{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/27334"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/27334","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Schur-class of finitely connected planar domains: the test-function approach","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 ∈ 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.","abstract_html":"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 ∈ 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.","abstract_has_math":false,"creators":["Guerra Huaman, Moises Daniel"],"institution":"Virginia Tech","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Ball, Joseph A."],"committee_members":["Hagedorn, George A.","Renardy, Michael J.","Kim, Jong Uhn"],"year":2011,"date_issued":"2011-04-18","date_published":"2011-04-18","updated_at":"2026-07-22T22:19:26Z","subjects":["completely positive kernel","extreme points.","Schur class","test functions"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-04262011-111257"],"render_values":[{"text":"etd-04262011-111257","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/27334","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Ball, Joseph A."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Hagedorn, George A.","Renardy, Michael J.","Kim, Jong Uhn"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Guerra Huaman, Moises Daniel"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T20:10:58Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T20:10:58Z","2011-05-12"]},{"key":"dc:date.issued","label":"Date","values":["2011-04-18"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. 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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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:title","label":"Title","values":["Schur-class of finitely connected planar domains: the test-function approach"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Ball, Joseph A."],"dc:contributor.committeemember":["Hagedorn, George A.","Renardy, Michael J.","Kim, Jong Uhn"],"dc:contributor.department":["Mathematics"],"dc:creator":["Guerra Huaman, Moises Daniel"],"dc:date.accessioned":["2014-03-14T20:10:58Z"],"dc:date.available":["2014-03-14T20:10:58Z","2011-05-12"],"dc:date.issued":["2011-04-18"],"dc:description.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 ∈ 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."],"dc:description.degree":["Ph. D."],"dc:identifier.other":["etd-04262011-111257"],"dc:identifier.uri":["http://hdl.handle.net/10919/27334"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["completely positive kernel","extreme points.","Schur class","test functions"],"dc:title":["Schur-class of finitely connected planar domains: the test-function approach"],"dc:type":["Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:26Z"}