Almost p -summing polynomials and multilinear mappings

Abstract

For several applications it is very useful to classify the linear or non-linear mappings by their summability properties. Absolutely summing operators and polynomials are prominent and classical examples of such setting. Here we are interested in the larger class of almost summing polynomials and we investigate their connections to other related notions of summability.Introduction. The successful theory of operator ideals has been a permanent inspiration for the study of special classes of multilinear mappings and homogeneous polynomials between Banach spaces. There are several methods to generate extensions of a given ideal a of linear operators to the class of multilinear mappings or to the class of polynomials. Among them there are the following two natural settings: (I) Select those multilinear mappings and polynomials which enjoy the same property shared by the linear operators belonging to a. Typical examples are certain compactness or summability properties. (II) Select those multilinear mappings and polynomials which have associated linear operators belonging to a. Concrete examples for this method are the factorization method and the linearization method, respectively, which will be introduced in Section 2 and 5 for the case of almost summing operators. For other operator ideals the methods (I) and (II) have been extensively studied in the past (see, e.g., Alencar/Matos [1], Aron/Lacruz/Ryan/Tonge [2], Botelho [3] and [4], Braunss [5], Braunss/Junek [6] and [7], Floret/Matos [10], Gonza  lez/ Gutie  rrez [11], Junek/Matos [12], Matos [13] and [14], Pietsch [15]). Here we are interested in connections between these methods for the case of all almost summing operators. In Section 2 we compare the method (I) and the factorization method. Next, in Section 3, a scale of almost p-summing multilinear mappings and homogeneous polynomials is generated by method (I). Finally, in Section 5 the linearization method is introduced and the resulting classes of polynomials and multilinear mappings are compared with the classes of almost psumming polynomials and multilinear mappings introduced before. The paper also provides a number of illustrative examples and counterexamples, which are presented in Section 4.

Almost summing mappings

“…The Theorem of Dvoretzky-Rogers for absolutely summing linear operators has natural versions for absolutely summing multilinear mappings and polynomials (see [9]). A linear Dvoretzky-Rogers Theorem for almost p-summing mappings can be found in [2,Ex 4.1] and tells us that if p > 1, then L al,p (E; E) = L(E; E) for every infinite dimensional Banach space E. In this section, we will show that we also have multilinear and polynomial versions for this result.…”

“…In [2,Proposition 5.1] it is shown that if E is an L ∞ space then L( 2 E; K) = L al,2 ( 2 E; K). Next corollary shows that the aforementioned result is still valid for vector valued n-linear mappings, for every n 2.…”

“…. × C(K) into a Banach space F is almost 2-summing, generalizing a recent result obtained in [2]. This paper also analyzes the connections of almost p-summing mappings and type/cotype and provides various examples of analytic almost p-summing mappings.…”

“…Using the polynomial version of this result, it is not hard to prove that (see [12,Theorem 4]) whenever F has finite cotype q, every bounded n-homogeneous (n 2) polynomial P : E → F is absolutely (q; 2)-summing and P as(q; 2) C q (F )K G 3 n−2 2 P λ n , where C q (F ) and K G are the cotype's constant of F and Grothendieck's constant, respectively.…”

Almost summing and multiple almost summing multilinear operators on l p spaces

“…and π as (U ) = inf {C | C as above}, see [3]. In analogy with the linear case in [10], see also [13,14], has introduced the concept of multiple almost summing operators.…”