Multilinear Operators on Banach Spaces

“…The concepts of mixed m(s, q)-summing sequences and multiple mixing (s, q, p)-summing operators were studied in [1,18,20]. Just reminding the main definition, for 1 ≤ p 1 , .…”

“…Several aspects of the theory of summing multilinear operators show that the class of multiple summing multilinear operators is one of the most suitable and useful approaches to the nonlinear theory of absolutely summing operators. Details can be found, e.g., in [1,5,9,12,16,22,23,19].…”

Transformations of sequence spaces by multipolinomials

“…In an attempt to mitigate such difficulty, we introduce a new approach for multipolynomials where each of them is associated with a multilinear application with domain E n 1 1 × • • • × E nm m acting on F . One of our first goals in this work is to investigate, under what conditions, the abovementioned application T :…”

“…. , p n )-mixing summing [1,8], the mid summing [4,6,10] and the almost summing [9]. As such constructs followed a similar script, there arose the natural concern of constructing an abstract approach for such multilinear operators and the homogeneous polynomials [3,10,11].…”

“…Following standard arguments, it is possible to prove that: Theorem 3.8. Suppose that γ s 1 (K) σ 1 (1) , . .…”