Composition operators on weighted Bergman spaces.

In the late 1960’s, E.A. Nordgren and J.V. Ryff studied composition operators on the Hardy space H2. They provided upper and lower bounds on the norms of general composition operators and gave the exact norm in the case where the symbol map is an inner function. Composition operators themselves, on various other spaces, have been studied by many authors since and much deep work has been done concerning them. Recently, however B.D. MacCluer and T. Kriete have developed the study of composition operators on very general weighted Bergman spaces of the unit disk in the complex plane. My starting point is this work.
Composition operators serve well to link the two areas of analysis, operator theory and complex function theory. The products of this link lie deep in complex analysis and are diverse indeed. These include a thorough study
of the Schr¨oeder functional equation:
' =
and its solutions, see [16] and the references therein, in fact some of the well known conjectures can be linked to composition operators. Nordgren, [12],has shown that the Invariant Subspace Problem can be solved by classifying
the minimal invariant subspaces of a certain composition operator on H2, and de Branges used composition operators to prove the Bieberbach conjecture. In this thesis, I use various methods from complex function theory to prove results concerning composition operators on weighted Bergman spaces of the unit disk, the main result is the confirmation of two conjectures of T. Kriete, which appeared in [7]. I also construct, in the final chapter, inner functions which map one arbitrary weighted Bergman space into another.