Toeplitz Operators on Bergman Spaces

Abstract

Let G be a bounded, open, connected, non-empty subset of the complex plane C . We put the usual two dimensional (Lebesgue) area measure on G and consider the Hilbert space L 2 ( G ) that consists of the complex-valued, measurable functions defined on G that are square integrable. The inner product on L 2 ( G ) is given by the norm ‖ h ‖ 2 of a function h in L 2 ( G ) is given by ‖ h ‖ 2 = (∫ G | h | 2 ) 1/2 . The Bergman space of G, denoted L a 2 ( G ), is the set of functions in L 2 ( G ) that are analytic on G . The Bergman space L a 2 ( G ) is actually a closed subspace of L 2 ( G ) (see [12 , Section 1.4]) and thus it is a Hilbert space. Let G denote the closure of G and let C(G) denote the set of continuous, complex-valued functions defined on G .