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We study the existence and characterization properties of compact Hermitianoperators C on a separable Hilbert space H such that ||C|| is less or equalthan || C + D ||, for all D in D(K(H)). This property is equivalent to || C ||= min{||C+D||: D in D(K(H))} = dist (C,D(K(H))), where D(K(H)) denotes thespace of compact diagonal operators in a fixed base of H and ||.|| is theoperator norm. We also exhibit a positive trace class operator that fails toattain the minimum in a compact diagonal.

Best approximation by diagonal compact operators