Compact operators on the weighted Bergman space A1(ψ)

We show that a bounded linear operator S on the weighted Bergman space A(1)(psi) is compact and the predual space A(0)(phi) of A(1)(psi) is invariant under S* if and only if Sk(z) -> 0 as z -> partial derivative D, where kz is the normalized reproducing kernel of A(1)(psi). As an application, we give conditions for an operator in the Toeplitz algebra to be compact.