Interpolation problems and Toeplitz operators on multiply connected domains

Abstract

LetR be a bounded domain in the complex plane bounded by n + 1 nonintersecting analytic Jordan curves, letE, F, andG be flat unitary vector bundles (in the sense of Abrahamse and Douglas) and let Θ:F → G and ψ:E → G be bounded analytic bundle maps. A condition is given for the existence of a bounded analytic map D:E → F such that ΘD = ψ, together with an estimate for ∥D∥∞. An interesting special case is the case whereE = G and ψ = I E , for which the condition involves a uniform lower bound for a class of Toeplitz operators overR, all of which are induced (formally) by the bundle map\(\begin{array}{*{20}c} N \\ \oplus \\ l \\ \end{array} \) Θ (N = rankE). When interpreted for a finite column of analytic scalar functions, this special case gives quantitative information on the corona theorem forR. The main tool is the Sz.Nagy-Foias commutant lifting theorem for regionsR recently obtained by the author.