  Using results from 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 $H^\infty$. We first
prove a generalization of a result of Carath\'eodory. 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 $\mathbb C^N$
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.
