Decompositions for multivariable Schur-class functions, Christoffel-Darboux type formulas, and related problems

Drexel University, Philadelphia PA

Investigators

Abstract

Woerdeman The proposal lies at the interface of multivariable complex analysis and multivariable operator theory. The setting of study is the operator-valued Schur class and its subclass, the Schur?Agler class. While the latter is more understood due to Agler?s seminal work, the multivariable Schur class remains largely unexplored. The proposed program is aimed to gain a novel insight into the structure of multivariable Schur-class functions and higher-dimensional analogs of the two-variable Christoffel?Darboux formula. The investigation is motivated by its ultimate goals which would be to describe the class of commuting tuples of contractions having unitary dilations, to obtain solvability criteria for the Nevanlinna?Pick interpolation problem in the multivariable Schur class, and to develop a theory of system realizations for this class of functions. Tools to be employed include the machinery of scattering systems (building momentum on PIs? recent work) and the technique of Schur complements of multivariable Toeplitz operators (with a parallel development of fast inversion/solver numerical algorithms for multivariable Toeplitz matrices). One of the themes in this program is a best approximation geometric problem, tied to an important special case of Paulsen?s conjecture on the best constant in the multivariable operator-valued linear von Neumann inequality. The project addresses several questions in the active area of multivariable interpolation and factorization problems. These questions are of current relevance to a variety of areas in science and engineering, which include, but are not limited to, system and control theory, filter design, signal and image processing, compressive sensing, and quantum computation. The main educational component of the project is the supervision of graduate students and the mentoring of undergraduates who will be supported by the Research Experiences for Undergraduates program. The results of the proposed research will be disseminated at several levels: through publications and presentations at national and international professional meetings, some of which will also be attended by researchers from other fields, such as computer science, physics and engineering; via formal and informal educational activities, including the weekly Analysis seminar at Drexel University run by the PIs and attended by both faculty and students; via visit exchanges with colleagues from other institutions for collaboration purposes; via the PIs? web sites and preprint servers.