Adjoints of Composition Operators with a Broader Class of Symbols

Erin E. M Rizzie, Purdue University

Abstract

This thesis contains a collection of results in the study of composition operators and multiple-valued weighted (and non-weighted) composition operators. The composition operator Cϕ on a space X of functions is defined by the equation Cϕ (f) = f ∘ ϕ, with symbol ϕ a self-map of the domain of the functions in X, and arises naturally in the study of general operator theory. The adjoints of such operators on the Hardy Hilbert space on the complex unit disk are known in the case of a rational symbol and involve a type of operator called a multiple-valued weighted composition operator. For the main result, we give adjoint formulas for composition operators with a broader class of symbols: namely, the single-valued square roots of certain functions without zeroes in the disk. We also present various results in the study of weighted composition operators and multiple-valued composition operators, including some results in compactness, adjoints, and point spectra of these operators.^ Chapter 1 is a general introduction to the study of composition operators, including some well-known results in functional analysis and composition operators necessary to our study here. Chapter 2 gives the known adjoint formulas for composition operators on the Hardy Hilbert space and defines multiple-valued weighted composition operators. Chapter 3 is a collection of results in the study of weighted and multiple-valued composition operators. Chapter 4 gives an adjoint formula for composition operators whose symbols are the single-valued square roots of linear fractional transformations that do not vanish in the disk. Chapter 5 looks at the adjoints of composition operators whose symbols are the single-valued square roots of quadratic functions that do not vanish in the disk.^

Degree

Ph.D.

Advisors

Carl C. Cowen, Purdue University.

Subject Area

Mathematics

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