Factorizations induced by complete Nevanlinna–Pick factors

We prove a factorization theorem for reproducing kernel Hilbert spaces whose kernel has a normalized complete Nevanlinna–Pick factor. This result relates the functions in the original space to pointwise multipliers determined by the Nevanlinna–Pick kernel and has a number of interesting applications. For example, for a large class of spaces including Dirichlet and Drury–Arveson spaces, we construct for every function f in the space a pluriharmonic majorant of |f|² with the property that whenever the majorant is bounded, the corresponding function f is a pointwise multiplier.