Best approximation by diagonal compact operators

We study the existence and characterization properties of compact Hermitian operators C on a Hilbert space H such that parallel to C parallel to <= parallel to C + D parallel to, for all D is an element of D(kappa(H)(h)) or equivalently parallel to C parallel to = min(D is an element of D(kappa(H)h)) parallel to C + D parallel to = dist(C, D(kappa(H)(h))) where D(kappa(H)(h)) denotes the space of compact self-adjoint diagonal operators in a fixed base of H and parallel to.parallel to is the operator norm. We also exhibit a positive trace class operator that fails to attain the minimum in a compact diagonal. (C) 2013 Elsevier Inc. All rights reserved.