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.

Abstract. We give a necessary and sufficient condition for a composition operator C φ on the Bloch space to have closed range. We show that when φ is univalent, it is sufficient to consider the action of C φ on the set of Möbius transforms. In this case the closed range property is equivalent to a specific sampling set satisfying the reverse Carleson condition.

Abstract. We characterize bounded and compact composition operators on weighted Dirichlet spaces. The method involves integral averages of the determining function for the operator, and the connection between composition operators on Dirichlet spaces and Toeplitz operators on Bergman spaces. We also present several examples and counter-examples that point out the borderlines of the result and its connections to other themes.

Abstract. We characterize the closed-range composition operators on weighted Bergman spaces in terms of the ranges of the inducing maps on the unit disc. The method uses Nevanlinna's counting function and Luecking's results on inequalities on Bergman spaces.

Abstract. We analyze the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a special class of operators that we call radial operators, an oscilation criterion is a sufficient condition under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit circle. We further study a special class of radial operators, i.e., Toeplitz operators with a radialLet L 2 a denote the Bergman space of functions analytic on the unit disk D. For a general bounded operator A on the Bergman space, the Berezin transform of A is the functionà defined byÃwhere ·, · denotes the inner product in L 2 a , and k z is the normalized evaluation function from L 2 a . The function k z is defined byand has the property that f,a . The Berezin transform is a function that is bounded by the norm of the operator. It is also easy to see that each bounded operator on L 2 a is uniquely determined by its Berezin transform. Thus, the behavior of the operator can be analyzed by exploring the corresponding Berezin transform. This idea can be employed in a much more general context where the space on which the operator acts is a so-called standard functional Hilbert space, an example of which is the Bergman space. For results on Berezin transforms on standard functional Hilbert spaces, see for example [6] and [10].Let P denote the Bergman projection fromThe Toeplitz operator is bounded whenever f ∈ L ∞ (D), but is not bounded for every f in L 1 (D). The boundedness and compactness of Toeplitz operators on the Bergman space have been of interest to mathematicians working in operator