On the invariant subspace problem via universal Toeplitz operators on the Hardy space $H^{2}(\mathbb{D}^{2})$

TL;DR

This paper investigates the invariant subspace problem on the Hardy space over the bidisk by analyzing universal Toeplitz operators, providing new conditions under which invariant subspaces exist.

Contribution

It introduces a method to find nontrivial invariant subspaces of certain Toeplitz operators on the Hardy space over the bidisk, advancing understanding of the invariant subspace problem.

Findings

Existence of nontrivial invariant subspaces for specific Toeplitz operators.

Sufficient conditions for the invariant subspace problem on the Hardy space.

Connection between universal operators and the invariant subspace problem.

Abstract

The Invariant Subspace Problem (ISP) for Hilbert spaces asks if every bounded linear operator has a non-trivial closed invariant subspace. Due to the existence of universal operators (in the sense of Rota) the ISP can be solved by proving that every minimal invariant subspace of a universal operator is one dimensional. In this paper, we obtain a nontrivial invariant subspace of $T^{*}_{\varphi}|_{M}$, where $T_{\varphi}$ is the Toeplitz operator on the Hardy space over the bidisk $H^{2}(\mathbb{D}^{2})$ induced by the symbol $\varphi\in H^{\infty}(\mathbb{D})$ and $M$ is a $T_{\varphi}^{*}$-invariant subspace. We use this fact to get sufficient conditions for the ISP.