Semi-commuting Toeplitz Operators on Fock-Sobolev spaces

Abstract

Let F 2,m (C) denote the Fock-Sobolev space of complex plane. In this paper, we complete characterize when the semi-commutator (T f , Tg] on F 2,m (C)(m > 0) is zero, where f and g are functions in the set of all finite linear combinations of kernel functions. We obtain (T f , T g ] = 0 if and only if at least one of f and g is a constant function. The result is different from the result of semi-commutator on Fock space

Bounded Hankel products on Fock-Sobolev spaces

“…In [5], they obtained H * f H g = 0 on F 2 doesn't imply that at least one of f and g is a constant. In particular, if f (z) = e 2απiz and g(z) = e z , then H * f H g = 0 on F 2 but E(f, g) is unbounded on C. But to our surprise, we have showed that there is actually no nontrivial functions f and g in D such that H * f H g = 0, see [8]. It is clear that…”

“…for all K ≥ 0, then f is a constant function, see the proof of Theorem 6 in [8]. So we assume there is a integer j ≥ 2 so that N1 i=1 a i A j i = 0.…”