Universal Toeplitz operators on the Hardy space over the polydisk

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 may be solved by describing the invariant subspaces of these operators alone. We characterize all 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 Caradus criterion for universality, that is, when $T_\phi^*$ is surjective and has infinite dimensional kernel. In particular if $\phi$ in a non-constant inner function 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-free on $\mathbb{T}^n$, then $T_\phi^*$ is universal for $H^2(\mathbb{D}^n)$. The analogs of these results for $n=1$ are not true.