Factorizations induced by complete Nevanlinna–Pick factors

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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.