Eric Koch, Ph.D. Candidate

Department of Mathematical Sciences

Composition Operators on Holomorphic Function Spaces

Committee Members:

• Dr. Zhijian Wu, Chair 
• Dr. Angel Muleshkov, Member
• Dr. Xin Li, Member
• Dr. Stephen Lepp, Graduate College Representative

Abstract

Our focus is differences of weighted composition operators on Hardy and Bergman spaces. Earl Berkson first motivated studying differences of composition operators in 1981. Since then many authors have dug into analyzing the properties of these operators on Hardy and Bergman spaces. For example, we already know that such operators are bounded on these spaces, but they are not always compact.  Authors have done much investigation into when these differences are compact. Authors have also investigated when these differences are Hilbert-Schmidt. These results rely on conditions to be met by both of the composition operators involved in the difference. Our contribution here is to show results about compactness and Hilbert-Schmidt of these weighted differences involving only one of the weighted composition operators. We also investigate conditions under which they may belong to the Schatten p-class of operators, for p=4, again only involving one of the weighted composition operators instead of both.