Operator Theoretic Methods in Nevanlinna-Pick Interpolation

Operator theory;Nevanlinna-Pick interpolation;Pure Mathematics

Hamilton, Ryan

University of Waterloo

关键词: Operator theory;    Nevanlinna-Pick interpolation;    Pure Mathematics;   

Others  :  https://uwspace.uwaterloo.ca/bitstream/10012/4305/1/Hamilton-Thesis-2008.pdf

瑞士|英语

来源: UWSPACE Waterloo Institutional Repository

【 摘 要 】

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 $f: mathbb{D} ightarrow mathbb{C}$ satisfying $f(z_i) = w_i$ for $1 leq i leq N$ and $vert f(z) vertleq 1$ 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.