Some characterizations of minimal compact normal operators

In this note, we consider properties of a compact normal operator A such that‖A‖⩽‖A+D‖for all D∈D(K(H)), or equivalently,‖A‖=dist(A,D(K(H))), where ‖⋅‖ denotes the usual operator norm and D(K(H)) is the subalgebra of diagonal compact operators in a fixed orthonormal basis of the Hilbert space H.