Symmetric norms and spaces of operators

We show that if (E,parallel to . parallel to (E)) is a symmetric Banach sequence space then the corresponding space S-E of operators on a separable Hilbert space, defined by T is an element of S-E if and only if (s(n)(T))(n=1)(infinity) is an element of E, is a Banach space under the norm parallel to T parallel to(SE) =parallel to (s(n)(T))(n=1)(infinity) parallel to E. Although this was proved for finite-dimensional spaces by von Neumann in 1937, it has never been established in complete generality in infinite-dimensional spaces; previous proofs have used the stronger hypothesis of full symmetry on E. The proof that parallel to . parallel to(SE) is a norm requires the apparently new concept of uniform Hardy-Littlewood majorization; completeness also requires a new proof. We also give the analogous results for operator spaces modelled on a semifinite von Neumann algebra with a normal faithful semi-finite trace.