On the fixed points of the Berezin transform -

Abstract

The Bergman space A2 consists of functions analytic and square integrable on a region of the complex plane. The Berezina transform T of a function in this space is defined as the Berezin transform of the Toeplitz operator. A question of interest is to determine fixed points of the Berezin transform. In this thesis, we present a partial study of work done on this question. We first consider the case where the region is the open unit disk D and present conditions for which Tu≤ u and Tu≥ u, where u is integrable. We then consider the more difficult case where the region is an annulus centered at the origin. In the case of a radial function, we present conditions implying either Tu ≥u, or Tu≤u.