{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/80999"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/80999","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Functions of subnormal operators","abstract":"If f is analytic in a neighborhood of ∂D = {z| |z|= 1} and if K = f(∂D), then C-K has only finitely many components; moreover, if U is a bounded simply connected region of the plane, then ∂U = Uj=0nj where each rj is a rectifiable Jordan curve and ri ∩ rj is a finite set whenever i ≠ j. Let μ be a positive regular Borel measure supported on ∂D and let m denote normalized Lebesgue measure on ∂D. If L is a compact set such that ∂L ⊂ K and R(L) is a Dirichlet algebra and if ν = μof⁻¹, then the Lebesgue decomposition of ν|∂V with respect to harmonic measure for L is ν|∂V = μaof⁻¹|∂V + μsof⁻¹|∂V where V = intL and μ = μa + μs is the Lebesgue decomposition of μ with respect to m. Applying Sarason’s process, we obtain P∞(ν) ≠ L∞(ν) if, and only if there is a Jordan curve r contained in K such that mof⁻¹|Γ << μaof⁻¹|Γ. If U is a unitary operator with scalar-valued spectral measure μ then f(U) is non-reductive if and only if there is a Jordan curve r ⊂ K such that mof⁻¹|Γ << μaof⁻¹|Γ. Let G be a bounded region of the plane and B(H) the algebra of bounded operators in the separable Hilbert space H. If π: H∞(G)→B(H) is a norm-continuous homomorphism such that π(1) = 1 and π(z) is pure subnormal then π is weak-star, weak-star continuous. Moreover, if S is a pure subnormal contraction, the S*n→0 sot.","abstract_html":"If f is analytic in a neighborhood of ∂D = {z| |z|= 1} and if K = f(∂D), then C-K has only finitely many components; moreover, if U is a bounded simply connected region of the plane, then ∂U = U<sub>j=0</sub><sup>n</sup r<sub>j</sub> where each r<sub>j</sub> is a rectifiable Jordan curve and r<sub>i</sub> ∩ r<sub>j</sub> is a finite set whenever i ≠ j. Let μ be a positive regular Borel measure supported on ∂D and let m denote normalized Lebesgue measure on ∂D. If L is a compact set such that ∂L ⊂ K and R(L) is a Dirichlet algebra and if ν = μof⁻¹, then the Lebesgue decomposition of ν|<sub>∂V</sub> with respect to harmonic measure for L is ν|<sub>∂V</sub> = μ<sub>a</sub>of⁻¹|<sub>∂V</sub> + μ<sub>s</sub>of⁻¹|<sub>∂V</sub> where V = intL and μ = μ<sub>a</sub> + μ<sub>s</sub> is the Lebesgue decomposition of μ with respect to m. Applying Sarason’s process, we obtain P<sup>∞</sup>(ν) ≠ L<sup>∞</sup>(ν) if, and only if there is a Jordan curve r contained in K such that mof⁻¹|<sub>Γ</sub> << μ<sub>a</sub>of⁻¹|<sub>Γ</sub>. If U is a unitary operator with scalar-valued spectral measure μ then f(U) is non-reductive if and only if there is a Jordan curve r ⊂ K such that mof⁻¹|<sub>Γ</sub> << μ<sub>a</sub>of⁻¹|<sub>Γ</sub>. Let G be a bounded region of the plane and B(H) the algebra of bounded operators in the separable Hilbert space H. If π: H<sup>∞</sup>(G)→B(H) is a norm-continuous homomorphism such that π(1) = 1 and π(z) is pure subnormal then π is weak-star, weak-star continuous. Moreover, if S is a pure subnormal contraction, the S<sup>*n</sup>→0 sot.","abstract_has_math":false,"creators":["Miller, Thomas L."],"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":["Olin, Robert F."],"committee_members":["Thomson, J.E.","Fletcher, Peter","Farkas, Daniel R.","Arnold, Jesse T."],"year":1982,"date_issued":"1982","date_published":"1982","updated_at":"2026-07-22T22:19:13Z","subjects":[],"languages":["en_US"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/80999","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Olin, Robert F."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Thomson, J.E.","Fletcher, Peter","Farkas, Daniel R.","Arnold, Jesse T."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Miller, Thomas L."