This thesis is based on some problems posed in \\cite{BaHa} and\\cite{NazarovShapiro}, and in some cases, we find thesolutions and investigate their properties.
Chapter 1 presents the prerequisites for the rest of the chapters.These materials can be found with more details in the books cited inthe references. In the last section, we find a concrete formula forthe adjoint of the \\mathbb{U}-automorphic composition operators, acting on the monomial basis for $H^2(\\mathbb{U})$; which is used as one of the major tools later in Chapter 3.
In Chapter 2, we introduce the class of Parametric Toepliz operators(PTOs) as the solutions to the operator-equation$T_{e^{-\\imath\\theta}}XT_{e^{\\imath\\theta}}=\\lambda X$ (for a givencomplex number \\lambda) on\\mathscr{B}\\big(H^{2}(\\partial\\mathbb{U})\\big), where$T_{e^{\\imath\\theta}}$ and $T_{e^{-\\imath\\theta}}$ are unilateralforward and backward shifts respectively. This operator-equation wasfirst introduced in \\cite{BaHa}, but not studied. We investigate thealgebraic and operator-theoretic properties of PTOs. In most cases,it is shown that PTOs behave in the same way as the classicalToeplitz operators on $H^2(\\partial\\mathbb{U})$.
In the first section of Chapter 3, after introducing the notions ofasymptotic Toeplitzness and asymptotic Hankelness, which were firstintroduced in \\cite{BaHa} and \\cite{Feintuch} respectively, we studysome of their algebraic properties along with a distance formula. Inthe next section, building on techniques developed byNazarov-Shapiro \\cite{NazarovShapiro} and the adjoint formula givenin Chapter 1, we show that the adjoint of a composition operator,induced by a unit disk-automorphism, is not strongly asymptoticallyToeplitz. This result answers Nazarov-Shapiro's question in\\cite{NazarovShapiro}. In the other direction, we also study theasymptotic Toeplitzness of the product of a composition operatorwith its adjoint, and Toeplitz-Composition operators.
","abstract_html":"<p>This thesis is based on some problems posed in \\cite{BaHa} and\\cite{NazarovShapiro}, and in some cases, we find thesolutions and investigate their properties.</p><p>Chapter 1 presents the prerequisites for the rest of the chapters.These materials can be found with more details in the books cited inthe references. In the last section, we find a concrete formula forthe adjoint of the <span>\\mathbb{U}</span>-automorphic composition operators, acting on the monomial basis for H2(\\mathbb{U}); which is used as one of the major tools later in Chapter 3.</p><p>In Chapter 2, we introduce the class of Parametric Toepliz operators(PTOs) as the solutions to the operator-equationTe-\\imathθXTe\\imathθ=\\lambda X (for a givencomplex number <span>\\lambda</span>) on<span>\\mathscr{B}\\big(H^{2}(\\partial\\mathbb{U})\\big)</span>, whereTe\\imathθ and Te-\\imathθ are unilateralforward and backward shifts respectively. This operator-equation wasfirst introduced in \\cite{BaHa}, but not studied. We investigate thealgebraic and operator-theoretic properties of PTOs. In most cases,it is shown that PTOs behave in the same way as the classicalToeplitz operators on H2(\\partial\\mathbb{U}).</p><p>In the first section of Chapter 3, after introducing the notions ofasymptotic Toeplitzness and asymptotic Hankelness, which were firstintroduced in \\cite{BaHa} and \\cite{Feintuch} respectively, we studysome of their algebraic properties along with a distance formula. Inthe next section, building on techniques developed byNazarov-Shapiro \\cite{NazarovShapiro} and the adjoint formula givenin Chapter 1, we show that the adjoint of a composition operator,induced by a unit disk-automorphism, is not strongly asymptoticallyToeplitz. This result answers Nazarov-Shapiro's question in\\cite{NazarovShapiro}. In the other direction, we also study theasymptotic Toeplitzness of the product of a composition operatorwith its adjoint, and Toeplitz-Composition operators.</p>","abstract_has_math":true,"creators":["Nikpour, Mehdi"],"institution":"University of Toledo","degree_name":"Doctor of Philosophy","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Cuckovic, Zeljko"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012","date_published":"2012","updated_at":"2026-07-24T03:36:08Z","subjects":["Mathematics","Hardy space","Composition operators","Toeplitz operators","Asymptotic Toeplitz operators"],"languages":["English"],"rights":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://rave.ohiolink.edu/etdc/view?acc_num=toledo1346951238","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Cuckovic, Zeljko"]},{"key":"dc:creator","label":"Author","values":["Nikpour, Mehdi"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2012"]},{"key":"dc:publisher","label":"Institution","values":["University of Toledo / OhioLINK"]},{"key":"dc:type","label":"Dc Type","values":["Electronic Thesis or 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":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Toledo"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Hardy space","Composition operators","Toeplitz operators","Asymptotic Toeplitz operators"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]},{"key":"dc:rights","label":"Dc Rights","values":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://rave.ohiolink.edu/etdc/view?acc_num=toledo1346951238"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This thesis is based on some problems posed in \\cite{BaHa} and\\cite{NazarovShapiro}, and in some cases, we find thesolutions and investigate their properties.
