M ar 1 99 4 BANACH SPACES WITH THE 2 - SUMMING PROPERTY

A Banach space X has the 2-summing property if the norm of every linear operator from X to a Hilbert space is equal to the 2-summing norm of the operator. Up to a point, the theory of spaces which have this property is independent of the scalar field: the property is self-dual and any space with the property is a finite dimensional space of maximal distance to the Hilbert space of the same dimension. In the case of real scalars only the real line and real ℓ 2 ∞ have the 2-summing property. In the complex case there are more examples; e.g., all subspaces of complex ℓ 3 ∞ and their duals. Some important classical Banach spaces; in particular, C(K) spaces, L 1 spaces, the disk algebra; as well as some other spaces (such as quotients of L 1 spaces by reflexive subspaces [K], [Pi]), have the property that every (bounded, linear) operator from the space into a Hilbert space is 2-summing. (Later we review equivalent formulations of the definition of 2-summing operator. Here we mention only that an operator T : X → ℓ 2 is 2-summing provided that for all operators u : ℓ 2 → X the composition T u is a Hilbert-Schmidt operator; moreover, the 2-summing norm π 2 (T) of T is the supremum of the Hilbert-Schmidt norm of T u as u ranges over all norm one operators u : ℓ 2 → X.) In this paper we investigate the isometric version of this property: say that a Banach space X has the 2-summing property provided that π 2 (T) = T for all operators T : X → ℓ 2. While the 2-summing property is a purely Banach space concept and our investigation lies purely in the realm of Banach space theory, part of the motivation for studying the 2-summing property comes from operator spaces. In [Pa], Paulsen defines for a Banach space X the parameter α(X) to be the supremum of the completely bounded norm of T as T ranges over all norm one operators from X into the space B(ℓ 2) of all bounded linear operators on ℓ 2 and asks which spaces X have the property that α(X) = 1. Paulsen's problem and study of α(X) is motivated by old results of von Neumann, Sz.-Nagy, Arveson, and Parrott as well as more recent research of Misra and Sastry. The connection …