  A certain kernel (sometimes called the Pick kernel) associated to Schur
functions on the disk is always positive semi-definite. A generalization of
this fact is well-known for Schur functions on the polydisk. In this article,
we show that the Pick kernel on the polydisk has a great deal of structure
beyond being positive semi-definite. It can always be split into two kernels
possessing certain shift invariance properties.
