Type of Document	Dissertation
Author	Tadesse, Abebaw
Author's Email Address	abt4@pitt.edu
URN	etd-05302006-090524
Title	Compact Composition Operators on the Hardy and Bergman Spaces
Degree	Doctor of Philosophy
Program	Mathematics
School	School of Arts and Sciences
Advisory Committee	Advisor Name Title Prof. Juan Manfredi Committee Chair Prof. Louis Chaparro Committee Member Prof. Chris Lennard Committee Member Prof. Frank Beatrous Committee Member
Keywords	Bergman Spaces bounded composition operators Composition Operators Hilbert Schmidt Classes Compact Composition operators Schatten Classes Î’-bounded composition operators Hardy Space
Date of Defense	2006-04-27
Availability	unrestricted
Abstract The thesis consists of three pieces of results on compact composition operators on the Hardy and Bergman spaces. In the first part, chapter 2, we re-formulate Lotto's conjecture on the weighted Bergman space [Special characters omitted.] ,(-1 < α < ∞), setting. We used the result of D. H. Luecking and K. H. Zhu (1992) to extend Zhu's solution (on the Hardy space H 2 ) to the weighted Bergman space [Special characters omitted.] . The results of this chapter has been published in Tadesse (2004). In the second part of the thesis, chapter 3, we investigate compact composition operators which are not Hilbert-Schmidt. We consider the class of examples (see B. Lotto (1998)) of composition operators C [straight phi] whose symbol [straight phi] is a Riemann map from the unit disk D onto the semi-disk with center (½, 0), radius ½ and, in general, onto a "crescent" shaped regions constructed based on this semi-disk (see also B. Lotto (1998)) We use the R. Riedel (1994) characterization of β-boundedness/compactness on H 2 to determine the range of values of β ∈ [Special characters omitted.] for which C [straight phi] is β-bounded/compact. Similar result also extends to composition operators acting on the weighted Bergman spaces [Special characters omitted.] (α ≥ -1) based on W.Smith (2003) characterization of β-boundednes/compactness on these spaces. In particular, we show that the class of Riemann maps under consideration gives example(s) of β-bounded composition operators C [straight phi] which fail to be β compact (0 < β < ∞) This was an open question raised by Hunziker and Jarchaw (1991)(Section 5.2). Our second result arises from our attempt to generalize these observations to relate Hilbert-Schmidt classes with β-bounded/compact operators. We prove a necessary condition for C [straight phi] to be Hilbert-Schmidt in terms of β-boundedness. Extending this result to the Schatten classes, we proved a necessary condition relating β-bounded composition operators with those that belong to the Schatten ideals. The results of this chapter has been presented at the January 2005 AMS joint meeting in Atlanta, Georgia, and they are under preparation for publication. In the last part of the thesis, Chapter 4, we characterized compact composition operators on the Hardy-Smirnov spaces over a simply connected domain. In the end, we gave an explicit example demonstrating the main results of this chapter for a simple geometry where an explicit and simplified expression for the Riemann map is known. The results of this chapter has been presented at the January 2006 AMS joint meeting in San Antonio, Texas, at the Analysis conference in honor of Prof. Vladmir Gurariy at Kent State University and it is also under preparation for publication.
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