A dilation theoretic approach to approximation by inner functions

Using results from the theory of operators on a Hilbert space, we prove approximation results for matrix-valued holomorphic functions on the unit disc and the unit bidisc. The essential tools are the theory of unitary dilation of a contraction and the realization formula for functions in the unit ball of H-infinity. We first prove a generalization of a result of Caratheodory. This generalization has many applications. A uniform approximation result for matrix-valued holomorphic functions which extend continuously to the unit circle is proved using the Potapov factorization. This generalizes a theorem due to Fisher. Approximation results are proved for matrix-valued functions for whom a naturally associated kernel has finitely many negative squares. This uses the Krein-Langer factorization. Approximation results for J-contractive meromorphic functions where J induces an indefinite metric on C-N are proved using the Potapov-Ginzburg theorem. Moreover, approximation results for holomorphic functions on the unit disc with values in certain other domains of interest are also proved.