Nearly invariant subspaces and kernels of Toeplitz operators on the Hardy space over the bidisk

TL;DR

This paper investigates the structure of nearly invariant subspaces and kernels of Toeplitz operators on the Hardy space over the bidisk, extending existing results to vector-valued spaces and operator tuples.

Contribution

It generalizes the characterization of nearly invariant subspaces and kernels of Toeplitz operators to vector-valued Hardy spaces and commutative isometric tuples.

Findings

Extended Chalendar, Chevrot, and Partington's results to vector-valued Hardy spaces.

Characterized kernels of Toeplitz operators as nearly invariant subspaces.

Showed kernels of general Toeplitz operators are nearly invariant.

Abstract

In this paper, the analysis of nearly invariant subspaces and kernels of Toeplitz operators on the Hardy space over the bidisk is developed. Firstly, we transcribe Chalendar, Chevrot and Partington's result to vector-valued Hardy space $H^{2}_{\ma{H}}(\mathbb{D})$ when $\ma{H}$ is an infinite dimensional separable complex Hilbert space. Secondly, we explore the definition of nearly invariant subspaces on Hardy space over the bidisk, and apply it to characterize kernels of Toeplitz operators. Finally, we define the nearly invariant subspaces for commutative isometric tuples, which allows us to show that the kernel of general Toeplitz operators is also nearly invariant.