Commuting Toeplitz operators on the harmonic Bergman space.

The Pluriharmonic Hardy Space and Toeplitz Operators

“…On the harmonic Bergman space b 2 (D), B. R. Choe and Y. J. Lee [3] proved that two Toeplitz operators with holomorphic symbols commute if and only if a nontrivial linear combination of the symbols is constant, this result has been extended to various domains such as the polydisk ( [5]) and the unit ball ( [11]). In [3,4] [4] showed that the answer to the first question is yes under some additional noncyclictiy hypothesis, and whether the noncyclicity hypotheses can be removed or not remains open.…”

“…In [3,4] [4] showed that the answer to the first question is yes under some additional noncyclictiy hypothesis, and whether the noncyclicity hypotheses can be removed or not remains open. We will be concerned with these two similar questions on h 2 (T 2 ).…”

Commuting Separately Quasihomogeneous Small Hankel Operators on Pluriharmonic Bergman Space

“…On the harmonic Bergman space, there were some studies focusing on the commuting Toeplitz operators with harmonic symbols [10,11] or quasihomogeneous symbols [12,13] and showed that the results obtained are also quite different from the case on the Hardy or Bergman space. Recently, [14] studied the algebraic properties of small Hankel operators on the harmonic Bergman space and got very different commutativity of small Hankel operators compared with the case of Toeplitz operators.…”

Commuting Toeplitz Operators on the Pluriharmonic Bergman Space

“…Recently, the present pluriharmonic case was also studied in [2], [7] and [8] and holomorphic symbols for commuting Toeplitz operators were completely characterized. In particular, the author and K. Zhu [7] proved the following: For nonconstant holomorphic symbols f , g, T f and T g commute on b…”