Toeplitz operators with pluriharmonic symbol on the unit ball

Abstract

A well-known result going back to M. Engliš shows that Toeplitz operators with bounded symbol on the Bergman space A2(D) cannot be characterized by a Brown–Halmos type operator identity. In the opposite direction we prove that Toeplitz operators with pluriharmonic symbol on the analytic functional Hilbert spaces Hm(B) with reproducing kernel Km(z,w)=(1−〈z,w〉)−m can be characterized by an operator identity that specializes to the Brown–Halmos condition in the case of the Hardy space on the unit disc. We thus extend results obtained by Louhichi and Olofsson for the standard weighted Bergman spaces on the unit disc to the analytic Besov spaces Hm(B) on the unit ball.