This research deals with multivariable operator theory, focusing attention on three areas: (i) algebraic conditions for existence, uniqueness, and localization of the support of representing measures for truncated moment problems (TMP); (ii) operator theory over Reinhardt domains, with special emphasis on spectral and structural properties of multivariable weighted shifts; and (iii) multivariable techniques in the study of block Toeplitz operators. Concerning the first area, the plan is to extend recent work on flat extensions of positive moment matrices and extremal moment problems (joint with L. Fialkow, H.M. Möller and S. Yoo), which has led to a general framework for the study of TMP. The principal investigator and his collaborators will develop new methods and techniques, and apply them to the case of cubic column relations associated with finite algebraic varieties, and to TMPs with recursively determinate moment matrices. The second area deals with multiplication operators on functional Hilbert spaces over Reinhardt domains. This part of the project will extend the study of the spectral picture of subnormal 2-variable weighted shifts to hyponormal ones, by applying previous results (joint with S.H. Lee and J. Yoon) and by employing the groupoid techniques developed in the principal investigator's work with P. Muhly and K. Yan. The third area deals with a multivariable approach to subnormality of Hilbert space operators, with special emphasis on scalar and block Toeplitz operators. The approach is to characterize 2-hyponormality, then k-hyponormality, and eventually subnormality. The principal investigator will develop further the ideas in his previous work with I.B. Jung, S.H. Lee, W.Y. Lee, S.S. Park, and M. Putinar. As a testing ground, he will search for a model theory for 2-hyponormal operators, a topic that leads to useful connections with J. Agler's abstract model theory. Using function-theoretic and multivariable operator theory techniques, the principal investigator has recently proved a version of Abrahamse's theorem for block Toeplitz operators (jointly with I.S. Hwang and W.Y. Lee). He will seek a more general version of this theorem, and the solution of a related subnormal Toeplitz completion problem.
Many problems in physics, mathematics, and engineering can be best described by representing complex physical entities as large arrays of numbers and mathematical symbols, called matrices. Matrices help us visualize how linear transformations act on vector spaces; determining their structure reveals important properties of the transformations. Hilbert space operators are infinite-dimensional generalizations of matrices. The generalization of a vector is often a function, and as a result, operators are frequently modeled as multiplications on spaces of functions. Part of this project involves finding such models for operators. Once the models are obtained, many basic structural questions about the operators become natural. Beginning in the 1950s, the study of subnormal operators has been highly successful, and its theory has made key contributions to areas such as functional analysis, quantum mechanics, and engineering. Similarly, the classes of scalar and block Toeplitz operators arise in a variety of areas of mathematics and physics. A separate part of the research deals with inverse problems that occur naturally in statistics, spectral analysis, geophysics, image recognition, global positioning tools, signal detection theory, and economics. The principal investigator's work on truncated moment problems has been applied in optimization theory, real algebraic geometry, numerical analysis, semidefinite programming, and sensor network localization. The project aims to resolve some outstanding problems in multivariable operator theory, at the same time creating recruitment and retention opportunities for women and minorities to pursue careers in mathematics and other STEM fields. Several questions in this project are written to generate research problems accessible to undergraduate and graduate students.
