Symmetries and reflections from composition operators in the disk

TL;DR

This paper investigates the properties of composition operators on the Hardy space of the disk, focusing on their symmetries, eigenspaces, and geometric relations, revealing their reflection-like behavior and structural characteristics.

Contribution

It provides a detailed analysis of the eigenspaces and symmetries of composition operators acting as reflections on the Hardy space, highlighting their geometric and algebraic properties.

Findings

Composition operators act as reflections with square equal to identity.

Eigenspaces of these operators are characterized and studied.

The relative positions of eigenspaces reveal geometric symmetries.

Abstract

We study the composition operators $C_a$ acting on the Hardy space $H^2$ of the unit disk, given by $C_af=f\circ\varphi_a$, where $$ \varphi_a(z)=\frac{a-z}{1-\bar{a}z}, $$ for $|a|<1$. These operators are reflections: $C_a^2=1$. We study their eigenspaces $N(C_a\pm 1)$, their relative position (i.e., the intersections between these spaces and their orthogonal complementes for $a\ne b$ in the unit disk) and the symmetries induced by $C_a$ and these eigenspaces.