## 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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### References:

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