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2017-12-06T15:20:42Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-12-06T15:20:42Z"]},{"key":"dc:date.issued","label":"Date","values":["1982"]},{"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. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/80999"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["If f is analytic in a neighborhood of ∂D = {z| |z|= 1} and if K = f(∂D), then C-K has only finitely many components; moreover, if U is a bounded simply connected region of the plane, then ∂U = Uj=0nj where each rj is a rectifiable Jordan curve and ri ∩ rj is a finite set whenever i ≠ j. Let μ be a positive regular Borel measure supported on ∂D and let m denote normalized Lebesgue measure on ∂D. If L is a compact set such that ∂L ⊂ K and R(L) is a Dirichlet algebra and if ν = μof⁻¹, then the Lebesgue decomposition of ν|∂V with respect to harmonic measure for L is ν|∂V = μaof⁻¹|∂V + μsof⁻¹|∂V where V = intL and μ = μa + μs is the Lebesgue decomposition of μ with respect to m. Applying Sarason’s process, we obtain P∞(ν) ≠ L∞(ν) if, and only if there is a Jordan curve r contained in K such that mof⁻¹|Γ << μaof⁻¹|Γ. If U is a unitary operator with scalar-valued spectral measure μ then f(U) is non-reductive if and only if there is a Jordan curve r ⊂ K such that mof⁻¹|Γ << μaof⁻¹|Γ. Let G be a bounded region of the plane and B(H) the algebra of bounded operators in the separable Hilbert space H. If π: H∞(G)→B(H) is a norm-continuous homomorphism such that π(1) = 1 and π(z) is pure subnormal then π is weak-star, weak-star continuous. Moreover, if S is a pure subnormal contraction, the S*n→0 sot."]},{"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":["Functions of subnormal operators"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Olin, Robert F."],"dc:contributor.committeemember":["Thomson, J.E.","Fletcher, Peter","Farkas, Daniel R.","Arnold, Jesse T."],"dc:contributor.department":["Mathematics"],"dc:creator":["Miller, Thomas L."],"dc:date.accessioned":["2017-12-06T15:20:42Z"],"dc:date.available":["2017-12-06T15:20:42Z"],"dc:date.issued":["1982"],"dc:description.abstract":["If f is analytic in a neighborhood of ∂D = {z| |z|= 1} and if K = f(∂D), then C-K has only finitely many components; moreover, if U is a bounded simply connected region of the plane, then ∂U = Uj=0nj where each rj is a rectifiable Jordan curve and ri ∩ rj is a finite set whenever i ≠ j. Let μ be a positive regular Borel measure supported on ∂D and let m denote normalized Lebesgue measure on ∂D. If L is a compact set such that ∂L ⊂ K and R(L) is a Dirichlet algebra and if ν = μof⁻¹, then the Lebesgue decomposition of ν|∂V with respect to harmonic measure for L is ν|∂V = μaof⁻¹|∂V + μsof⁻¹|∂V where V = intL and μ = μa + μs is the Lebesgue decomposition of μ with respect to m. Applying Sarason’s process, we obtain P∞(ν) ≠ L∞(ν) if, and only if there is a Jordan curve r contained in K such that mof⁻¹|Γ << μaof⁻¹|Γ. If U is a unitary operator with scalar-valued spectral measure μ then f(U) is non-reductive if and only if there is a Jordan curve r ⊂ K such that mof⁻¹|Γ << μaof⁻¹|Γ. Let G be a bounded region of the plane and B(H) the algebra of bounded operators in the separable Hilbert space H. If π: H∞(G)→B(H) is a norm-continuous homomorphism such that π(1) = 1 and π(z) is pure subnormal then π is weak-star, weak-star continuous. Moreover, if S is a pure subnormal contraction, the S*n→0 sot."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/80999"],"dc:language.iso":["en_US"],"dc:publisher":["Virginia Polytechnic Institute and State University"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Functions of subnormal 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":"2026-07-22T22:19:13Z"}