Almost invariant subspaces of shift operators and products of Toeplitz and Hankel operators

TL;DR

This paper characterizes almost invariant subspaces of shift operators using Toeplitz and Hankel operator products, revealing their equivalence for forward and backward shifts and providing explicit descriptions.

Contribution

It introduces a new approach to describe almost invariant subspaces via Toeplitz and Hankel operators, simplifying previous formulations and establishing their equivalence for forward and backward shift operators.

Findings

Explicit forms of almost invariant subspaces derived from Toeplitz and Hankel operators

Equivalence of almost invariant subspaces for forward and backward shift operators

Simplified characterization of nearly backward shift invariant subspaces

Abstract

In this paper we formulate the almost invariant subspaces theorems of backward shift operators in terms of the ranges or kernels of product of Toeplitz and Hankel operators. This approach simplifies and gives more explicit forms of these almost invariant subspaces which are derived from related nearly backward shift invariant subspaces with finite defect. Furthermore, this approach also leads to the surprising result that the almost invariant subspaces of backward shift operators are the same as the almost invariant subspaces of forward shift operators which were treated only briefly in literature.