Operator Theory and Complex Analysis

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Proposal: DMS-9970376Principal Investigator: Nathan S. FeldmanAbstract: Feldman will conduct research in operator theory and its interplay with complex analysis. Primary emphasis will be placed on studying special classes of linear operators that are close to normal operators, such as subnormal, hyponormal, and essentially subnormal operators. Questions regarding various forms of cyclicity for cosubnormal and cohyponormal operators will be investigated, as will the relationship between cyclicity and the existence of intertwining maps. Other topics to be explored include: the existence of hyperinvariant subspaces for subnormal operators; "generalized eigenvectors" for adjoints of subnormal operators; the principal function for a subnormal operator, and its relations with such classes of operators as Toeplitz operators, Hankel operators, and self-commutators.Operator theory is a natural extension of the ideas of matrices and linear algebra to infinite dimensions. In simple terms, an operator is just an infinite array or matrix of numbers and is used in linear problems involving infinitely many unknown variables. Although many intractable real world problems are not linear, one can frequently "linearize" a given problem to produce a new problem that is both linear and solvable and that has a solution which closely approximates -- hence provides valuable information about -- the solution of the original physical problem. Perhaps the simplest example of this is the idea of the derivative from calculus, which is used to approximate a curve by a straight line (something that is linear). In a multivariable setting, the precise analogue of the derivative is a linear operator. Recognizing how useful and powerful a tool the derivative of even a single variable function has become in all areas of science, one can readily appreciate how important it is to understand fully the structure of linear operators. While some operators can be rather "pathological" in character, the subnormal operators that are the objects of study in this project have a great deal of structure: they are very natural themselves, and they arise in natural ways. Operator theory and linear algebra have had a profound impact on all branches of mathematics, physics, and engineering. It is hoped that Feldman's research will likewise exert an influence not only on pure mathematics but on the applied sciences as well.