If ~ is an analytic function mapping the unit disk D into itself, the composition operator C~ is the operator on H 2 given by C~,f = fo!o. The structure of the composition operator C~ is usually complex, even if the function ~ is fairly simple. In this paper, we consider composition operators whose symbol !o is a linear fractional transformation mapping the disk into itself. That is, we will assume throughout that az+b ~(z) --cz + d for some complex numbers a, b, c, d such that ~ maps the unit disk D into itself. For this restricted class of examples, we address some of the basic questions of interest to operator theorists, including the computation of the adjoint. For any ~ that maps the disk into itself, it is known that C~ is a bounded operator, and some general properties of C~ have been established (see for example, [15], [12], [17], [13], [10], [3], [11], [14], and [16]). However, not all questions that would be considered basic by operator theorists are understood. For example, for general ~, no convenient description of C~* is known and it is not known how to compute IIC~[[ (although order of magnitude estimates are available [3]). J. S. Shapiro (see [16]) has completely answered the question "When is C~ compact?" Although the general answer is complicated, ff to is a linear fractional transformation C~ is compact if and only if ~ maps the closed unit disk into the open disk. It follows from this that for a linear fractional ~, all powers of C~ are non-compact if and only if ~ has a fixed point on the unit circle. The first section illustrates the diversity of this class of examples by showing there are eight distinct classes on the basis of spectral information alone. Much of the spectral information depends on the behavior of lo near the Denjoy-Wolff point,, the unique fixed point & of ~ in the clo~ed disk such that [~'(&)l -1.In the second section of the paper, we find that in the linear fractional case C* is the product of Toeplitz operators and another composition operator. From this computation, we derive I[C~[I in certain cases and give a short proof of the subnormality of
