Kernels with complete Nevanlinna-Pick factors and the characteristic function

The Sz.-Nagy Foias characteristic function for a contraction has had a rejuvenation in recent times due to a number of authors. Such a classical object relates to an object of very contemporary interest, viz., the complete Nevanlinna-Pick kernels. Indeed, a unitarily invariant kernel on the unit ball admits a characteristic function if and only if it is a complete Nevanlinna-Pick kernel. However, what has captured our curiosity are the recent advancements in constructing characteristic functions for kernels that do not have complete Nevanlinna-Pick property. In such cases, the reproducing kernel Hilbert space which has served as the domain of the multiplication operator has always been the vector-valued Drury-Arveson space (thus the Hardy space in case of the unit disc). We present a unified framework for deriving characteristic functions for kernels that allow a complete Nevanlinna-Pick factor. Notably, our approach not only encapsulates all previously documented cases but also achieves a remarkable level of generalization, thereby expanding the concept of the characteristic function substantially. We also provide an explanation for the prominence of the Drury-Arveson kernel in all previously established results by showing that the Drury-Arveson kernel was the natural choice inherently suitable for those situations.