  We study the existence and characterization properties of compact Hermitian
operators C on a separable Hilbert space H such that ||C|| is less or equal
than || 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 the
space of compact diagonal operators in a fixed base of H and ||.|| is the
operator norm. We also exhibit a positive trace class operator that fails to
attain the minimum in a compact diagonal.
