Integral Equations and Operator Theory, Vol. 4/2, 1981 © Birkh~user Verlag, CH--4010 Basel (Switzerland), 1981 INTERPOLATION PROBLEMS AND TOEPLITZ OPERATORS ON MULTIPLY CONNECTED DOMAINS Joseph A. Ball* Let R be a bounded domain in the complex plane bounded by n + 1 nonintersecting analytic Jordan curves, let E, F, and G be flat unitary vector bundles (in the sense of Abrahamse and Douglas) and let O: F~ G and ~: E ~ G be bounded analytic bundle maps. A condition is given for the existence of a bounded analytic map D: E---" F such that OD = ~, together with an estimate for I]D[~o. An interesting special case is the case where E = G and •---IE, for which the condition involves a uniform lower bound for a class of Toeplitz operators over R, all of which are induced (formally) by the N bundle map @ 0 (N = rank E). When interpreted for a finite column of ana- l lytic scalar functions, this special case gives quantitative information on the corona theorem for R. The main tool is the Sz.Nagy-Foias commutant lifting theorem for regions R recently obtained by the author. 1. Introduction Let R be a bounded domain in the complex plane bounded by n + 1 nonintersecting analytic Jordan curves. To keep the exposition as self- contained as possible, we review some definitions from [4]. Let C1 ..... Cn be n *Research supported by National Science Foundation Grant No. MCS 77-00966.