In this talk, we   focus on the weighted Hardy spaces of polynomial growth, which cover the   classical Hardy space, weighted Bergman spaces, weighted Dirichlet spaces and   much broader. We discuss the boundedness of the composition operators with   symbols of analytic automorphisms of unit open disk (finite Blaschke product   and analytic functions on the unit closed disk) acting on weighted Hardy   spaces of polynomial growth. Furthermore, we could compute the K_0-groups of   the commutant algebras of those multiplication operators. Then, the Jordan   decomposition theorem and similar classification for the representation of   analytic functions on the unit closed disk as multiplication operators are   obtained. Moreover, we also study the norms, spectra and (semi-)Fredholmness   of composition operators induced by disc automorphisms. This is a joint work   with Prof. Chunlan Jiang.