{"id":{"repo_id":"calpoly","oai_identifier":"oai:digitalcommons.calpoly.edu:theses-4066"},"canonical_url":"https://search.dev.ndltd.org/etd/calpoly/oai:digitalcommons.calpoly.edu:theses-4066","repository":{"repo_id":"calpoly","name":"Cal Poly","base_url":"https://digitalcommons.calpoly.edu/do/oai/"},"display":{"title":"On the Numerical Range of Compact Operators","abstract":"<p>One of the many characterizations of compact operators is as linear operators which<br />can be closely approximated by bounded finite rank operators (theorem 25). It is<br />well known that the numerical range of a bounded operator on a finite dimensional<br />Hilbert space is closed (theorem 54). In this thesis we explore how close to being<br />closed the numerical range of a compact operator is (theorem 56). We also describe<br />how limited the difference between the closure and the numerical range of a compact<br />operator can be (theorem 58). To aid in our exploration of the numerical range of<br />a compact operator we spend some time examining its spectra, as the spectrum of a<br />bounded operator is closely tied to its numerical range (theorem 45). Throughout,<br />we use the forward shift operator and the diagonal operator (example 1) to illustrate<br />the exceptional behavior of compact operators.</p>","abstract_html":"&lt;p&gt;One of the many characterizations of compact operators is as linear operators which&lt;br /&gt;can be closely approximated by bounded finite rank operators (theorem 25). It is&lt;br /&gt;well known that the numerical range of a bounded operator on a finite dimensional&lt;br /&gt;Hilbert space is closed (theorem 54). In this thesis we explore how close to being&lt;br /&gt;closed the numerical range of a compact operator is (theorem 56). We also describe&lt;br /&gt;how limited the difference between the closure and the numerical range of a compact&lt;br /&gt;operator can be (theorem 58). To aid in our exploration of the numerical range of&lt;br /&gt;a compact operator we spend some time examining its spectra, as the spectrum of a&lt;br /&gt;bounded operator is closely tied to its numerical range (theorem 45). Throughout,&lt;br /&gt;we use the forward shift operator and the diagonal operator (example 1) to illustrate&lt;br /&gt;the exceptional behavior of compact operators.&lt;/p&gt;","abstract_has_math":false,"creators":["Dabkowski, Montserrat"],"institution":null,"degree_name":"MS in Mathematics","degree_level":null,"degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Linda Patton","Mathematics","College of Science and Mathematics"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-06-01T07:00:00Z","date_published":"2022-06-01T07:00:00Z","updated_at":"2026-07-24T01:32:29Z","subjects":["Numerical Range","Compact Operator","Spectra","Eigenvalue","Analysis","Other Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["10.15368/theses.2022.54"],"render_values":[{"text":"10.15368/theses.2022.54","href":"https://doi.org/10.15368/theses.2022.54","code":true}]}]},"links":{"outbound_url":"https://digitalcommons.calpoly.edu/theses/2492","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Linda Patton","Mathematics","College of Science and Mathematics"]},{"key":"dc:creator","label":"Author","values":["Dabkowski, Montserrat"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.available","label":"Dc Date Available","values":["2022-06-09T07:00:00Z"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["MS in Mathematics"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Numerical Range","Compact Operator","Spectra","Eigenvalue","Analysis","Other Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://digitalcommons.calpoly.edu/theses/2492","10.15368/theses.2022.54"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["<p>One of the many characterizations of compact operators is as linear operators which<br />can be closely approximated by bounded finite rank operators (theorem 25). It is<br />well known that the numerical range of a bounded operator on a finite dimensional<br />Hilbert space is closed (theorem 54). In this thesis we explore how close to being<br />closed the numerical range of a compact operator is (theorem 56). We also describe<br />how limited the difference between the closure and the numerical range of a compact<br />operator can be (theorem 58). To aid in our exploration of the numerical range of<br />a compact operator we spend some time examining its spectra, as the spectrum of a<br />bounded operator is closely tied to its numerical range (theorem 45). Throughout,<br />we use the forward shift operator and the diagonal operator (example 1) to illustrate<br />the exceptional behavior of compact operators.</p>"]},{"key":"dc:title","label":"Title","values":["On the Numerical Range of Compact Operators"]}]}],"canonical_facts":{"dc:contributor":["Linda Patton","Mathematics","College of Science and Mathematics"],"dc:creator":["Dabkowski, Montserrat"],"dc:date.available":["2022-06-09T07:00:00Z"],"dc:description.abstract":["<p>One of the many characterizations of compact operators is as linear operators which<br />can be closely approximated by bounded finite rank operators (theorem 25). It is<br />well known that the numerical range of a bounded operator on a finite dimensional<br />Hilbert space is closed (theorem 54). In this thesis we explore how close to being<br />closed the numerical range of a compact operator is (theorem 56). We also describe<br />how limited the difference between the closure and the numerical range of a compact<br />operator can be (theorem 58). To aid in our exploration of the numerical range of<br />a compact operator we spend some time examining its spectra, as the spectrum of a<br />bounded operator is closely tied to its numerical range (theorem 45). Throughout,<br />we use the forward shift operator and the diagonal operator (example 1) to illustrate<br />the exceptional behavior of compact operators.</p>"],"dc:identifier":["https://digitalcommons.calpoly.edu/theses/2492","10.15368/theses.2022.54"],"dc:subject":["Numerical Range","Compact Operator","Spectra","Eigenvalue","Analysis","Other Mathematics"],"dc:title":["On the Numerical Range of Compact Operators"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_name":["MS in Mathematics"]},"updated_at":"2026-07-24T01:32:29Z"}