This thesis is based on some problems posed in 1 and 27, and in some cases, we find the solutions 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 in the references. In the last section, we find a concrete formula for the adjoint of the U-automorphic composition operators, acting on the monomial basis for H2(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 Te–i&thetas; XTei&thetas; = λX (for a given complex number λ) on B( H2∂U)), where Tei&thetas; and Te–i&thetas; are unilateral forward and backward shifts respectively. This operator-equation was first introduced in 1, but not studied. We investigate the algebraic and operator-theoretic properties of PTOs. In most cases, it is shown that PTOs behave in the same way as the classical Toeplitz operators on H2(∂ U).
In the first section of Chapter 3, after introducing the notions of asymptotic Toeplitzness and asymptotic Hankelness, which were first introduced in 1 and 18 respectively, we study some of their algebraic properties along with a distance formula. In the next section, building on techniques developed by Nazarov-Shapiro 27 and the adjoint formula given in Chapter 1, we show that the adjoint of a composition operator, induced by a unit disk-automorphism, is not strongly asymptotically Toeplitz. This result answers Nazarov-Shapiro's question in 27. In the other direction, we also study the asymptotic Toeplitzness of the product of a composition operator with its adjoint, and Toeplitz-Composition operators.
|Commitee:||Akaki, Tikaradze, Cuckovic, Zeljko, Denis, White, A., Shapiro, Joel, H.|
|School:||The University of Toledo|
|School Location:||United States -- Ohio|
|Source:||DAI-B 78/11(E), Dissertation Abstracts International|
|Keywords:||Asymptotic toeplitz operators, Composition operators, Hardy space, Toeplitz operators|
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