| 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 A2a ,(-1 < alpha < infinity), setting. We used the result of D. H. Luecking and K. H. Zhu (1992) to extend Zhu's solution (on the Hardy space H2) to the weighted Bergman space A2a . 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 &phis; whose symbol &phis; 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 beta-boundedness/compactness on H2 to determine the range of values of beta ∈ R for which C&phis; is beta-bounded/compact. Similar result also extends to composition operators acting on the weighted Bergman spaces A2a (alpha ≥ -1) based on W.Smith (2003) characterization of beta-boundednes/compactness on these spaces. In particular, we show that the class of Riemann maps under consideration gives example(s) of beta-bounded composition operators C&phis; which fail to be beta compact (0 < beta < infinity) 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 beta-bounded/compact operators. We prove a necessary condition for C&phis; to be Hilbert-Schmidt in terms of beta-boundedness. Extending this result to the Schatten classes, we proved a necessary condition relating beta-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. |
