Toeplitz Operators with Bmo Symbols and the Berezin Transform

We prove that the boundedness and compactness of the Toeplitz operator on the Bergman space with a BMO 1 symbol is completely determined by the boundary behaviour of its Berezin transform. This result extends the known results in the cases when the symbol is either a positive L 1-function or an L ∞ function. 1. Introduction. Toeplitz operators are one of the most widely studied classes of concrete operators. The study of their behavior on the Hardy and Bergman spaces has generated an extensive list of results in the operator theory and in the theory of function spaces. One of the latest approaches in this area is the use of the Berezin transform as a determining factor of the behaviour of the Toeplitz operator (see [1, 2, 6, 8, 10]). This method is motivated by its connections with quantum physics and noncommutative geometry. We start with a few of the basic definitions. For more details and references, see [3, 9]. The Bergman space L