An Extremal Problem in Interpolation Theory

If z1, z2, . . . , zn are complex numbers in the open unit disk D and A1, A2, . . . , An are N ×N matrices, let F denote the family of analytic functions, bounded in D, such that for each F ∈ F , F (zk) = Ak, k = 1, 2, . . . , n. We consider ρ = infF∈F supz∈D |F (z)|sp where | · |sp denotes the spectral radius. H. Bercovici has raised the question whether this infimum is attained. We will show that the answer is affirmative for N ≤ 3, and we point out at the obstructions to generalize this result to the case N > 3.