  We introduce a formal $\hbar$-differential operator $\Delta$ that generates
higher Koszul brackets on the algebra of (pseudo)differential forms on a
$P_{\infty}$-manifold. Such an operator was first mentioned by Khudaverdian and
Voronov in \texttt{arXiv:1808.10049}. (This operator is an analogue of the
Koszul--Brylinski boundary operator $\partial_P$ which defines Poisson homology
for an ordinary Poisson structure.)
  Here we introduce $\Delta=\Delta_P$ by a different method and establish its
properties. We show that this BV type operator generating higher Koszul
brackets can be included in a one-parameter family of BV type formal
$\hbar$-differential operators, which can be understood as a quantization of
the cotangent $L_{\infty}$-bialgebroid. We obtain symmetric description on both
$\Pi TM$ and $\Pi T^*M$.
  For the purpose of the above, we develop in detail a theory of formal
$\hbar$-differential operators and also of operators acting on densities on
dual vector bundles. In particular, we have a statement about operators that
can be seen as a quantization of the Mackenzie--Xu canonical diffeomorphism.
Another interesting feature is that we are able to introduce a grading, not a
filtration, on our algebras of operators. When operators act on objects on
vector bundles, we obtain a bi-grading.
