{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71247"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71247","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Approximation of Generalized Schur Complements (Ladder Network)","abstract":"The Schur complement of a linear operator on a finite dimensional Hilbert space is discussed and six generalizations are presented. Conditions are established under which these generalizations are equivalent. The infinite dimensional case is then considered and counterexamples are given to show that not all the generalizations extend in a natural way. An approximating sequence is shown to exist for the Schur complement matrix equation. The sequences which work as approximating sequences are classified. Finally, an application to infinite ladder networks of operators is given.","abstract_html":"The Schur complement of a linear operator on a finite dimensional Hilbert space is discussed and six generalizations are presented. Conditions are established under which these generalizations are equivalent. The infinite dimensional case is then considered and counterexamples are given to show that not all the generalizations extend in a natural way. An approximating sequence is shown to exist for the Schur complement matrix equation. The sequences which work as approximating sequences are classified. Finally, an application to infinite ladder networks of operators is given.","abstract_has_math":false,"creators":["Butler, Charles Allen"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:17Z","date_published":"2014-12-16T06:18:17Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8711776"],"render_values":[{"text":"(UMI)AAI8711776","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71247","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Butler, Charles Allen"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:17Z","10000-01-01","1987"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71247","(UMI)AAI8711776"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The Schur complement of a linear operator on a finite dimensional Hilbert space is discussed and six generalizations are presented. Conditions are established under which these generalizations are equivalent. The infinite dimensional case is then considered and counterexamples are given to show that not all the generalizations extend in a natural way. An approximating sequence is shown to exist for the Schur complement matrix equation. The sequences which work as approximating sequences are classified. Finally, an application to infinite ladder networks of operators is given.","Made available in DSpace on 2014-12-16T06:18:17Z (GMT). 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The infinite dimensional case is then considered and counterexamples are given to show that not all the generalizations extend in a natural way. An approximating sequence is shown to exist for the Schur complement matrix equation. The sequences which work as approximating sequences are classified. Finally, an application to infinite ladder networks of operators is given.","Made available in DSpace on 2014-12-16T06:18:17Z (GMT). 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