Asymptotic Versions of Operators and Operator Ideals

Summary

The goal of this note is to introduce new classes of operator ideals, and, moreover, a new way of constructing such classes through an application to operators of the asymptotic structure recently introduced by Maurey, Milman, and Tomczak-Jaegermann in Op. Th. Adv. Appl. 77 (1995), 149-175.

1. Preliminaries

1.1. Notation. We follow standard Banach-space theory notation, as outlined in [LT_z]. Throughout this note X will be an infinite dimensional Banach-space with a shrinking basis

The notation [X]_n will stand for the head subspace (span and for the tail subspace (the closure of span P_n and P_>n are the coordinate-orthogonal projections on these subspaces respectively.

A vector is called a block if it has finite support, that is, if it is a finite linear combination of elements of the basis. The blocks v and ware said to be consecutive (υ < w) if the support of υ (the set of elements of the basis that form υ as a linear combination) ends before the support of w begins. is the collection of all n-tuples of consecutive normalised blocks; thus means normalised finite support vectors.

1.2. An intuitive introduction to asymptotic structure. The language of asymptotic structure has been introduced to study the essentially infinite dimensional structure of Banach-spaces, and to help bridge between finite dimensional and infinite dimensional theories. This approach does generalise spreading models, but takes an essentially different view. Formally introduced in [MMiT], it has already been studied, extended and applied in [KOS], [OTW], [T], [W1] and [W2]; it is closely related to the new surge of results in infinite dimensional Banach-space theory, and especially to [G] and [MiT].