Sequences of ideal norms


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.

DOI Code: 10.1285/i15900932v10supn2p411

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