Weighted interpolation over W*-algebras

<p>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 <em>W</em>*-correspondences provides an appropriate context for generalizing this classic question. Their reproducing kernel <em>W</em>*-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 <em>W</em>*-correspondence associated with a certain <em>W</em>*-version of the classic Szegö kernel may be identified with their primary operator algebra of interest, the Hardy algebra.</p> <p>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 <em>W</em>*-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.</p> <p>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 <em>W</em>*-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 <em>W</em>*-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 <em>W</em>*-Szegö kernel in its light.</p>