Chapter 1 presents the prerequisites for the rest of the chapters.These materials can be found with more details in the books cited inthe references. In the last section, we find a concrete formula forthe adjoint of the \\mathbb{U}-automorphic composition operators, acting on the monomial basis for $H^2(\\mathbb{U})$; which is used as one of the major tools later in Chapter 3.
In Chapter 2, we introduce the class of Parametric Toepliz operators(PTOs) as the solutions to the operator-equation$T_{e^{-\\imath\\theta}}XT_{e^{\\imath\\theta}}=\\lambda X$ (for a givencomplex number \\lambda) on\\mathscr{B}\\big(H^{2}(\\partial\\mathbb{U})\\big), where$T_{e^{\\imath\\theta}}$ and $T_{e^{-\\imath\\theta}}$ are unilateralforward and backward shifts respectively. This operator-equation wasfirst introduced in \\cite{BaHa}, but not studied. We investigate thealgebraic and operator-theoretic properties of PTOs. In most cases,it is shown that PTOs behave in the same way as the classicalToeplitz operators on $H^2(\\partial\\mathbb{U})$.
In the first section of Chapter 3, after introducing the notions ofasymptotic Toeplitzness and asymptotic Hankelness, which were firstintroduced in \\cite{BaHa} and \\cite{Feintuch} respectively, we studysome of their algebraic properties along with a distance formula. Inthe next section, building on techniques developed byNazarov-Shapiro \\cite{NazarovShapiro} and the adjoint formula givenin Chapter 1, we show that the adjoint of a composition operator,induced by a unit disk-automorphism, is not strongly asymptoticallyToeplitz. This result answers Nazarov-Shapiro's question in\\cite{NazarovShapiro}. In the other direction, we also study theasymptotic Toeplitzness of the product of a composition operatorwith its adjoint, and Toeplitz-Composition operators.
"]},{"key":"dc:format","label":"Dc Format","values":["application/pdf","p.100","654.2 KB"]},{"key":"dc:title","label":"Title","values":["Toeplitzness of Composition Operators and Parametric Toeplitzness"]}]}],"canonical_facts":{"dc:contributor":["Cuckovic, Zeljko"],"dc:creator":["Nikpour, Mehdi"],"dc:date":["2012"],"dc:description":["This thesis is based on some problems posed in \\cite{BaHa} and\\cite{NazarovShapiro}, and in some cases, we find thesolutions and investigate their properties.
Chapter 1 presents the prerequisites for the rest of the chapters.These materials can be found with more details in the books cited inthe references. In the last section, we find a concrete formula forthe adjoint of the \\mathbb{U}-automorphic composition operators, acting on the monomial basis for $H^2(\\mathbb{U})$; which is used as one of the major tools later in Chapter 3.
In Chapter 2, we introduce the class of Parametric Toepliz operators(PTOs) as the solutions to the operator-equation$T_{e^{-\\imath\\theta}}XT_{e^{\\imath\\theta}}=\\lambda X$ (for a givencomplex number \\lambda) on\\mathscr{B}\\big(H^{2}(\\partial\\mathbb{U})\\big), where$T_{e^{\\imath\\theta}}$ and $T_{e^{-\\imath\\theta}}$ are unilateralforward and backward shifts respectively. This operator-equation wasfirst introduced in \\cite{BaHa}, but not studied. We investigate thealgebraic and operator-theoretic properties of PTOs. In most cases,it is shown that PTOs behave in the same way as the classicalToeplitz operators on $H^2(\\partial\\mathbb{U})$.
In the first section of Chapter 3, after introducing the notions ofasymptotic Toeplitzness and asymptotic Hankelness, which were firstintroduced in \\cite{BaHa} and \\cite{Feintuch} respectively, we studysome of their algebraic properties along with a distance formula. Inthe next section, building on techniques developed byNazarov-Shapiro \\cite{NazarovShapiro} and the adjoint formula givenin Chapter 1, we show that the adjoint of a composition operator,induced by a unit disk-automorphism, is not strongly asymptoticallyToeplitz. This result answers Nazarov-Shapiro's question in\\cite{NazarovShapiro}. In the other direction, we also study theasymptotic Toeplitzness of the product of a composition operatorwith its adjoint, and Toeplitz-Composition operators.
"],"dc:format":["application/pdf","p.100","654.2 KB"],"dc:identifier":["http://rave.ohiolink.edu/etdc/view?acc_num=toledo1346951238"],"dc:language":["English"],"dc:publisher":["University of Toledo / OhioLINK"],"dc:rights":["unrestricted","This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws."],"dc:subject":["Mathematics","Hardy space","Composition operators","Toeplitz operators","Asymptotic Toeplitz operators"],"dc:title":["Toeplitzness of Composition Operators and Parametric Toeplitzness"],"dc:type":["Electronic Thesis or Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["University of Toledo"]},"updated_at":"2026-07-24T03:36:08Z"}