Berezin Transform and Toeplitz Operators on Bergman Spaces Induced by Regular Weights

Given a regular weight omega and a positive Borel measure mu on the unit disc D, the Toeplitz operator associated with mu is T-mu(f) (z) = integral(D) f (zeta) <(B-z(omega)(zeta))over bar> d(mu)(zeta), where B-z(omega) are the reproducing kernels of the weighted Bergman space A(omega)(2). The primary purpose of this paper is to study the interrelationships between the Toeplitz operator T-mu, Carleson measures, and the Berezin transform (T) over tilde (mu)(z) = < T-mu(B-z(omega)), B-z(omega)> (A omega 2) / parallel to B-z(omega)parallel to(2) (A omega 2). We provide descriptions of bounded and compact operators T-mu : A(omega)(p) -> A(omega)(q) 1 < q, p < infinity, as well as, Schatten class Toeplitz operators T-mu on A(omega)(2). The last mentioned characterization is applied to study Schatten class composition operators on A(omega)(2).