Abstract. We prove that the essentially normal composition operators with closed range on the space $H^2$ must be normal. We also give some other partial answers to the following general question. Is every noncompact essentially normal composition operator on $ H^2$ normal?

AMS Subject Classification (1991): 47B38, 47B15

Received December 10, 1997 and in final form August 26, 1998. (Registered under 2685/2009.)