On permutations, convex hulls, and normal operators

AbstractA 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,…,αn and β1,…, βn respectively, and if A − B is also normal, then ‖A − B‖ ⩽ maxσ ϵ Sn ‖ diag(αk −βσ(k))‖ for any unitarily invariant norm on L(H)