Abstract. For any analytic map $\varphi\colon {\msbm D}\rightarrow{\msbm D}$, the composition operator $C_{\varphi }$ is bounded on the Hardy space $H^2$, but there is no known procedure for precisely computing its norm. This paper considers the situation where $\varphi $ is a linear fractional map. We determine the conditions under which $\|C_{\varphi }\|$ is given by the action of either $C_{\varphi }$ or $C_{\varphi }^{\ast }$ on the normalized reproducing kernel functions of $H^2$. We also introduce a new set of conditions on $\varphi $ under which we can calculate $\|C_{\varphi }\|$; moreover, we identify the elements of $H^2$ on which such an operator $C_{\varphi }$ attains its norm. Several specific examples are provided.

AMS Subject Classification (1991): 47B33

Received February 20, 2002, and in revised form June 18, 2002. (Registered under 2933/2009.)