Analytic approximation of rational matrix functions V. PellerV. Vasyunin 47B35superoptimal approximationHankel operatorMcMillan degreerational matrix function For a rational matrix function $\Phi$ with poles outside the unit circle, we estimate the degree of the unique superoptimal approximation $\mathcal{A}\Phi$ by matrix functions analytic in the unit disk. We obtain sharp estimates in the case of $2 \times 2$ matrix functions. It turns out that "generically`" $\deg \mathcal{A}Phi \le \deg\Phi - 2$. We prove that for an arbitrary $2 \times 2$ rational function $\Phi$, $\deg\mathcal{A}\Phi \le 2\deg\Phi - 3$ whenever $\deg\Phi \ge 2$. On the other hand, for $k \ge 2$, we construct a $2 \times 2$ matrix function $\Phi$, for which $\deg\Phi = k$, while $\deg\mathcal{A}Phi = 2k - 3$. Moreover, we conduct a detailed analysis of the situation when the inequality $\deg\mathcal{A}\Phi \le \deg\Phi - 2$ can violate and obtain best possible results. Indiana University Mathematics Journal 2007 text pdf 10.1512/iumj.2007.56.3075 10.1512/iumj.2007.56.3075 en Indiana Univ. Math. J. 56 (2007) 1913 - 1937 state-of-the-art mathematics http://iumj.org/access/