Beurling's Theorem And Invariant Subspaces For The Shift On Hardy Spaces

TL;DR

This paper characterizes invariant subspaces of the shift operator on Hardy spaces over certain domains, linking geometric properties of the domain to the structure of these subspaces.

Contribution

It provides necessary and sufficient conditions involving domain connectivity, harmonic measures, and polynomial density for the invariant subspaces to be generated by bounded analytic functions.

Findings

Invariant subspaces are characterized by domain geometry and harmonic measure conditions.

Conditions include perfect connectivity of components and mutual singularity of harmonic measures.

Every invariant subspace corresponds to multiplication by a bounded analytic function with specific boundary behavior.

Abstract

Let $G$ be a bounded open subset in the complex plane and let $H^{2}(G)$ denote the Hardy space on $G$. We call a bounded simply connected domain $W$ perfectly connected if the boundary value function of the inverse of the Riemann map from $W$ onto the unit disk $D$ is almost 1-1 rwith respect to the Lebesgure on $\partial D$ and if the Riemann map belongs to the weak-star closure of the polynomials in $H^{\infty}(W)$. Our main theorem states: In order that for each $M\in Lat(M_{z})$, there exist $u\in H^{\infty}(G)$ such that $ M = \vee\{u H^{2}(G)\}$, it is necessary and sufficient that the following hold: 1) Each component of $G$ is a perfectly connected domain. 2) The harmonic measures of the components of $G$ are mutually singular. 3) % $P^{\infty}(\omega) The set of polynomials is weak-star dense in $ H^{\infty}(G)$. \noindent Moreover, if $G$ satisfies these conditions, then every $M\in Lat(M_{z})$ is of the form $u H^{2}(G)$, where %$u\in H^{\infty}(G)$ and the restriction of $u$ to each of the components of $G$ is either an inner function or zero.