Document Type

Dissertation

Date of Degree

Summer 2015

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Paul S. Muhly

Abstract

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

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 W*-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.

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 W*-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 W*-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 W*-Szegö kernel in its light.

Public Abstract

An interpolation problem is one where we are given two collections of points, {x1,…., xn} and {y1,…., yn} and asked if we can find a certain function that sends x1 to y1, x2 to y2, etc. For example, starting with {1, 2} and {3, 4}, is there a polynomial that sends 1 to 3 and 2 to 4? To solve this interpolation problem, we might draw the points (1, 3) and (2, 4) in an xy-coordinate plane and choose our polynomial to be the linear function whose graph is the line connecting the points. By a similar procedure, we could find a quadratic polynomial that interpolates {1, 2, 3} and {1, 0, 1}.

A classic interpolation problem concerns finding an “analytic" function, loosely thought of as a polynomial of infinite degree, that interpolates two collections of complex numbers of absolute value less than one. In the early twentieth century, G. Pick and R. Nevanlinna separately solved the problem. More recent problems of similar flavor are often referred to as Nevanlinna-Pick interpolation problems.

Such a problem arises in the study of the weighted Hardy algebras developed by Paul S. Muhly and Baruch Solel. The setting is more general than that of the original problem of Nevanlinna and Pick. For instance, the two collections of points, {x1,…., Xn} and {y1,…., yn}, consist not of real or complex numbers, but bounded linear operators between Hilbert spaces. In the course of the thesis, we formulate and solve this weighted W*-version of the classic Nevanlinna-Pick interpolation problem.

Keywords

publicabstract, Commutant Lifting, Hardy Algebra, Nevanlinna-Pick, Parrott's Lemma, Reproducing Kernel, W*-Correspondence

Pages

viii, 293 pages

Bibliography

Includes bibliographical references (pages 290-293).

Copyright

Copyright 2015 Jennifer Rose Good

Included in

Mathematics Commons

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