On permutations,convex hulls,and normal operators

A spectral characterization is obtained for those normal operators which belong to the convex hull of the unitary orbit of a given normal operator on a finite-dimensional space. This is used to prove the following: if A and B are normal operators on an n-dimensional complex Hilbert space H with eigenvalues given by α₁,…,α_n and β₁,…, β_n respectively, and if A ? B is also normal, then $\text{6A ? B6 ?}\text{max}\text{σ ? S}{}_{\mathrm{n}}\text{6}\text{diag}\text{(α}{}_{\mathrm{k}}\text{?β}{}_{\mathrm{\sigma (k)}}\text{)6}$ for any unitarily invariant norm on L(H).