On the Invariant Subspace Problem via Universal Toeplitz Operators on the Hardy Space Over the Bidisk

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 work, we obtain conditions for \(T^{*}_{\varphi }|_{M}\) $T_{\phi }^{\ast }{|}_M$ to have a non-trivial subspace where \(M\subset H^{2}({\mathbb {D}}^{2})\) $M\subset H^2({\mathbb{D}}^2)$ is an invariant subspace of the Toeplitz operator \(T_{\varphi }^{*}\) $T_{\phi }^{\ast }$ on the Hardy space over the bidisk \(H^{2}({\mathbb {D}}^{2})\) $H^2({\mathbb{D}}^2)$ induced by the symbol \(\varphi \in H^{\infty }({\mathbb {D}})\) $\phi \in H^{\infty }(\mathbb{D})$ . We then use this fact to obtain sufficient conditions for the ISP to be true.