Beurling's theorem for the Hardy operator on $L^2[0,1]$

TL;DR

This paper characterizes the invariant subspaces of the Hardy operator on L^2[0,1], showing they are limits of sequences of finite-dimensional spaces spanned by monomials, thus providing a structural understanding.

Contribution

It establishes a new description of invariant subspaces for the Hardy operator on L^2[0,1], linking them to limits of finite-dimensional monomial-spanned spaces.

Findings

Invariant subspaces are limits of finite-dimensional monomial spaces.

Provides a structural characterization of the Hardy operator's invariant subspaces.

Connects invariant subspaces to polynomial approximation sequences.

Abstract

We prove that the invariant subspaces of the Hardy operator on $L^2[0,1]$ are the spaces that are limits of sequences of finite dimensional spaces spanned by monomial functions.