The numerical range of a composition operator with conformal automorphism symbol

Let p∈U,αp(z):=p-z1-p¯z(z∈U) and let Cαp:H2→H2 be the composition operator induced by the conformal automorphism αp. In this paper we show that the closure of the numerical range of Cαp,W¯(Cαp), is an ellipse with foci at ±1 and major axis 21-|p|2.