Commuting Toeplitz operators with pluriharmonic symbols on the Fock space

Semi-commuting Toeplitz Operators on Fock-Sobolev spaces

“…Let P be the space of all holomorphic polynomials on C. Suppose h 1 , h 2 ∈ D and h 3 , h 4 ∈ P. If f = h 1 h 3 and g = h 2 h 4 , then, by the proof of Theorem 6 and the proof of Theorem 4.1 in [2], we can see that Theorem 6 is also holds. By what was proved in the previous paragraph, we give a conjecture: suppose f and g be functions in F 2,m (C)(m = 0), then H * f H g = 0 if and only if at least one of f and g is a constant function.…”

“…So, it is difficult to characterize the commuting Toeplitz operators on Fock space. There is only one result for the semi-commuting Toeplitz operators with non-radial symbols on Fock space F 2 , see [2].…”

“…The operator U ω is a key to consider the operators on F 2 , the results of [8,2] are based on the property of U ω . However, the translations do not have such good property on Fock-Sobolev space F 2,m , since the kernel of Fock-Sobolev is very complicated.…”

Algebraic Properties of Toeplitz Operators on the Pluri-harmonic Fock Space

“…The commuting problem for Toeplitz operators on the Hardy, Bergman and Fock spaces have generated a lot of research in recent years (see for example Brown andHalmos (1964), Axler andCucković (1991), Cucković and Rao (1998), Dong and Zhou (2009), Choe and Lee (1993), Guan et al (2013), Yan and Liu (2013), Bauer and Le (2011), Bauer andLee (2011), Bauer andIssa (2012) and Bauer et al (2015)). In (Brown and Halmos, 1964), it is shown for the Hardy space on the unit circle that the Toeplitz operators T f and T g commute if and only if either (1) f and g are holomorphic or (2) f and g are anti-holomorphic or (3) one is a linear function of the other.…”

“…For the case of the Fock space, (Bauer and Lee, 2011), showed that if f and g have atmost exponential growth and f is radial and non-constant then T f and T g commute implies that g is radial. In (Bauer et al, 2015), the case when the symbols are pluri-harmonic and satifies some exponential growth condition has been studied.…”