Reproducing Kernels and Function Theory

University Of Alabama Tuscaloosa, Tuscaloosa AL

Investigators

Abstract

Abstract: The proposer plans to examine function theory problems, invariant subspace problems, and model theory on certain reproducing kernel Hilbert spaces of analytic functions on domains in Cn. The main cases concern, but are not limited to, spaces whose reproducing kernel has one positive square. Recent commutant lifting theorems and Beurling-Lax-Halmos type theorems for such spaces provide the rationale for connecting these topics. Among the function theory questions considered are corona theorems for various multiplier algebras of operators on reproducing kernel spaces. In addition, using Hilbert space methods, the principal investigator expects to derive new estimates for the H2(D2)-corona theorem on the bidisk and consider vector valued corona theorems. The invariant subspace topics which will be pursued include consequences of the Beurling-type invariant subspace theory for reproducing kernel Hilbert spaces with Nevanlinna-Pick kernels. In addition, a concrete model theory of commuting n-tuples related to that of Agler and Athavale will be developed. One of the unifying themes of the proposed work is the systematic use of control theory ideas in the context of reproducing kernel Hilbert spaces. Such spaces provide a fundamental notion for connecting problems in, for example, classical mechanics, quantum mechanics, circuit theory, and signal analysis. Some of the proposed problems may be viewed as inverse problems or "partial knowledge" problems. For example, to effect a desired output, how should one decide on an appropriate input? Interpolation type problems have long been studied for the stabilization of systems in feedback control in one dimension. Much of the emphasis of the proposed research will be on the multidimensional theory. The proposer expects this viewpoint to be productive in developing new techniques for the inverse problems being considered.