{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/28807"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/28807","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Structure of Invariant Subspaces for Left-Invertible Operators on Hilbert Space","abstract":"This dissertation is primarily concerned with studying the invariant subspaces of left-invertible, weighted shifts, with generalizations to left-invertible operators where applicable. The two main problems that are researched can be stated together as When does a weighted shift have the one-dimensional wandering subspace property for all of its closed, invariant subspaces? This can fail either by having a subspace that is not generated by its wandering subspace, or by having a subspace with an index greater than one. For the former we show that every left-invertible, weighted shift is similar to another weighted shift with a residual space, with respect to being generated by the wandering subspace, of dimension $n$, where $n$ is any finite number. For the latter we derive necessary and sufficient conditions for a pure, left-invertible operator with an index of one to have a closed, invariant subspace with an index greater than one. We use these conditions to show that if a closed, invariant subspace for an operator in a class of weighted shifts has a vector in $l^1$, then it must have an index equal to one, and to produce closed, invariant subspaces with an index of two for operators in another class of weighted shifts.","abstract_html":"This dissertation is primarily concerned with studying the invariant subspaces of left-invertible, weighted shifts, with generalizations to left-invertible operators where applicable. The two main problems that are researched can be stated together as When does a weighted shift have the one-dimensional wandering subspace property for all of its closed, invariant subspaces? This can fail either by having a subspace that is not generated by its wandering subspace, or by having a subspace with an index greater than one. For the former we show that every left-invertible, weighted shift is similar to another weighted shift with a residual space, with respect to being generated by the wandering subspace, of dimension $n$, where $n$ is any finite number. For the latter we derive necessary and sufficient conditions for a pure, left-invertible operator with an index of one to have a closed, invariant subspace with an index greater than one. We use these conditions to show that if a closed, invariant subspace for an operator in a class of weighted shifts has a vector in <span class=\"etd-inline-math\">l<sup>1</sup></span>, then it must have an index equal to one, and to produce closed, invariant subspaces with an index of two for operators in another class of weighted shifts.","abstract_has_math":true,"creators":["Sutton, Daniel Joseph"],"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":["Sun, Shu-Ming","Klaus, Martin","Johnson, Martin E."],"year":2010,"date_issued":"2010-08-12","date_published":"2010-08-12","updated_at":"2026-07-22T22:18:52Z","subjects":["Index","Wandering Subspace","Invariant Subspace","Weighted Shift","Left-Invertible"],"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-08262010-161822"],"render_values":[{"text":"etd-08262010-161822","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/28807","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":["Sun, Shu-Ming","Klaus, Martin","Johnson, Martin E."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Sutton, Daniel Joseph"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T20:15:44Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T20:15:44Z","2010-09-15"]},{"key":"dc:date.issued","label":"Date","values":["2010-08-12"]},{"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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The two main problems that are researched can be stated together as When does a weighted shift have the one-dimensional wandering subspace property for all of its closed, invariant subspaces? This can fail either by having a subspace that is not generated by its wandering subspace, or by having a subspace with an index greater than one. For the former we show that every left-invertible, weighted shift is similar to another weighted shift with a residual space, with respect to being generated by the wandering subspace, of dimension $n$, where $n$ is any finite number. For the latter we derive necessary and sufficient conditions for a pure, left-invertible operator with an index of one to have a closed, invariant subspace with an index greater than one. We use these conditions to show that if a closed, invariant subspace for an operator in a class of weighted shifts has a vector in $l^1$, then it must have an index equal to one, and to produce closed, invariant subspaces with an index of two for operators in another class of weighted shifts."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:title","label":"Title","values":["Structure of Invariant Subspaces for Left-Invertible Operators on Hilbert Space"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Ball, Joseph A."],"dc:contributor.committeemember":["Sun, Shu-Ming","Klaus, Martin","Johnson, Martin E."],"dc:contributor.department":["Mathematics"],"dc:creator":["Sutton, Daniel Joseph"],"dc:date.accessioned":["2014-03-14T20:15:44Z"],"dc:date.available":["2014-03-14T20:15:44Z","2010-09-15"],"dc:date.issued":["2010-08-12"],"dc:description.abstract":["This dissertation is primarily concerned with studying the invariant subspaces of left-invertible, weighted shifts, with generalizations to left-invertible operators where applicable. The two main problems that are researched can be stated together as When does a weighted shift have the one-dimensional wandering subspace property for all of its closed, invariant subspaces? This can fail either by having a subspace that is not generated by its wandering subspace, or by having a subspace with an index greater than one. For the former we show that every left-invertible, weighted shift is similar to another weighted shift with a residual space, with respect to being generated by the wandering subspace, of dimension $n$, where $n$ is any finite number. For the latter we derive necessary and sufficient conditions for a pure, left-invertible operator with an index of one to have a closed, invariant subspace with an index greater than one. We use these conditions to show that if a closed, invariant subspace for an operator in a class of weighted shifts has a vector in $l^1$, then it must have an index equal to one, and to produce closed, invariant subspaces with an index of two for operators in another class of weighted shifts."],"dc:description.degree":["Ph. D."],"dc:identifier.other":["etd-08262010-161822"],"dc:identifier.uri":["http://hdl.handle.net/10919/28807"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Index","Wandering Subspace","Invariant Subspace","Weighted Shift","Left-Invertible"],"dc:title":["Structure of Invariant Subspaces for Left-Invertible Operators on Hilbert Space"],"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":"2026-07-22T22:18:52Z"}