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

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.

Theorems & Definitions (27)

• Definition 1
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 5
• proof
• Definition 6
• Example 7
• Proposition 8
• proof