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

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