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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 may be solvedby describing the invariant subspaces of these operators alone. We characterizeall anaytic Toeplitz operators $T_\phi$ on the Hardy space $H^2(\mathbb{D}^n)$over the polydisk $\mathbb{D}^n$ for $n>1$ whose adjoints satisfy the Caraduscriterion for universality, that is, when $T_\phi^*$ is surjective and hasinfinite dimensional kernel. In particular if $\phi$ in a non-constant innerfunction on $\mathbb{D}^n$, or a polynomial in the ring$\mathbb{C}[z_1,\ldots,z_n]$ that has zeros in $\mathbb{D}^n$ but is zero-freeon $\mathbb{T}^n$, then $T_\phi^*$ is universal for $H^2(\mathbb{D}^n)$. Theanalogs of these results for $n=1$ are not true.

Universal Toeplitz operators on the Hardy space over the polydisk