MEAN GROWTH OF KOENIGS EIGENFUNCTIONS

In 1884, G. Koenigs solved Schroeder's functional equation f o phi = l ambda f in the following context: phi is a given holomorphic function mapping the open unit disk U into itself and fixing a point a is an el ement of U, f is holomorphic on U, and lambda is a complex scalar. Koe nigs showed that if 0 < \phi'(a)\ < 1, then Schroeder's equation for p hi has a unique holomorphic solution a satisfying sigma o phi = phi'(a )sigma and sigma'(0) = 1; moreover, he showed that the only other solu tions are the obvious ones given by constant multiples of powers of si gma. We call the Koenigs eigenfunction of phi. Motivated by fundamenta l issues in operator theory and function theory, we seek to understand the growth of integral means of Koenigs eigenfunctions. For 0 < p < i nfinity, we prove a sufficient condition for the Koenigs eigenfunction of phi to belong to the Hardy space H-p and show that the condition i s necessary when phi is analytic on the closed disk. For many mappings phi the condition may be expressed as a relationship between phi'(a) and derivatives of phi at points on partial derivative U that are fixe d by some iterate of phi. Our work depends upon a formula we establish for the essential spectral radius of any composition operator on the Hardy space H-p.