Linear Fractional Composition Operators on H 2

If φ is an analytic function mapping the unit disk D into itself, the composition operator Cφ is the operator on H 2 given by Cφf = f ◦φ. The structure of the composition operator Cφ 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 φ(z) = az + b cz + d for some complex numbers a, b, c, d such that φ 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 φ that maps the disk into itself, it is known that Cφ is a bounded operator, and some general properties of Cφ 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 φ, no convenient description of C∗ φ is known and it is not known how to compute ‖Cφ‖ (although order of magnitude estimates are available [3]). J. S. Shapiro (see [16]) has completely answered the question “When is Cφ compact?” Although the general answer is complicated, if φ is a linear fractional transformation Cφ is compact if and only if φ maps the closed unit disk into the open disk. It follows from this that for a linear fractional φ, all powers of Cφ are non-compact if and only if φ 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 φ near the Denjoy-Wolff point, the unique fixed point α̂ of φ in the closed disk such that |φ′(α̂)| ≤ 1. In the second section of the paper, we find that in the linear fractional case C∗ φ is the product of Toeplitz operators and another composition operator. From this computation, we derive ‖Cφ‖ in certain cases and give a short proof of the subnormality of C∗ φ when φ is a hyperbolic inner linear fractional transformation (see also [14, 5]). Finally, the class of linear fractional transformations for which Cφ is hyponormal or subnormal is identified. The class of composition operators is related to other areas of operator theory in somewhat surprising ways. For example, Deddens [6] established a connection between the discrete Cesaro operator and Cφ where φ(z) = sz + 1 − s for 0 < s < 1 and showed that therefore C∗ φ is subnormal for these φ. In addition, commutants of many analytic Toeplitz operators are generated by composition and multiplication operators. Although this paper makes progress in answering some basic questions about linear fractional composition operators, there are still problems to be considered. For example, computing the norm is still unsolved except in special cases and exact conditions for unitary equivalence and similarity are not known. It is hoped that the results here will point the way toward results about more general composition operators, both on H2 and on related Hilbert spaces of analytic functions. Supported in part by National Science Foundation Grant DMS 8300883.