Composition Operators On The Weighted Bergman Spaces

In this paper, we study the compact operators on the weighted Bergman space, the compact composition operators on the weighted Bergman space and the compact product of a composition operator with another one's adjoint on the Hardy space.In the first chapter, we introduce the development of the composition operator theory, give some definitions which will be used in the paper, and give the main results of the paper.In the second chapter, we prove that if a bounded operator S satisfies some integrable conditions, then S is compact on the weighted Bergman space A~p(φ) of the unit disk for 1＜p＜∞if and only if the Berezin transform of S vanishes on the boundary of the unit disk.In the third chapter, under some conditions we show that a composition operator C_ψis compact on the weighted Bergman space A~p(φ) for 1＜p＜∞of the open unit ball in C~n if and only if (1-[z[~2)/(1-|φ(z)|~2)→0 as |z|→1~-.In the fourth chapter, for two analytic self-maps of B_n, under some conditions we showthatif(N_φ(φ(z))N_ψ(ψ(z)))/(log 1／|φ(z)|log 1／|ψ(z)|)→0 as|z|→1, then on the weighted Hardy space H~2(B_n) C_ψ~*C_φis compact.