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Mean growth of Koenigs eigenfunctions. (English) Zbl 0870.30018

Summary: In 1884, G. Koenigs solved Schroeder’s functional equation \[ f\circ \varphi = \lambda f \] in the following context: \(\varphi\) is a given holomorphic function mapping the open unit disk \(U\) into itself and fixing a point \(a\in U\), \(f\) is holomorphic on \(U\), and \(\lambda\) is a complex scalar. Koenigs showed that if \(0 < |\varphi '(a)|< 1\), then Schroeder’s equation for \(\varphi\) has a unique holomorphic solution \(\sigma\) satisfying \[ \sigma \circ\varphi = \varphi '(a) \sigma \qquad \text{and}\qquad \sigma '(0) = 1; \] moreover, he showed that the only other solutions are the obvious ones given by constant multiples of powers of \(\sigma\). We call \(\sigma\) the Koenigs eigenfunction of \(\varphi\). Motivated by fundamental issues in operator theory and function theory, we seek to understand the growth of integral means of Koenigs eigenfunctions. For \(0 < p < \infty\), we prove a sufficient condition for the Koenigs eigenfunction of \(\varphi\) to belong to the Hardy space \(H^p\) and show that the condition is necessary when \(\varphi\) is analytic on the closed disk. For many mappings \(\varphi\) the condition may be expressed as a relationship between \(\varphi '(a)\) and derivatives of \(\varphi\) at points on \(\partial U\) that are fixed by some iterate of \(\varphi\). Our work depends upon a formula we establish for the essential spectral radius of any composition operator on the Hardy space \(H^p\).

MSC:

30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
47B38 Linear operators on function spaces (general)
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