Analytic approximation of matrix functions in

We consider the problem of approximation of matrix functions of class L^p on the unit circle by matrix functions analytic in the unit disk in the norm of L^p, 2≤p<∞. For an m×n matrix function Φ in L^p, we consider the Hankel operator , 1/p+1/q=1/2. It turns out that the space of m×n matrix functions in L^p splits into two subclasses: the set of respectable matrix functions and the set of weird matrix functions. If Φ is respectable, then its distance to the set of analytic matrix functions is equal to the norm of H_Φ. For weird matrix functions, to obtain the distance formula, we consider Hankel operators defined on spaces of matrix functions. We also describe the set of p-badly approximable matrix functions in terms of special factorizations and give a parametrization formula for all best analytic approximants in the norm of L^p. Finally, we introduce the notion of p-superoptimal approximation and prove the uniqueness of a p-superoptimal approximant for rational matrix functions.