Banach spaces with the $2$-summing property

Abstract

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 ${\ell }_{\infty }^2$ have the $2$-summing property. In the complex case there are more examples; e.g., all subspaces of complex ${\ell }_{\infty }^3$ and their duals.