The double commutants of the direct sum of certain upper triangular matrice

Let A_l = [a_ij(l)] be a strictly upper triangular n×n matrix on Cⁿ with ^ajj+1^(l) ≠ 0 for j=1, 2, ..., n -1 and R=⊕^k_i=1 A_i In this paper, we prove that if S is the double commutants of R, then there are polynomials p_i (Z)=Σ^n-1_j=0 C_ij Z^j of degree less than n-1 such that S is a diagonal matrix with the form S=diag(p₁(A₁), p₂(A₂), ..., p_k(A_k)).