Complete Nevanlinna–Pick Kernels

We give a new treatment of Quigginʹs and McCulloughʹs characterization of complete Nevanlinna–Pick kernels. We show that a kernel has the matrix-valued Nevanlinna–Pick property if and only if it has the vector-valued Nevanlinna–Pick property. We give a representation of all complete Nevanlinna–Pick kernels, and show that they are all restrictions of a universal complete Nevanlinna–Pick kernel.