  For spaces of analytic functions defined on an open set in $\mathbb{C}^n$
that satisfy certain nice properties, we show that operators that preserve
shift-cyclic functions are necessarily weighted composition operators. Examples
of spaces for which this result holds true consist of the Hardy space
$H^p(\mathbb{D}^n) \, (0 < p < \infty)$, the Drury-Arveson space
$\mathcal{H}^2_n$, and the Dirichlet-type space $\mathcal{D}_{\alpha} \,
(\alpha \in \mathbb{R})$. We focus on the Hardy spaces and show that when $1
\leq p < \infty$, the converse is also true. The techniques used to prove the
main result also enable us to prove a version of the Gleason-Kahane-\.Zelazko
theorem for partially multiplicative linear functionals on spaces of analytic
functions in more than one variable.
