The authors show that if (\(E, \| \cdot \| _{E}\)) is a symmetric Banach sequence space (i.e., \(E\) is a sequence space invariant under permutations such that (\(E, \| \cdot \| _{E}\)) is complete and has the property that \(f \in E\), \(g^\ast \leq f^\ast\) implies \(g \in E\) and \(\| g\| _E \leq \| f\| _E\)), then the corresponding space \({\mathcal S}_E\) of operators on a separable Hilbert space, defined by \(T\in{\mathcal S}_E\) if and only if \(\left(s_n(T)\right)^\infty_{n=1} \in E\), is a Banach space under the norm \(\| T\| _{{\mathcal S}_E}=\| \left(s_n(T)\right)^\infty_{n=1}\| _E\). Although this was proved for finite-dimensional spaces by J.von Neumann in [Rev.Tomsk Univ.1, 286–299 (1937; Zbl 0017.09802)], it has never been established in complete generality in infinite-dimensional spaces; previous proofs have used the stronger hypothesis of full symmetry on \(E\), defined in terms of Hardy–Littlewood majorization and equivalent to the requirement that \(E\) is a \(1\)-interpolation space between \(\ell_1\) and \(\ell_\infty\), or the hypothesis of relative full symmetry, i.e., that \(E\) is a closed subspace of a fully symmetric space.
In the classical literature, the definition of a symmetric Banach sequence space very often involves some additional properties which imply full symmetry or relative full symmetry. However, as shown by the authors and others, there are examples of Banach sequence spaces which are not (relatively) fully symmetric, e.g., the Marcinkiewicz sequence space \({\mathcal L}^{1,\infty}\). Thus the known results in the literature are far from a complete solution of the problem of the extension of von Neumann’s result to infinite dimensions.
The proof that \(\| \cdot\| _{{\mathcal S}_E}\) is a norm requires the apparently new fundamental concept of uniform Hardy–Littlewood majorization, somewhat related to calculations of Banach envelopes of some weak type spaces; completeness also requires a new proof. The authors also give the analogous results for operator spaces modelled on a semifinite von Neumann algebra with a normal faithful semi-finite trace and prove some generalizations to \(p\)-convex quasi-Banach function spaces.