A PROBLEM IN LINEAR MATRIX APPROXIMATION

Let A be a normal operator in B(H), H a complex Hilbert space, and let R(A) = greater than or less than {AX - XA:X is an element of B(H)} be the commutator subspace of B(H) associated with A. If B in B(H) commu tes with A, then B is orthogonal to R(A), 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 a t their result from a more approximation theoretical point of view in the finite dimensional setting; in particular, we characterize all ele ments of best approximation of B in R(A), and prove that the metric pr ojection of H onto R(A) is continuous.