Methods of Hankel and Toeplitz Operators in Noncommutative Analysis

Michigan State University, East Lansing MI

Investigators

Abstract

PI: Vladimir Peller Proposal Number 0200712 Abstract The principal investigator will use vectorial Hankel and Toeplitz operators in noncommutative analysis. In spite of the recent progress of approximation theory for matrix and operator functions there are still many open problems important in applications such as control theory, multivariate stationary processes, etc. The principal investigator will look for a sharp estimate of the degree of the superoptimal approximant of a rational matrix function and sharp estimates of the resolvents of Toeplitz and Wiener-Hopf operators acting on spaces of vector functions, study properties of Wiener-Hopf, thematic, and canonical factorizations of matrix functions, study different regularity conditions for multivariate stationary processes in spectral terms, study Hankel-Schur multipliers, study Schatten--von Neumann properties of certain integral operators arising from Schrodinger operators. Among operators acting on spaces of analytic functions there are two classes that play an extremely important role. These are Hankel operators and Toeplitz operators. The theory of such operators has been rapidly developing for the last 20 years. Recently it has become clear that to satisfy the needs of control theory, prediction theory and other applications, it is necessary to study Toeplitz and Hankel operators on spaces of vector functions. This leads to a considerably more complicated theory, The principal investigator has obtained recently interesting results that required a new technique. He is going to continue to work in this direction and apply Hankel and Toeplitz operators on spaces of vector functions to different problems in noncommutative analysis.