Semi-commuting Toeplitz operators on Fock-Sobolev spaces

Semi-Commutants of Toeplitz Operators on Fock-Sobolev Spaces

“…In fact, one of the p and f is a constant function if f is a polynomial by Theorem 9 in [15]. This is the desired result.…”

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We make a progress towards describing the semi-commutants of Toeplitz operators on Fock-Sobolev spaces of negative orders. We generalize the results in [2,15]. 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.

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“…Suppose f and g are functions in D. It is easy to check that T z z = T z T z . Using Proposition 2 and the main result of [15], we have…”

“…Proof. In fact, we only need to consider the "Step 1"( the proof of the main result in [15]). By the "Step 1", Theorem 3.5 in [2] and the proof of Proposition 2, one can see that a j b l = 2ηπi for each j and l if…”

“…In fact, the semi-commutativity of Toeplitz operators with general symbols is still an open problem in Fock and Fock-Sobolev space, so it is difficult to complete characterize the bounded semi-commutant on Fock spaces and Fock-Sobolev spaces. There are two natural problems, which arise from the main results of [2,12,15]. Problem A.…”