# A Beurling Theorem for almost-invariant subspaces of the shift operator

## TL;DR

This paper characterizes nearly-invariant subspaces with finite defect for the backward shift on Hardy space, extending classical theorems to describe almost-invariant subspaces for the shift operator and its adjoint.

## Contribution

It provides a complete characterization of nearly-invariant subspaces of finite defect for the backward shift, generalizing previous results by Hitt and Sarason.

## Key findings

- Characterization of nearly-invariant subspaces with finite defect
- Description of almost-invariant subspaces for the shift and its adjoint
- Extension of classical theorems to broader subspace classes

## Abstract

A complete characterization of nearly-invariant subspaces of finite defect for the backward shift operator acting on the Hardy space is provided in the spirit of Hitt and Sarason's theorem. As a corollary we describe the almost-invariant subspaces for the shift and its adjoint.