Operator Theoretic Methods in Nevanlinna-Pick Interpolation

Abstract

This Master's thesis will develops a modern approach to complex interpolation problems studied by Carath\'{e}odory, Nevanlinna, Pick, and Schur in the early $20^{th}$ century. The fundamental problem to solve is as follows: given complex numbers $z_1,z_2,...,z_N$ of modulus at most $1$ and $w_1,w_2,...,w_N$ additional complex numbers, what is a necessary and sufficiency condition for the existence of an analytic function $f: \mathbb{D} \rightarrow \mathbb{C}$ satisfying $f(z_i) = w_i$ for $1 \leq i \leq N$ and $\vert f(z) \vert \leq 1$ for each $z \in \mathbb{D}$? The key idea is to realize bounded, analytic functions (the algebra $H^\infty$) as the \emph{multiplier algebra} of the Hardy class of analytic functions, and apply dilation theory to this algebra.

This operator theoretic approach may then be applied to a wider class of interpolation problems, as well as their matrix-valued equivalents. This also yields a fundamental distance formula for $H^\infty$, which provides motivation for the study of completely isometric representations of certain quotient algebras.

Our attention is then turned to a related interpolation problem. Here we require the interpolating function $f$ to satisfy the additional property $f'(0) = 0$. When $z_i =0$ for some $i$, we arrive at a special case of a problem class studied previously. However, when $0$ is not in the interpolating set, a significant degree of complexity is inherited. The dilation theoretic approach employed previously is not effective in this case. A more function theoretic viewpoint is required, with the proof of the main interpolation theorem following from a factorization lemma for the Hardy class of analytic functions. We then apply the theory of completely isometric maps to show that matrix interpolation fails when one imposes this constraint.