Abstract
If ϕ is an analytic function mapping the unit diskD into itself, the composition operatorC ϕ is the operator onH 2 given byC ϕf=foϕ. The structure of the composition operatorC ϕ is usually complex, even if the function ϕ is fairly simple. In this paper, we consider composition operators whose symbol ϕ is a linear fractional transformation mapping the disk into itself. That is, we will assume throughout that $$\varphi \left( z \right) = \frac{{az + b}}{{cz + d}}$$ for some complex numbersa, b, c, d such that ϕ maps the unit diskD 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.
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