Bounded composition operators on weighted Bergman spaces

We answer two questions asked by T. L. Kriete (1998, in Contemp. Math., Vol. 213, pp. 73–91, Amer. Math. Soc., Providence) concerning bounded composition operators on weighted Bergman spaces of the unit disk. The main result is the following: if Gi = e− hi, for i = 1, 2, are weight functions in a certain range for which h′1(r)/h′2(r) → ∞ as r → 1 then there is a self-map of the unit disk such that the induced composition operator C maps A2G2 boundedly into itself but does not map A2G1 into itself.