Invariance Under Bounded Analytic Functions

TL;DR

This paper extends Beurling's theorem to various mathematical contexts including uniform algebras, Lebesgue spaces, and BMOA, providing new proofs and generalizations.

Contribution

It generalizes Beurling's theorem to uniform algebras, Lebesgue spaces, and BMOA, and offers simplified and new proofs for classical theorems.

Findings

Extension of Beurling's theorem to uniform algebras and Lebesgue spaces.

Simplified proof of Beurling's theorem in uniform algebras.

New proof of Helson-Lowdenslager theorem for compact abelian groups.

Abstract

In a recent paper, M. Raghupathi has extended the famous theorem of Beurling to the context of subspaces that are invariant under the class of subalgebras of $H^\infty$ of the form $IH^\infty$, where $I$ is an inner function. In this paper, we provide analouges of the above mentioned $IH^\infty$ related extension of Beurling's theorem to the context of uniform algebras, on compact abelian groups with ordered duals, the Lebesgue space on the real line and in the setting of the space $BMOA$. We also provide a significant simplification of the proof of the Beurling's theorem in the setting of uniform algebras and a new proof of the Helson-Lowdenslager theorem that generalizes Beurling's theorem in the context of compact abelian groups with ordered duals.