DOI

10.17077/etd.30i9m4qn

Document Type

Dissertation

Date of Degree

Spring 2017

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Muhly, Paul S.

First Committee Member

Chifan, Ionut

Second Committee Member

Curto, Raul E.

Third Committee Member

Iovanov, Miodrag C.

Fourth Committee Member

Jorgensen, Palle E. T.

Abstract

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.

Public Abstract

Suppose you are given a finite number of points (x1, y1), … , (xn, yn) in the xy-plane. The problem of finding a function that goes through the given points is called an interpolation problem. In the early 1900s, mathematicians were interested in an interpolation problem in which the desired function was required to satisfy certain conditions. While not overly restrictive, the conditions on the function made it hard to find in some cases and impossible in others. Georg Pick and Rolf Nevanlinna were the first to solve this problem. Instead of finding a formula for the function, they gave an easy way to check if it exists. Since then, theorems about interpolation problems of this form have been called Nevanlinna-Pick theorems, even if the setting is much more general than the original setting.

In this thesis, we compare two very general Nevanlinna-Pick theorems, one due to Constantinescu and Johnson and the other due to Muhly and Solel. Though the theorems seem similar, their connection is obfuscated by the authors’ notation and context. First we reformulate Constantinescu-Johnson’s theorem in the setting of Muhly-Solel’s result. Our proof, modeled after Constantinescu-Johnson’s, employs the so-called displacement equation. When we compare our new theorem to Muhly-Solel’s, we find fundamental differences. Nevertheless, we identify certain additional assumptions on the original data points which guarantee that the two theorems will yield the same result.

Keywords

displacement equation, Nevanlinna-Pick interpolation, noncommutative Hardy algebra, W*-correspondence

Pages

vii, 84 pages

Bibliography

Includes bibliographical references (pages 83-84).

Copyright

Copyright © 2017 Rachael M. Norton

Included in

Mathematics Commons

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