Properties of Commutativity of Dual Toeplitz Operators on the Orthogonal Complement of Pluriharmonic Dirichlet Space over the Ball

Abstract

We completely characterize the pluriharmonic symbols for (semi)commuting dual Toeplitz operators on the orthogonal complement of the pluriharmonic Dirichlet space in Sobolev space of the unit ball. We show that, for and pluriharmonic functions, = on (D ℎ ) ⊥ if and only if and satisfy one of the following conditions: (1) both and are holomorphic;(2) both and are holomorphic; (3) there are constants and , both not being zero, such that + is constant.

Dual Toeplitz Operators on the Orthogonal Complement of the Fock–Sobolev Space

“…On the setting of the orthogonal complement of the Bergman space, Stroethof and Zheng [12] frst characterized (semi-)commuting dual Toeplitz operators on the unit disk and their results were extended to the unit ball or unit polydisk as in [7,[13][14][15][16] and reference therein. Later, the corresponding problems have been studied on the Dirichlet spaces and Hardy-Sobolev spaces of the unit disk or unit ball as in [17][18][19][20][21].…”