Beurling and Model subspaces invariant under a universal operator

TL;DR

This paper characterizes certain invariant subspaces of the Hardy space under a specific affine composition operator, linking to the invariant subspace problem and classical operators in functional analysis.

Contribution

It provides a characterization of Beurling and Model subspaces invariant under a universal affine composition operator on Hardy space.

Findings

Identifies invariant subspaces under the operator $C_{_a}$.

Connects the invariant subspace problem with the structure of these subspaces.

Highlights the operator's universal properties and their mathematical significance.

Abstract

In this article, we characterize the Beurling and Model subspaces of the Hardy-Hilbert space $H^2(\mathbb{D})$ invariant under the composition operator $C_{\phi_a}f=f\circ\phi_a$, where $\phi_a(z) = az + 1 - a$ for $a \in (0,1)$ is an affine self-map of the open unit disk $\mathbb{D}$. These operators have universal translates (in the sense of Rota) and have attracted attention recently due to their connection with the Invariant Subspace Problem (ISP) and the classical Ces\`aro operator.