Abstract. We explore a condition under which the ideal of polynomials generated by an ideal of multilinear mappings between Banach spaces is a global holomorphy type. After some examples and applications, this condition is studied in its own right. A final section provides applications to the ideals formed by multilinear mappings and polynomials which are absolutely (p; q)-summing at every point.Introduction. Special classes of homogeneous polynomials between Banach spaces have been studied by taking two different approaches. On the one hand, inspired by the dual theory of polynomials, L. Nachbin [19] introduced holomorphy types as classes of polynomials which are uniformly stable under differentiation. On the other hand, as a natural consequence of the successful theory of operator ideals, A. Pietsch [25] introduced the notion of ideals of multilinear mappings, which was immediately adapted to polynomials.Both notions have been widely studied, holomorphy types as a branch of infinite-dimensional holomorphy (see the references cited in [12, p. 135]) and ideals of multilinear mappings/polynomials as a branch of Banach space theory (see [3,9] and the references therein). Many outstanding examples, such as nuclear and compact polynomials, are simultaneously holomorphy types and ideals of polynomials. In this paper we try to give a unified treatment of the subject, exploring the interplay between these two notions, mainly trying to identify when an ideal of multilinear mappings generates a (global) holomorphy type. Recently, some particular ideals of polynomials have been proved to be global holomorphy types, for example: everywhere absolutely (p; q)-summing polynomials (M. Matos [17]), strongly almost q-summing

Using complex interpolation we prove new inclusion and coincidence theorems for multiple (fully) summing multilinear and holomorphic mappings. Among several other results we show that continuous n-linear forms on cotype 2 spaces are multiple (2; q k , ..., q k )-summing, where 2 k−1 < n ≤ 2 k , q 0 = 2 and q k+1 = 2q k 1+q k for k ≥ 0.

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.

No abstract

No abstract