Methods of Hankel and Toeplitz Operators in Noncommutative Function Theory

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Proposal: DMS-9970561Principal Investigator: Vladimir V. PellerAbstract: V.V. Peller is going to continue to work on Hankel and Toeplitz operators and their applications in different fields of mathematics: control theory, prediction theory, approximation theory. It has become clear that it is especially important in applications to study Hankel and Toeplitz operators with matrix-valued symbols. The development of the theory of Hankel and Toeplitz operators with matrix-valued symbols depends on further development of noncommutative function theory (the theory of matrix-valued or operator-valued functions). On the other hand, many problems arising in noncommutative function theory can be solved with the help of Hankel and Toeplitz operators having matrix-valued symbols. In particular, V.V. Peller is going to study the following topics: spectral characterizations of completely regular vectorial stationary processes; characterization of unitary-valued functions of vanishing mean oscillation (and other classes) in terms of Wiener-Hopf factorizations; unitary interpolants of continuous matrix functions as well as of matrix functions in other function classes; the relationship between Wiener-Hopf factorizations and thematic factorizations; similarity to a unitary operator; estimates of resolvents; and the existence of nontrivial invariant subspaces for Toeplitz operators with matrix-valued symbols.Hankel and Toeplitz operators are certain special classes of operators which have proved to be very helpful in many fields of mathematics and for applications in control theory and electrical engineering. V.V. Peller is going to continue his research on applications of these interesting operators to the theory of functions whose values are matrices rather than real or complex numbers. The theory of such functions is considerably more delicate than classical function theory because the values of such functions do not have to commute; i.e., the order of the factors in a product cannot be changed without risk of changing the value of the product. On the other hand, such matrix-valued functions are extremely important in applications. For example, they appear in a natural way in systems theory when one must deal with multiple inputs and multiple outputs. V.V. Peller is going to study several important problems that involve matrix-valued functions (with technical names such as factorization problems, approximation problems, interpolation problems, etc.). The solution of these problems would lead in turn to new results on Hankel and Toeplitz operators as well as to numerous new concrete applications.