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

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 $T_{\phi }^{\ast }{\mathrm{\mid }}_M$, where $T_{\phi }$ is the Toeplitz operator on the Hardy space over the bidisk $H^2({\mathbb{D}}^2)$ induced by the symbol $\phi \in H^{\mathrm{\infty }}(\mathbb{D})$ and $M$ is a $T_{\phi }^{\ast }$-invariant subspace. We use this fact to get sufficient conditions for the ISP.