Best approximation by diagonal compact operators

We study the existence and characterization properties of compact Hermitian operators C   on a Hilbert space HH such that‖C‖⩽‖C+D‖,for all D∈D(K(H)h) or equivalently‖C‖=minD∈D(K(H)h)‖C+D‖=dist(C,D(K(H)h)) where D(K(H)h)D(K(H)h) denotes the space of compact self-adjoint diagonal operators in a fixed base of HH and ‖.‖‖.‖ is the operator norm. We also exhibit a positive trace class operator that fails to attain the minimum in a compact diagonal.