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Linear fractional composition operators on H2

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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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Authors and Affiliations

  1. Department of Mathematics, Purdue University, 47907, West Lafayette, Indiana, USA

    Carl C. Cowen

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  1. Carl C. Cowen
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Supported in part by National Science Foundation Grant DMS 8300883.

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Cowen, C.C. Linear fractional composition operators on H2 . Integr equ oper theory 11, 151–160 (1988). https://doi.org/10.1007/BF01272115

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