Bounded composition operators on weighted Bergman spaces

We answer two questions asked by T. L. Kriete (1998, bl Contemp. Math., Vol . 213, pp. 73-91, Amer. Math. Sec., Providence) concerning bounded composit ion operators on weighted Bergman spaces of the unit disk. The main result is the following: if G(i) = e(-hi), for i = 1, 2, are weight functions in a certain range for which h(1)'(r)/h(2)'(r) --> infinity as r --> 1 then the re is a self-map of the unit disk such that the induced composition operato r C-phi maps A(G2)(2) boundedly into itself but does not map A(G1)(2) into itself. (C) 2001 Academic Press.