  Let $\mathcal P:=\mathcal P_{m\times n}$ denote the quantized coordinate ring
of the space of $m\times n$ matrices, equipped with natural actions of the
quantized enveloping algebras $U_q(\mathfrak{gl}_m)$ and
$U_q(\mathfrak{gl}_n)$. Let $\mathcal L$ and $\mathcal R$ denote the images of
$U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$ in $\mathrm{End}(\mathcal
P)$, respectively. We define a $q$-analogue of the algebra of
polynomial-coefficient differential operators inside $\mathrm{End}(\mathcal
P)$, henceforth denoted by $\mathcal{PD}$, and we prove that $\mathcal L\cap
\mathcal{PD}$ and $\mathcal{R}\cap \mathcal{PD}$ are mutual centralizers inside
$\mathcal{PD}$. Using this, we establish a new First Fundamental Theorem of
invariant theory for $U_q(\mathfrak{gl}_n)$. We also compute explicit formulas
in terms of $q$-determinants for generators of the intersections with
$\mathcal{PD}$ of the images of the Cartan subalgebras of
$U_q(\mathfrak{gl}_m)$ and $U_q(\mathfrak{gl}_n)$.
