Rearrangement-Invariant Functionals with Applications to Traces on Symmetrically Normed Ideals

We present a construction of singular rearrangement invariant functionals on Marcinkiewicz function/operator spaces. The functionals constructed differ from all previous examples in the literature in that they fail to be symmetric. In other words, the functional ϕ fails the condition that if (Hardy-Littlewood-Polya submajorization) and 0 ≤ x, y, then 0 ≤ ϕ(x) ≤ ϕ(y). We apply our results to singular traces on symmetric operator spaces (in particular on symmetrically-normed ideals of compact operators), answering questions raised by Guido and Isola.