Sequences of ideal norms

Abstract

There is a host of possibilities to associate with every (bounded linear) operator T, acting between Banach spaces, a scalar sequence such that all maps are ideal norms. The asymptotic behaviour of as can be used to define various subclasses of operatore. The most simple condition is that where . Tris yiehis a 1-parameter scale of Banach operator ideals. In what follows, this construction will be applied in some concrete cases. In particular, we let where J{E \atop M} denotes the canonical embedding from the subspace M into E. Note that is the natural dimensional gradation of the Hilbertian operator norm in the sense of A. Pelczynski ([30], p. 165) and N. Tomczak-Jagermann ([46] and [48], p. 175). Taking the infimum over all with we get an index which can be used to measure the > of the operator T. Our main purpose is to show that several sequences of concrete ideal norms have the same asymptotic behaviour. This solves a problem posed in ([48],p. 210). We also give some applications to the geometry of Banach spaces. Conceming the basic definitions and various results from the theory of operator ideals, the reader is referred to my monographs [31] and [32]. The notation is adopted from the latter. The present paper is a revised and extended version of my preprint [36]. This revision became necessary when I observed that its main result was already contained in Remark 13.4 of G. Pisier's book [43]; see 5.3 below.