Operator $K$-Positivity Preserver

We characterize positivity preserving maps $T:B(\mathcal{H}{)}_h\otimes \mathbb{R}[x_1,\dots ,x_n]\to B(\mathcal{H}{)}_h\otimes \mathbb{R}[x_1,\dots ,x_n]$ on ${\mathbb{R}}^n$ and on compact sets $K\subseteq {\mathbb{R}}^n$. This also characterizes local operator moment sequences and general operator moment sequences via positivity preserving maps.