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A Problem in Linear Matrix Approximation

Let A be a normal operator in ℬ(H), H a complex Hilbert space, and let ℛ A = ≷ {AX - XA:X ∈ ℬ(H)} be the commutator subspace of ℬ(H) associated with A. If B in ℬ(H) commutes with A, then B is orthogonal to ℛA with respect to the spectral norm; i.e., the null operator is an element of best approximation of B in ℛ A. This was proved by J. Anderson in 1973 and extended by P. J. Maher with respect to the Schatten p-norm recently. 
 
 
 
We take a look at their result from a more approximation theoretical point of view in the finite dimensional setting; in particular, we characterize all elements of best approximation of B in RA and prove that the metric projection of H onto ℛA is continuous.