{"id":{"repo_id":"maryland","oai_identifier":"oai:drum.lib.umd.edu:1903/14691"},"canonical_url":"https://search.dev.ndltd.org/etd/maryland/oai:drum.lib.umd.edu:1903/14691","repository":{"repo_id":"maryland","name":"University of Maryland","base_url":"https://api.drum.lib.umd.edu/server/oai/request"},"display":{"title":"Operational Calculus","abstract":"The Operational Calculus is a construction used for analyzing the behavior of linear operators that arise in the study of ordinary and partial differential equations. Given a linear operator T and a class of functions F, one rigorously defines a new operator f(T) for each f in F and establishes properties of the transformation f -> f(T), among which is that, if F is an algebra of functions, then the transformation induces an algebra homomorphism from F to the algebra of bounded linear operators on a Banach space. This paper begins with a discussion of an operational calculus for compact symmetric operators. This motivates the construction of the Dunford operational calculus for general bounded linear operators. Next, a treatment for bounded symmetric operators is provided, together with a rigorous presentation of all background material. All this is the basis of an operational calculus for unbounded symmetric operators T on a complex Hilbert space. This latter construction is based on a representation theorem of Riesz and Lorch for unbounded self-adjoint operators: the presentation is simpler and more illuminating than the customary one.","abstract_html":"The Operational Calculus is a construction used for analyzing the behavior of linear operators that arise in the study of ordinary and partial differential equations. Given a linear operator T and a class of functions F, one rigorously defines a new operator f(T) for each f in F and establishes properties of the transformation f -> f(T), among which is that, if F is an algebra of functions, then the transformation induces an algebra homomorphism from F to the algebra of bounded linear operators on a Banach space. This paper begins with a discussion of an operational calculus for compact symmetric operators. This motivates the construction of the Dunford operational calculus for general bounded linear operators. Next, a treatment for bounded symmetric operators is provided, together with a rigorous presentation of all background material. All this is the basis of an operational calculus for unbounded symmetric operators T on a complex Hilbert space. This latter construction is based on a representation theorem of Riesz and Lorch for unbounded self-adjoint operators: the presentation is simpler and more illuminating than the customary one.","abstract_has_math":false,"creators":["Sedberry, Trevor Lear"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Mathematics","school":null,"contributors":[],"advisors":["Fitzpatrick, Patrick M"],"committee_chairs":[],"committee_members":[],"year":2013,"date_issued":"2013","date_published":"2013","updated_at":"2026-07-24T03:02:08Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1903/14691","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Fitzpatrick, Patrick M"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Sedberry, Trevor Lear"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2013-10-10T05:37:25Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2013-10-10T05:37:25Z"]},{"key":"dc:date.issued","label":"Date","values":["2013"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1903/14691"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The Operational Calculus is a construction used for analyzing the behavior of linear operators that arise in the study of ordinary and partial differential equations. Given a linear operator T and a class of functions F, one rigorously defines a new operator f(T) for each f in F and establishes properties of the transformation f -> f(T), among which is that, if F is an algebra of functions, then the transformation induces an algebra homomorphism from F to the algebra of bounded linear operators on a Banach space. This paper begins with a discussion of an operational calculus for compact symmetric operators. This motivates the construction of the Dunford operational calculus for general bounded linear operators. Next, a treatment for bounded symmetric operators is provided, together with a rigorous presentation of all background material. All this is the basis of an operational calculus for unbounded symmetric operators T on a complex Hilbert space. This latter construction is based on a representation theorem of Riesz and Lorch for unbounded self-adjoint operators: the presentation is simpler and more illuminating than the customary one."]},{"key":"dc:title","label":"Title","values":["Operational Calculus"]}]}],"canonical_facts":{"dc:contributor.advisor":["Fitzpatrick, Patrick M"],"dc:contributor.department":["Mathematics"],"dc:creator":["Sedberry, Trevor Lear"],"dc:date.accessioned":["2013-10-10T05:37:25Z"],"dc:date.available":["2013-10-10T05:37:25Z"],"dc:date.issued":["2013"],"dc:description.abstract":["The Operational Calculus is a construction used for analyzing the behavior of linear operators that arise in the study of ordinary and partial differential equations. 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This latter construction is based on a representation theorem of Riesz and Lorch for unbounded self-adjoint operators: the presentation is simpler and more illuminating than the customary one."],"dc:identifier.uri":["http://hdl.handle.net/1903/14691"],"dc:title":["Operational Calculus"],"dc:type":["Thesis"]},"updated_at":"2026-07-24T03:02:08Z"}