Date of Award

1-1-2010

Document Type

Dissertation

Degree Name

Ph.D.

Organizational Unit

College of Natual Science and Mathematics

First Advisor

Alvaro Arias, Ph.D.

Second Advisor

James Haglar

Third Advisor

Frédéric Latrémolière

Fourth Advisor

Nicholas Ormes

Fifth Advisor

Ramakrishna Thurimella

Keywords

Commutant lifting, Fock space, Interpolation, Multivariable operator theory, Noncommutative domain, Weighted shifts

Abstract

The classical Carathéodory interpolation problem is the following: let n be a natural number, a0, a1, . . . , aN be complex numbers, and D the unit disk. When does there exist an analytic function F : D → C and complex numbers aN+1, aN+2, . . . such that F(z) = a0 + a1z + a2z2 + . . . + aNzN + aN+1zN+1 + . . . and ||F||∞ < 1? In 1967, Sarason used operator theory techniques to give an elegant solution to the Carathéodory interpolation problem. In 1968, Sz.-Nagy and Foias extended Sarason's approach into a commutant lifting theorem. Both the theorem and the technique of the proof have become standard tools in control theory. In particular, the commutant lifting theorem approach lends itself to a wide range of generalizations. This thesis concerns one such generalization.

Arias presented generalizations of the original commutant lifting theorem relating to the full Fock space. Popescu then refined the approach by introducing domain algebras. While Arias and Popescu focused on module maps from one noncommutative domain algebra to itself, we present a unitarily equivalent representation which allows us to easily generalize their results to module maps between different noncommutative domain algebras. We use a renorming technique to prove that as long as the “formal identity map” is bounded, we can lift module maps between different noncommutative domains generated by finite positive regular free holomorphic functions. Finally, we apply the lifting theorem to create projective resolutions.

Publication Statement

Copyright is held by the author. User is responsible for all copyright compliance.

Rights Holder

Jonathan Von Stroh

Provenance

Received from ProQuest

File Format

application/pdf

Language

en

File Size

90 p.

Discipline

Mathematics

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