Banach spaces with the 2-summing property

Abstract

A Banach space X X has the 2 2 -summing property if the norm of every linear operator from X X to a Hilbert space is equal to the 2 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 \ell _\infty ^2 have the 2 2 -summing property. In the complex case there are more examples; e.g., all subspaces of complex ℓ ∞ 3 \ell _\infty ^3 and their duals.