Finite-dimensional model spaces invariant under composition operators

TL;DR

This paper characterizes finite-dimensional model spaces in the Hardy space that remain invariant under composition operators, revealing the role of finite cyclic groups and prime factorizations in their structure.

Contribution

It provides a complete characterization of finite-dimensional invariant model spaces under composition operators, linking group theory and prime factorization to their structure.

Findings

Finite-dimensional model spaces invariant under composition operators are characterized.

Finite cyclic groups and prime factorizations are key to understanding these invariant spaces.

The structure of invariant subspaces is explicitly described.

Abstract

Finite-dimensional model spaces are quotient spaces of the Hardy space on the open unit disc, determined by finite Blaschke products. Composition operators, on the other hand, act by composing Hardy space functions with analytic self-maps of the open unit disc. Both are classical and well-studied objects in the theory of analytic function spaces. In this paper, we present a complete characterization of finite-dimensional model spaces that are invariant under composition operators. Finite cyclic groups and the prime factorizations of natural numbers play a crucial role in understanding the structure of such invariant subspaces and the associated analytic self-maps.