Abstract. We study several properties of composition operators acting on the Dirichlet space, and related weighted Dirichlet spaces, in the disk. In particular we consider the question of when $C_{\varphi }$ has closed range on these spaces.

AMS Subject Classification (1991): 47B38

Received June 3, 1996 and in revised form November 12, 1996. (Registered under 6112/2009.)