Fourier multipliers and pseudo-differential operators on Fock-Sobolev spaces

Any bounded linear operator $T$ on $L^2({\mathbb{R}}^n)$ gives rise to the operator $S=B\circ T\circ B^{\ast }$ on the Fock space $\mathcal{F}({\text{C}}^n)$ where $B$ is the Bargmann transform. In this article we identify those $S$ which correspond to Fourier multipliers and pseudo-differential operators on $L^2({\mathbb{R}}^n)$ and study their boundedness on the Fock-Sobolev spaces ${\mathcal{F}}^{s,2}({\text{C}}^n)\mathrm{.}$