Multivariable operator theory and analytic operator spaces

Wayne State University, Detroit MI

Investigators

Abstract

ABSTRACT: The proposed research involves the study of higher-order Hankel forms, related analytic function theory on product domains, operator space structures on algebras of analytic multipliers and related similarity/dilation problems. One of the main goals in the study of higher-order Hankel forms is to provide a unified framework in which to understand the structure theory and, at the same time, fully exploit the interface with cohomology, algebraic geometry, analytic function theory and multivariable operator theory. A main goal of the proposed operator space theory is to determine necessary and sufficient conditions on a multiplier algebra to be completely isomorphic to the minimal operator space structure in terms of the underlying reproducing kernel and the "canonical model" associated to the algebra. The study of Hankel forms (or operators) arises in prediction theory, systems theory, interpolation and control theory. Higher-order analogues of these forms arose in a purely mathematical framework, namely, group representations. However, non-commutative analogues of these forms live on weighted Fock space and are closely connected to the model theory of theleft and right creation operators from particle physics. The theory of operator algebras is the mathematical framework of quantum mechanics while the study of operator space structures on operator algebras is the "quantization" of this theory.