Date of Award
College of Natual Science and Mathematics
Alvaro Arias, Ph.D.
Commutant lifting, Fock space, Interpolation, Multivariable operator theory, Noncommutative domain, Weighted shifts
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.
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Jonathan Von Stroh
Received from ProQuest
Von Stroh, Jonathan, "Lifting Module Maps Between Different Noncommutative Domain Algebras" (2010). Electronic Theses and Dissertations. 677.