Nonlinear absolutely summing mappings

Abstract

A mapping f , defined on an open subset A of a Banach space E, with values in another Banach space F , such that (f (a + xj) − f (a)) ∞ j=1 is absolutely summable in F , whenever (xj) ∞ j=1 is unconditionally summable (respectively, absolutely summable) in E, is called absolutely summing (respectively, regularly summing) at the point a ∈ A. It is proved that f is regularly summing at a if, and only if, there are M > 0 and δ > 0, such that f (a+x)−f (a) ≤ M x , for all x ≤ δ. This result has as a consequence a characterization of absolutely summing mappings by means of inequalities. This result is analogous to the well know characterization of the linear absolutely summing mappings. Several results and examples show that the existence of non-linear absolutely summing mappings is not a rare phenomena. A Dvoretzky-Rogers Theorem for n-homogeneous polynomials is proved.

On scalar-valued nonlinear absolutely summing mappings

“…The proof of theorem 1, in [4], is credited to A. Defant and J. Voigt. The case m = 2 of Theorem 2 was previously proved by Botelho [2] and is the unique known coincidence result for dominated polynomials.…”

Characterization of lr-dominated m-linear operators

“…A proof of the general case, including vector-valued operators, can be found in [14,Proposition 3.1]. In fact, we have the following theorem which is the main result of this section.…”

“…This terminology was introduced in the commutative case by Pietsch [19] for scalar valued mappings. The reader interested by previous work on this and related properties can consult [3,5,6,7,8,13,14,16,17].…”

“…Motivated by the importance of this theory, several authors have developed and studied many concepts relating to summability (we mention for example [13,14,16,18] among so many authors). Regarding this, it is natural to try to develop analogous in the noncommutative case.…”

Dominated polynomials on infinite dimensional spaces

“…The theory has been developed by several authors, and among the advances obtained thus far we mention: Pietsch-type domination/factorization theorems ( [11], Geiss [26], Pérez-García [38]), different types of absolutely summing multilinear mappings and polynomials (Achour and Mezrag [1], Carando and Dimant [15], Ç aliskan and Pellegrino [14], Dimant [23], Pellegrino and Souza [36], Pérez-García [38]), Grothendieck-type theorems (Bombal et al [4], Pérez-García and Villanueva [41]), coincidence/inclusion/composition theorems (Alencar and Matos [2], Botelho et al [8], Pérez-García [39,37], Popa [44]), connections with the geometry of Banach spaces ( [5], Floret and Matos [25], Meléndez and Tonge [33], [35], Pérez-García [40]), interplay with other multi-ideals and polynomial ideals (Botelho et al [7], [12], Cilia and Gutiérrez [17], Jarchow et al [27], Matos [29]), estimates for absolutely summing norms (Aron et al [3], [13], Choi et al [16], Defant and Sevilla-Peris [21], Zalduendo [45]), extensions of the theory to more general nonlinear mappings (Junek, Matos and Pellegrino [28], Matos [30,31], Matos and Pellegrino [32]). …”