Approximation by analytic matrix functions
by
Vladimir Peller
Michigan State University

The talk is based on joint results with L. Baratchart and F. Nazarov. For 2 ≤ p < ∞, we consider a problem of approximating in the norm of L^p a matrix function F on the unit circle by matrix functions analytic in the unit disc. It turns out that the space of matrix functions in L^p splits into two massive subsets: the set of respectable functions and the set of weird functions. For respectable m×n matrix functions F the distance o the set of analytic functions is equal to the norm of the Hankel operator H_F from H^q(Cⁿ) to H²_-(C^m), where 1/p+1/q=1/2. For weird functions F the distance is greater than the norm of the Hankel operator. We have found another distance formula that works for all matrix functions in L^p. We also consider related factorization formulae, describe all best approximants and characterize badly approximable matrix functions.

Date received: May 13, 2008