Some differential and integral equations with applications to Toeplitz operators (Q1865905)

Let \(B_n\) be the open unit ball of the complex \(n\)-space \(\mathbb C^n\) and let \(H(B_n)\) denote the space of all holomorphic functions on \(B_n\). Let \(D\) be the radial derivative of \(f\in H(B_n)\) and \(P\) be the standard Bergman projection on \(B_n\). The authors solve the differential equation \(fD(g)=gD(f)\) and system of integral equations \(fP(\overline z_kg)=gP(\overline z_kf)\), \(1\leq k\leq n\), to characterize holomorphic symbols of commuting Toeplitz operators on the pluriharmonic Bergman space. Pluriharmonic symbols of normal Toeplitz operators and zero semi-commutators for certain classes of Toeplitz operators are characterized.