Toeplitzness of Composition Operators and Parametric Toeplitzness

<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-equationT_e^-\imathθXT_e^\imathθ=\lambda X (for a givencomplex number <span>\lambda</span>) on<span>\mathscr{B}\big(H^{2}(\partial\mathbb{U})\big)</span>, whereT_e^\imathθ and T_e^-\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>

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor dc:publisher

University of Toledo

Year dc:date

2012

Author dc:creator

• Nikpour, Mehdi

Contributors dc:contributor

• Cuckovic, Zeljko

• Mathematics
• Hardy space
• Composition operators
• Toeplitz operators
• Asymptotic Toeplitz operators

Statement 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.

Language dc:language

English

Repository record dc:identifier

http://rave.ohiolink.edu/etdc/view?acc_num=toledo1346951238

OAI identifier oai:identifier

oai:etd.ohiolink.edu:toledo1346951238