Guerra Huaman, Moises Daniel2014-03-142014-03-142011-04-18etd-04262011-111257http://hdl.handle.net/10919/27334We 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 ∈ 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.In Copyrightcompletely positive kernelextreme points.Schur classtest functionsDissertationhttp://scholar.lib.vt.edu/theses/available/etd-04262011-111257/