Finite sums of Toeplitz products on the polydisk (Q735077)

The authors study operators \(S\) of the form \[ S=T_\lambda+\sum_{k=1}^N T_{u_k}T_{v_k}, \] where \(u_k, v_k\) are pluriharmonic functions, \(\lambda\) is an \(n\)-harmonic function on the polydisk \(D^n\), and \(T_u f=P(uf)\) is the Toeplitz operator with symbol \(u\) in the Bergman space \(A^2(D^n)\). The main results concern necessary and sufficient conditions for such an operator to have finite rank or to be a compact operator. As one possible application, the authors show that the sum of a certain number (which depends on and increases with \(n\)) of semicommutators of Toeplitz operators with pluriharmonic symbols cannot be compact unless it is a zero operator.