Analytic approximation of rational matrix functions

For a rational matrix function Phi with poles outside the unit circle, we estimate the degree of the unique superoptimal approximation A Phi by matrix functions analytic in the unit disk. We obtain sharp estimates in the case of 2 x 2 matrix functions. It turns out that "generically" deg A Phi <= deg Phi - 2. We prove that for an arbitrary 2 x 2 rational function Phi, deg A Phi <= 2 deg Phi - 3 whenever deg Phi >= 2. On the other hand, for k >= 2, we construct a 2 x 2 matrix function Phi, for which deg Phi = k, while deg A Phi = 2k - 3. Moreover, we conduct a detailed analysis of the situation when the inequality deg A Phi <= deg Phi - 2 can violate and obtain best possible results.