Abstract. If $\phi $ is an analytic function taking the unit disk into itself then the composition operator $C_\phi $ can be defined on the Hardy space $H^p(D)$ by $C_\phi(f)=f\circ\phi $. In this work it is shown that if some power of $C_\phi $ is compact and $\phi $ has a nonzero derivative at its unique fixed point inside the disk, then $C_\phi $ is similar to $C_\psi $ if and only if the similarity can be induced by an invertible composition operator.

AMS Subject Classification (1991): 47B38, 47B07

Keyword(s): Composition operator

Received October 14, 1992. (Registered under 5562/2009.)