Holomorphy types and ideals of multilinear mappings

Abstract

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

Hypercyclicity of convolution operators on spaces of entire functions

“…-Note that the case k = 0 implies that the constant C of condition 3.2(a) is greater than or equal to 1. From [4,Theorem 3.2] it follows that every coherent sequence {A k (E)} k∈N0 is a holomorphy type with constant σ = C. So,…”

“…Recently, Carando, Dimant and Muro [6] proved some far-reaching results -including a solution to a problem posed in [2] -that encompass as particular cases several of the above mentioned results. The main tool they use are the so-called coherent sequences of homogeneous polynomials, introduced by themselves in [7] based on properties of polynomials ideals previously studied in [5,4].…”

$${\mathcal A}$$ A -compact mappings

“…Carando, Dimant and Muro [9] (see also [7]) introduce the concept of coherent sequences of n-homogeneous Banach ideals of polynomials associated with Banach ideal of operators as follows. Given a Banach operator ideal A, the sequence (A n ) n of n-homogeneous polynomials is coherent and associated with A if there exist positive constants A and B such that for every Banach spaces X and Y the following conditions hold:…”

Composition results for strongly summing and dominated multilinear operators

“…Without any claim of completeness we mention: absolutely summing multilinear operators, see [1,2,9,24,37]; multiple summing multilinear operators, see [3,8,28,32]; dominated multilinear operators, see [21,27,30,31,33]. For the theory of polynomials in Banach spaces and their applications the interested reader can consult [17,18,29] and for the connection between holomorphy types and ideals of multilinear mappings we recommend the reader [7].…”