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 the Fock–Sobolev space of nonnegative orders. We generalize the results in Bauer et al. (J Funct Anal 268:3017, 2015) and Qin (Bull Sci Math 179:103156, 2022). For the certain symbol spaces, we obtain two Toeplitz operators can semi-commute only in the trivial case, which is different from what is known for the classical Fock space. As an application, we consider the conjecture which was shown to be false for the Fock space in Ma et al. (J Funct Anal 277:2644–2663, 2019). The main result of this paper says that there is a fundamental difference between the geometries of the Fock and Fock–Sobolev space.