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

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