Commuting dual Toeplitz operators with pluriharmonic symbols (Q1763407)

Let \(L_a^2(B_n)\) be the Bergman space on the unit ball \(B_n\) of complex \(n\)-space. Let \(P\) be the Hilbert space orthogonal projection from \(L^2(B_n)\) onto \(L_a^2(B_n)\). Given a symbol function \(f\in L^\infty(B_n)\), the Toeplitz operator \(T_f\) is defined by \(T_fg=P(fg)\) for \(g\in L^2_a(B_n)\). Such Toeplitz operators have been extensively studied over the last decades. Associated with Toeplitz operators are dual Toeplitz operators \(S_f\) defined by \(S_fg=(I-P)(fg)\) for \(g\in L^2_a(B_n)^\perp\). Dual Toeplitz operators have been getting more attention recently in connection with the study of Toeplitz operators. In this paper, the author characterizes commuting dual Toeplitz operators with pluriharmonic symbols as well as semi-commuting dual Toeplitz operators with pluriharmonic symbols. The results show that dual Toeplitz operators with pluriharmonic symbols can commute or semi-commute only in the trivial cases.