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 $\mathcal{F}(\C^n)$ 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 $\mathcal{F}^{s,2}(\C^n).$