Model spaces invariant under composition operators

TL;DR

This paper investigates the invariance of finite-dimensional model spaces within the Hardy space under composition operators, focusing on affine and linear fractional transformations to understand their structural properties.

Contribution

It characterizes when finite-dimensional model spaces are invariant under composition operators, especially for affine and linear fractional transformations.

Findings

Finite-dimensional model spaces can be invariant under certain composition operators.

Invariance depends on the type of transformation, with specific results for affine and linear fractional cases.

The paper provides criteria for invariance in these special cases.

Abstract

Given a holomorphic self-map $\varphi$ of $\D$ (the open unit disc in $\mathbb{C}$), the composition operator $C_{\varphi} f = f \circ \varphi$, $f \in H^2(\mathbb{\D})$, defines a bounded linear operator on the Hardy space $H^2(\mathbb{\D})$. The model spaces are the backward shift-invariant closed subspaces of $H^2(\mathbb{\D})$, which are canonically associated with inner functions. In this paper, we study model spaces that are invariant under composition operators. Emphasis is put on finite-dimensional model spaces, affine transformations, and linear fractional transformations.