ANALYTIC APPROXIMATION OF MATRIX FUNCTIONS IN L p L. BARATCHART, F.L. NAZAROV, AND V.V. PELLER Abstract. 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 H Φ : H q (C n ) → H 2 - (C m ), 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 approx- imants 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. 1. Introduction The classical problem of analytic approximation of functions on the unit circle T is for a given function ϕ ∈ L ∞ , to find a best H ∞ approximant to ϕ, i.e., to find a bounded analytic function ψ in the unit disk D such that ‖ϕ − ψ‖ L ∞ (T) = dist L ∞(ϕ,H ∞ ). A standard compactness argument shows that such a best approximant always exists, though it is not necessarily unique in general. However, under certain mild assumptions the best approximation is indeed unique. For example, this happens if ϕ is continuous which was proved for the first time in [Kha]. We refer the reader to [Pe1] for a comprehensive study of the problem of best uniform approximation by analytic functions. It turns out that this approximation problem is closely related to Hankel oper- ators on the Hardy class H 2 . For a function ϕ ∈ L ∞ the Hankel operator H ϕ : H 2 → H 2 − def = L 2 ⊖ H 2 The second author is partially supported by NSF grant DMS 0501067, the third author is partially supported by NSF grant DMS 0700995. 1