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

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.

The manuscript requires withdrawal due to the identification of minor flaws in the proof. These issues need to be addressed through further modification and analysis to ensure the validity of the conclusions