Semi-commutants of Toeplitz Operators on Fock-Sobolev space of Nonnegative orders

We make a progress towards describing the semi-commutants of Toeplitz operators on Fock-Sobolev spaces of nonnegative orders. We generalize the results in \cite{Bauer1,Qin}. For the certain symbol spaces, we obtain two Toeplitz operators can semi-commute only in the trivial cases, which is different from what is known for the classical Fock spaces. As an application, we consider the conjecture which was shown to be false for Fock space in \cite{MA}. The main results of this paper say that there is the fundamental difference between the geometries of Fock and Fock-Sobolev space.