Products of Bergman space Toeplitz operators on the polydisk

Abstract

Motivated by recent works of Ahern andCucković on the disk, we study the generalized zero product problem for Toeplitz operators acting on the Bergman space of the polydisk. First, we extend the results to the polydisk. Next, we study the generalized compact product problem. Our results are new even on the disk. As a consequence on higher dimensional polydisks, we show that the generalized zero and compact product properties are the same for Toeplitz operators in a certain case.

m-Berezin Transform on the Polydisk

“…We will obtain a useful and sharp approximate identity of the m-Berezin transforms (Theorem 3.7), which has been used to study compact products of Toeplitz operators [7].…”

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

“…This problem becomes more complicated on the Hardy space over the unit sphere and Bergman space over unit ball in C n . In (Choe et al, 2007), this problem was resolved for Toeplitz operators with plurihamonic symbols on the Bergman space of the polydisk while (Dong and Zhou, 2009), were able to determine when the product of two radial Toeplitz operators is a Toeplitz operator. For the case of Fock space, things are a lot more different.…”

Finite Sums of Toeplitz Products on the Polydisk

“…As an application of our results above, we obtain the following result. The assertion (b) below with N = 1 has been known; see [11,Theorem 8].…”

“…1.1 to be compact. The case N = 1, but with general λ ∈ L ∞ , was studied in [11]; see the remark at the end of Section 5. Pu j Pv j is boundary n-harmonic.…”

“…In the theory of Toeplitz operators on higher dimensional polydisks, it has been known that the zero and compact properties often coincide, especially when pluriharmonic symbols are considered, as one can see from results in [8], [11], [15], [17] and [24]. For example, we have the following characterizations of compact (semi)commutators with pluriharmonic symbols u, v ∈ ph ∞ : ([15] and [8]);…”