The classical Nevanlinna-Pick interpolation theorem, proved in 1915 by Pick and in 1919 by Nevanlinna, gives a condition for when there exists an interpolating function in H∞(D) for a specified set of data in the complex plane. In 1967, Sarason proved his commutant lifting theorem for H∞(D), from which an operator theoretic proof of the classical Nevanlinna-Pick theorem followed. Several competing noncommutative generalizations arose as a consequence of Sarason's result, and two strategies emerged for proving generalized Nevanlinna-Pick theorems: via a commutant lifting theorem or via a resolvent, or displacement, equation.
We explore the difference between these two approaches. Specifically, we compare two theorems: one by Constantinescu-Johnson from 2003 and one by Muhly-Solel from 2004. Muhly-Solel's theorem is stated in the highly general context of W*-correspondences and is proved via commutant lifting. Constantinescu-Johnson's theorem, while stated in a less general context, has the advantage of an elegant proof via a displacement equation. In order to make the comparison, we first generalize Constantinescu-Johnson's theorem to the setting of W*-correspondences in Theorem 3.0.1. Our proof, modeled after Constantinescu-Johnson's, hinges on a modified version of their displacement equation. Then we show that Theorem 3.0.1 is fundamentally different from Muhly-Solel's. More specifically, interpolation in the sense of Muhly-Solel's theorem implies interpolation in the sense of Theorem 3.0.1, but the converse is not true. Nevertheless, we identify a commutativity assumption under which the two theorems yield the same result.
In addition to the two main theorems, we include smaller results that clarify the connections between the notation, space of interpolating maps, and point evaluation employed by Constantinescu-Johnson and those employed by Muhly-Solel. We conclude with an investigation of the relationship between Theorem 3.0.1 and Popescu's generalized Nevanlinna-Pick theorem proved in 2003.