Multivariable Interpolation Problems

Abstract

In this dissertation, we solve multivariable Nevanlinna-Pick type interpolation problems. Particularly, we consider the left tangential interpolation problems on the commutative or noncommutative unit ball. For the commutative setting, we discuss left-tangential operator-argument interpolation problems for Schur-class multipliers on the Drury-Arveson space and for the noncommutative setting, we discuss interpolation problems for Schur-class multipliers on Fock space. We apply the Krein-space geometry approach (also known as the Grassmannian Approach). To implement this approach J-versions of Beurling-Lax representers for shift-invariant subspaces are required. Here we obtain these J-Beurling-Lax theorems by the state-space method for both settings. We see that the Krein-space geometry method is particularly simple in solving the interpolation problems when the Beurling-Lax representer is bounded. The Potapov approach applies equally well whether the representer is bounded or not.