{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/50020"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/50020","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Boundary value and Wiener-Hopf problems for abstract kinetic equations with nonregular collision operators","abstract":"We study the linear abstract kinetic equation T𝜑(x)′=-A𝜑(x) in the half space {x≥0} with partial range boundary conditions. The function <i>ψ</i> takes values in a Hilbert space H, T is a self adjoint injective operator on H and A is an accretive operator. The first step in the analysis of this boundary value problem is to show that T⁻¹A generates a holomorphic bisemigroup. We prove two theorems about perturbation of bisemigroups that are interesting in their own right. The second step is to obtain a special decomposition of H which is equivalent to a Wiener-Hopf factorization. The accretivity of A is crucial in this step. When A is of the form \"identity plus a compact operator\", we work in the original Hilbert space. For unbounded A’s we consider weak solutions in a larger space H<sub>T</sub>, which has a natural Krein space structure. Using the Krein space geometry considerably simplifies the analysis of the question of unique solvability.","abstract_html":"We study the linear abstract kinetic equation T𝜑(x)′=-A𝜑(x) in the half space {x≥0} with partial range boundary conditions. The function &lt;i&gt;ψ&lt;/i&gt; takes values in a Hilbert space H, T is a self adjoint injective operator on H and A is an accretive operator. The first step in the analysis of this boundary value problem is to show that T⁻¹A generates a holomorphic bisemigroup. We prove two theorems about perturbation of bisemigroups that are interesting in their own right. The second step is to obtain a special decomposition of H which is equivalent to a Wiener-Hopf factorization. The accretivity of A is crucial in this step. When A is of the form &quot;identity plus a compact operator&quot;, we work in the original Hilbert space. For unbounded A’s we consider weak solutions in a larger space H&lt;sub&gt;T&lt;/sub&gt;, which has a natural Krein space structure. Using the Krein space geometry considerably simplifies the analysis of the question of unique solvability.","abstract_has_math":false,"creators":["Ganchev, Alexander Hristov"],"institution":"Virginia Polytechnic Institute and State University","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Greenberg, William"],"committee_members":["Ball, Joseph A.","Hagedorn, George","Quinn, Frank","Slawny, Joseph","Zweifel, P.F."],"year":1986,"date_issued":"1986","date_published":"1986","updated_at":"2026-07-22T22:20:09Z","subjects":[],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/50020","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Greenberg, William"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Ball, Joseph A.","Hagedorn, George","Quinn, Frank","Slawny, Joseph","Zweifel, P.F."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Ganchev, Alexander Hristov"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-08-13T14:39:02Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-08-13T14:39:02Z"]},{"key":"dc:date.issued","label":"Date","values":["1986"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute and State University"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"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 first step in the analysis of this boundary value problem is to show that T⁻¹A generates a holomorphic bisemigroup. We prove two theorems about perturbation of bisemigroups that are interesting in their own right. The second step is to obtain a special decomposition of H which is equivalent to a Wiener-Hopf factorization. The accretivity of A is crucial in this step. When A is of the form \"identity plus a compact operator\", we work in the original Hilbert space. For unbounded A’s we consider weak solutions in a larger space H<sub>T</sub>, which has a natural Krein space structure. Using the Krein space geometry considerably simplifies the analysis of the question of unique solvability."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Boundary value and Wiener-Hopf problems for abstract kinetic equations with nonregular collision operators"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Greenberg, William"],"dc:contributor.committeemember":["Ball, Joseph A.","Hagedorn, George","Quinn, Frank","Slawny, Joseph","Zweifel, P.F."],"dc:contributor.department":["Mathematics"],"dc:creator":["Ganchev, Alexander Hristov"],"dc:date.accessioned":["2014-08-13T14:39:02Z"],"dc:date.available":["2014-08-13T14:39:02Z"],"dc:date.issued":["1986"],"dc:description.abstract":["We study the linear abstract kinetic equation T𝜑(x)′=-A𝜑(x) in the half space {x≥0} with partial range boundary conditions. The function <i>ψ</i> takes values in a Hilbert space H, T is a self adjoint injective operator on H and A is an accretive operator. The first step in the analysis of this boundary value problem is to show that T⁻¹A generates a holomorphic bisemigroup. We prove two theorems about perturbation of bisemigroups that are interesting in their own right. The second step is to obtain a special decomposition of H which is equivalent to a Wiener-Hopf factorization. The accretivity of A is crucial in this step. When A is of the form \"identity plus a compact operator\", we work in the original Hilbert space. For unbounded A’s we consider weak solutions in a larger space H<sub>T</sub>, which has a natural Krein space structure. Using the Krein space geometry considerably simplifies the analysis of the question of unique solvability."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/50020"],"dc:publisher":["Virginia Polytechnic Institute and State University"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Boundary value and Wiener-Hopf problems for abstract kinetic equations with nonregular collision operators"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:20:09Z"}