The Composition operator on hardy space H Induced by 9(z)=sz+t where |s| 1, 1 and | s | + 1.

We study in this paper the composition operator of induced by the function 9(z)=sz+t where |s| 1, t 1 and |s| + 1. We characterize the normal composition operator C9 on Hardy space H2 and other related classes of operators. In addition to that we study the essential normality of C9 and give some other partial results which are new to the best of our knowledge