Polynomial ideals from a nonlinear viewpoint

Abstract

Classes of homogeneous polynomials between Banach spaces have been studied in the last three decades from the perspective of the so-called ideal property: if a polynomial P belongs to a class Q, then the composition u • P • v of P with linear operators u and v belongs to Q as well. In an attempt to explore the nonlinearity of the subject in a more consistent way, and taking into account recent results in the field, in this paper we propose the study of classes of homogeneous polynomials that are stable under the composition with homogeneous polynomials. Some important classes justify the study of the intermediate concept of classes of polynomials Q such that if P belongs to Q, u is a linear operator and Q is a homogeneous polynomial, then u • P • Q belongs to Q.

Hyper-ideals of multilinear operators and two-sided polynomial ideals generated by sequence classes

“…• [11,14] Let (C n ) ∞ n=1 be a sequence of positive real numbers such that C n ≥ 1 for every n ∈ N and C 1 = 1. We say that (Q,…”

“…Proof. Using that (L X;Y , • X;Y ) is a Banach hyper-ideal (Theorem 2.3) and that a polynomial P belongs to P X;Y if and only if there is A in L X;Y such that A = P (Theorem 3.1), it follows from [11,Theorem 4.3]…”

“…Composition multi-ideals are always hyper-ideals [10], composition polynomial ideals are always hyper-ideals [11] and sometimes they are two-sided ideals [13]. To make sure that we are creating new hyper and two-sided ideals in this paper, in this section we check that the multi-ideals L X;Y and the polynomial ideals P X;Y are not composition ideals in general.…”

Generalized adjoints and applications to composition operators

“…Operator ideals will be taken in the sense of Pietsch [10,18], ideals of homogeneous polynomials (polynomial ideals) in the sense of García and Floret [13], two-sided polynomial ideals in the sense of [8] and polynomial hyper-ideals in the sense of [7]. For the sake of the reader, we recall these concepts next.…”

Ideals of polynomials between Banach spaces revisited

“…The natural extension to multilinear operators and polynomials was designed by Pietsch some years later in [25]. Nowadays, ideals of polynomials and multilinear operators are explored by several authors in different directions (see, for instance, [1,5,6,8,9,10,11,12,13,14,20]). In this paper we are mainly interested in the theory of ideals of polynomials and ideals of multilinear operators between Banach spaces.…”