Date of Award
4-2016
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
First Advisor
Carl C. Cowen
Committee Chair
Carl C. Cowen
Committee Member 1
Steven R. Bell
Committee Member 2
Marius Dadarlat
Committee Member 3
Donatella Danielli
Abstract
This thesis contains a collection of results in the study of the adjoint of a composition operator and its kernel in weighted Hardy spaces, in particular, the classical Hardy, Bergman, and Dirichlet spaces. In 2006, Cowen and Gallardo-Gutiérrez laid the groundwork for an explicit formula for the adjoint of a composition operator with rational symbol acting on the Hardy space, and in 2008, Hammond, Moorhouse, and Robbins established such a formula. In 2014, Goshabulaghi and Vaezi obtained analogous formulas for the adjoint of a composition operator in the Bergman and Dirichlet spaces. While it is known that the kernel of the adjoint of a composition operator whose symbol is not univalent on the complex unit disk is infinite-dimensional, no classification has been given for functions in this kernel.
Chapter 1 introduces the relevant definitions in the study of composition operators and their adjoints. Chapter 2 provides the background for results obtained by Cowen and Gallardo-Gutiérrez, and Hammond, Moorhouse, and Robbins in the Hardy space. The results by Goshabulaghi and Vaezi for the Bergman and Dirichlet spaces are also given. Chapter 3 contains explicit descriptions of the kernel of the adjoint of a composition operator in a particular class on general weighted Hardy spaces. Chapter 4 uses the adjoint formula by Hammond, Moorhouse, and Robbins to give a functional equation that characterizes functions in the kernel of the adjoint of a composition operator with a rational symbol of degree two on the Hardy space. Chapters 5 and 6 use the adjoint formulas by Goshabulaghi and Vaezi to prove some results about the kernels of adjoints of composition operators on the Bergman and Dirichlet spaces.
Recommended Citation
Miller, Brittney Rachele, "Kernels of adjoints of composition operators on Hilbert spaces of analytic functions" (2016). Open Access Dissertations. 680.
https://docs.lib.purdue.edu/open_access_dissertations/680