Trace class criteria for Toeplitz and composition operators on small Bergman spaces

We characterize the Schatten class Toeplitz operators induced by a positive Borel measure on the unit disc and the reproducing kernel of the Bergman space Aω2, where ω   is a radial weight satisfying the doubling property ∫r1ω(s)ds≤C∫1+r21ω(s)ds. By using this, we describe the Schatten class composition operators. We also discuss basic properties of composition operators acting from Aωp to Avq.