The finite sum of finite products of Toeplitz operators on the polydisk

A limit theorem is established for a finite sum of finite products of Toeplitz operators on the Hardy space of the polydisk. As a consequence we show that the product of six Toeplitz operators with pluriharmonic symbols is compact iff the product equals zero iff one of these Toeplitz operators equals zero. © 2005 Elsevier Inc. All rights reserved