On multi-ideals and polynomial ideals of Banach spaces: a new approach to coherence and compatibility

Abstract

What is an adequate extension of an operator ideal I to the polynomial and multilinear settings? This question motivated the appearance of the concepts of coherent sequences of polynomial ideals and compatibility of a polynomial ideal with an operator ideal, introduced by D. Carando el al. We propose a different approach by consideringis a multi-ideal, instead of considering just polynomial ideals. Our approach ends a discomfort caused by the previous theory: for real scalars the canonical sequence (P k ) ∞ k=1 of continuous k-homogeneous polynomials is not coherent according to the definition of Carando et al.We apply these new notions to test the factorization method and different classes that generalise the concept of absolutely summing operator.

The surjective hull of a polynomial ideal

“…The examples we are about to give concern two of the most studied (perhaps the two most studied) polynomial generalizations of the ideal of absolutely p ‐summing linear operators; namely, the ideals of multiple p ‐summing polynomials and of p ‐dominated polynomials (in both and one can find the two definitions).…”

Coherence and compatibility: a stronger approach

“…In Section 1 we formally present this definition and deduce some general properties of this new concept, comparing especially with the approach presented in [22]. In the following sections, through a series of examples, we show that well-known classes satisfy this new concept with some kind of particularity.…”

“…One question that arises at this point is: Should we develop the study of the coherence and compatibility of multilinear mappings and homogenous polynomials independently or should we compare these classes in any sense? Our approach starts from the principle, defended by Pellegrino and Ribeiro in [22], that yes, we must do a comparative and unified study of these classes. It would theoretically be possible to develop this study in more general classes, but due to the lack of applicability and also due to the good properties of the ideals, we will work with classes of multi-ideal mappings and classes of homogeneous polynomials ideals.…”

Nonexistence of certain universal polynomials between Banach spaces

“…The notion of ideal of k-homogeneous polynomials is well known and has been widely studied in the literature (see, for instance, [6, §3] or the recent paper [24]). …”