Essentially Normal Composition Operators
Katz, Mor, Mathematics - Graduate School of Arts and Sciences, University of Virginia
Kriete, Thomas, Department of Mathematics, University of Virginia
In this dissertation we prove a simple criterion for essential normality of composition operators induced by maps in a large class of functions. Additionally, we construct essentially normal composition operators which have arbitrary even order of contact with the unit circle at one point. To do so, we rely on results from three distinct areas. We use results and techniques of Agler-Lykova-Young to construct rational analytic self-maps of the unit disk with specifed Taylor coefficients at a given boundary point. This allows us to decompose a composition operator modulo the ideal of compact operators into a sum of rationally induced composition operators based on results of Kriete-Moorhouse. The adjoint of each rationally induced composition operator is then studied using results and ideas of Bourdon-Shapiro. Essential normality is then characterized, beginning with a single summand in the decomposition and continuing to the more general case. Finally, we construct essentially normal composition operators which have arbitrary even order of contact with the unit circle at one point.
PHD (Doctor of Philosophy)
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