Constructing hyper-ideals of multilinear operators between Banach spaces

Abstract

In view of the fact that some classical methods to construct multi-ideals fail in constructing hyper-ideals, in this paper we develop two new methods to construct hyper-ideals of multilinear operators between Banach spaces. These methods generate new classes of multilinear operators and show that some important well studied classes are Banach or p-Banach hyper-ideals.

Polynomial ideals from a nonlinear viewpoint

“…Reasoning as in the proof of [16,Theorem 3.3] with the help of the series criterion for polynomial hyper-ideals (Theorem 2.5), we get the Theorem 6.2. Let I be a p-Banach operator ideal.…”

“…Proof. We shall just check the hyper-ideal property (the remaining conditions follow the same steps of the proof of [16,Theorem 2.6]). Let P ∈ P(X ; Y)( n E; F ), Q ∈ P( m G; F ) and t ∈ L(F ; H).…”

“…Let (X ; Y) be the multilinear hyper-ideal determined by the sequence functors X and Y (cf. [16,Definition 2.4]). Our next goal is to compare the class P(X ; Y) with the classes P (X ;Y) and P (X ;Y) .…”

“…By L I we denote the multilinear hyper-ideal generated by the boundedness method (see [16,Section 3]), that is, a multilinear operator A ∈ L(E 1 , . .…”

“…Observe that this kind of composition also makes sense for multilinear operators. The corresponding concept for multilinear operators, called multilinear hyper-ideals, was studied in [15,16], a study from which we shall take advantage repeatedly.…”