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 (Formula presented.). We first prove a generalization of a result of Carathéodory. 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 (Formula presented.) -contractive meromorphic functions where (Formula presented.) induces an indefinite metric on (Formula presented.) 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.