Title page for ETD etd-04262011-111257


Type of Document Dissertation
Author Guerra Huaman, Moises Daniel
Author's Email Address moisesgg@vt.edu
URN etd-04262011-111257
Title Schur-class of finitely connected planar domains: the test-function approach
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Ball, Joseph A. Committee Chair
Hagedorn, George A. Committee Member
Kim, Jong Uhn Committee Member
Renardy, Michael J. Committee Member
Keywords
  • completely positive kernel
  • extreme points.
  • Schur class
  • test functions
Date of Defense 2011-04-18
Availability unrestricted
Abstract
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.

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