Kernels of adjoints of composition operators on Hilbert spaces of analytic functions
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.
Cowen, Purdue University.
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