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# Composition of Inner Functions

Published:2013-02-13
Printed: Apr 2014
• J. Mashreghi,
Département de mathématiques et de statistique, Université Laval, Québec, QC, Canada G1K 7P4
• M. Shabankhah,
Department of Engineering Science, College of Engineering, University of Tehran, Tehran, 11155-4563, Iran
 Format: LaTeX MathJax PDF

## Abstract

We study the image of the model subspace $K_\theta$ under the composition operator $C_\varphi$, where $\varphi$ and $\theta$ are inner functions, and find the smallest model subspace which contains the linear manifold $C_\varphi K_\theta$. Then we characterize the case when $C_\varphi$ maps $K_\theta$ into itself. This case leads to the study of the inner functions $\varphi$ and $\psi$ such that the composition $\psi\circ\varphi$ is a divisor of $\psi$ in the family of inner functions.
 Keywords: composition operators, inner functions, Blaschke products, model subspaces
 MSC Classifications: 30D55 - ${H}^p$-classes30D05 - Functional equations in the complex domain, iteration and composition of analytic functions [See also 34Mxx, 37Fxx, 39-XX] 47B33 - Composition operators

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