# Schröder Equation in Several Variables and Composition Operator

### Cinzia Bisi

Università degli Studi della Calabria, Arcavacata Di Rende, Italy### Graziano Gentili

Università degli Studi di Firenze, Italy

## Abstract

Let $\varphi$ be a holomorphic self-map of the open unit ball $\B$ of $\C^n$ such that $\varphi(0)=0$ and that the differential $d \varphi_{0}$ of $\varphi$ at $0$ is non singular. The study of the Schr\"oder equation in several complex variables

is naturally related to the theory of composition operators on Hardy spaces of holomorphic maps on $\B$ and to the theory of discrete, complex dynamical systems. An extensive use of the infinite matrix which represents the composition operator associated to the map $\varphi$ leads to a simpler approach, and provides new proofs, to results of existence of solutions for the Schr\"oder equation.

## Cite this article

Cinzia Bisi, Graziano Gentili, Schröder Equation in Several Variables and Composition Operator. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 17 (2006), no. 2, pp. 125–134

DOI 10.4171/RLM/458