Power bounded composition operators on weighted Dirichlet spaces

In this paper, we study power bounded composition operators on weighted Dirichlet spaces D alpha. As applications, we give the necessary and sufficient conditions for the composition operators to be Riesz operator on D-alpha, when C-phi is power bounded on D-beta, for some 0 < beta < alpha. For alpha > 1, we completely characterize the Riesz composition operators on D-alpha. Moreover, we investigate the functions f is an element of D alpha, when f & phi(n) is convergent or lim(n ->infinity) f & phi(n) = 0, in D-alpha. Some of the techniques developed in the paper are not new but lead to new results.