  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.
