We study the structure of the set of extreme points of the compact convex set of matrix-valued
holomorphic functions with positive real part on a finitely-connected planar domain R normalized to have value equal to the identity matrix at some prescribed point t0 in R.
This leads to an integral representation for such functions more general than what would be expected from the result for the scalar-valued case. After Cayley transformation, this leads to a integral Agler decomposition for the matrix Schur class over R (holomorphic contractive matrix-valued functions over R). Application of a general theory of abstract Schur-class generated by a collection of test functions leads to a transfer-function realization for the matrix Schur-class over R, extending results known up to now only for the scalar case. We also explain how these results provide a new perspective for the dilation theory for Hilbert space operators having R as a spectral set.
