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 $\big \Vert T \big \Vert = A_1(T) ≤ A_2(T) ≤ ...$ such that all maps $A_n : T \rightarrow A_n(T)$ are ideal norms. The asymptotic behaviour of $A_n(T)$ as $n \rightarrow ∈fty$ can be used to define various subclasses of operatore. The most simple condition is that $\sup {n^{- \rho}}{A_n(T)} < ∈fty$ where $\rho ≥ 0$ . 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 $H_n(T) := sup \bigl\{ \big\Vert TJ{E \atop M}|H \big\Vert : M ⊆ E, dim(M) ≤ n \bigr\}$ where J{E \atop M} denotes the canonical embedding from the subspace M into E. Note that $(H_n)$ is the natural dimensional gradation of the Hilbertian operator norm $\big\Vert |H \big\Vert$ in the sense of A. Pelczynski ([30], p. 165) and N. Tomczak-Jägermann ([46] and [48], p. 175). Taking the infimum over all $\rho ≥ 0$ with $sup n^\rho H_n(T) < ∈fty$ we get an index $h(T) ⊂ [0,1/2]$ which can be used to measure the <<Hilbertness>> 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.