Mean growth of Koenigs eigenfunctions

Abstract

In 1884, G. Koenigs solved Schroeder's functional equation f o a = Af in the following context: ( is a given holomorphic function mapping the open unit disk U into itself and fixing a point a E U, f is holomorphic on U, and A is a complex scalar. Koenigs showed that if 0 equation for p has a unique holomorphic solution cr satisfying a Co p = '(a)a and o(0) = 1; moreover, he showed that the only other solutions are the obvious ones given by constant multiples of powers of a. We call a the Koenigs eigenfunction of (. Motivated by fundamental issues in operator theory and function theory, we seek to understand the growth of integral means of Koenigs eigenfunctions. For 0 of ( to belong to the Hardy space HP and show that the condition is necessary when ( is analytic on the closed disk. For many mappings ( the condition may be expressed as a relationship between (p'(a) and derivatives of ( at points on OU that are fixed by some iterate of (. Our work depends upon a formula we establish for the essential spectral radius of any composition operator on the Hardy space HP. DEPARTMENT OF MATHEMATICS, WASHINGTON AND LEE UNIVERSITY, LEXINGTON, VIRGINIA 24450 E-mail address: pbourdonfwlu.edu DEPARTMENT OF MATHEMATICS, MICHIGAN STATE UNIVERSITY, EAST LANSING, MICHIGAN 48824 E-mail address: shapirofmath. msu. edu This content downloaded from 157.55.39.211 on Sat, 25 Jun 2016 06:33:26 UTC All use subject to http://about.jstor.org/terms