An operator-theoretic formulation of the interpolation problem posed by Nevanlinna and Pick in the early twentieth century asks for conditions under which there exists a multiplier of a reproducing kernel Hilbert space that interpolates a specified set of data. Paul S. Muhly and Baruch Solel have shown that their theory for operator algebras built from W*-correspondences provides an appropriate context for generalizing this classic question. Their reproducing kernel W*-correspondences are spaces of functions that generalize the reproducing kernel Hilbert spaces. Their Nevanlinna-Pick interpolation theorem, which is proved using commutant lifting, implies that the algebra of multipliers of the reproducing kernel W*-correspondence associated with a certain W*-version of the classic Szegö kernel may be identified with their primary operator algebra of interest, the Hardy algebra. To provide a context for generalizing another familiar topic in operator theory, the study of the weighted Hardy spaces, Muhly and Solel have recently expanded their theory to include operator-valued weights. This creates a new family of reproducing kernel W*-correspondences that includes certain, though not all, classic weighted Hardy spaces. It is the purpose of this thesis to generalize several of Muhly and Solel's results to the weighted setting and investigate the function-theoretic properties of the resulting spaces. We give two principal results. The first is a weighted version of Muhly and Solel's commutant lifting theorem, which we obtain by making use of Parrott's lemma. The second main result, which in fact follows from the first, is a weighted Nevanlinna-Pick interpolation theorem. Other results, several of which follow from the two primary results, include the construction of an orthonormal basis for the nonzero tensor product of two W*-corrrespondences, a double commutant theorem, the identification of several function-theoretic properties of the elements in the reproducing kernel W*-correspondence associated with a weighted W*-Szegö kernel as well as the elements in its algebra of mutlipliers, and the presentation of a relationship between this algebra of multipliers and a weighted Hardy algebra. In addition, we consider a candidate for a W*-version of the complete Pick property and investigate the aforementioned weighted W*-Szegö kernel in its light.

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.