The 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) multivariable techniques in the detection of subnormality, esp. for Toeplitz operators on the unit circle, including an approach to the Lifting Problem for Commuting Subnormals (LPCS); and (iii) operator theory over Reinhardt domains, with special attention given to the spectral and structural properties of multivariable weighted shifts. Concerning the first area, we plan to extend recent work on flat extensions of positive moment matrices (joint with L. Fialkow and H.M. Möller), which has led to a general framework for the study of TMP. We plan to apply these methods beyond the extremal case, to obtain algebraic and geometric invariants for solubility, to further develop an appropriate analogue of the Riesz-Haviland Theorem, and to investigate the duality between TMP and degree-bounded representations of polynomials nonnegative on a prescribed semialgebraic set. The second area deals with a multivariable approach to LPCS and with subnormality for Toeplitz operators. Building on work of C. Cowen for the case of hyponormal Toeplitz operators, our approach is to first characterize 2-hyponormality, then k-hyponormality, and eventually subnormality. We would also like to develop further the ideas in recent joint work with J. Yoon and S.H. Lee to search for necessary and sufficient conditions for two commuting subnormal operators to admit a joint normal extension, including some useful connections with Agler's abstract model theory. The third area deals with structural and spectral properties of multiplication operators on functional Hilbert spaces over Reinhardt domains. We plan to extend the study of the spectral picture of subnormal multivariable weighted shifts to hyponormal ones, exploiting recent results (joint with J. Yoon) which highlight some of the pathology that arises when a Berger measure is absent, and using the groupoid techniques introduced in joint work with P. Muhly.
Hilbert space operators are infinite generalizations of matrices. The infinite generalization of a vector is frequently a function and for this reason Hilbert space operators are frequently modeled as the operator of multiplication on a space of functions. Part of this project involves finding such models for operators or tuples of operators. Once such models are obtained many basic questions about the structure of these operators become more natural. A separate part of the research deals with inverse problems, esp. moment problems, which are related to power moments of mass distributions, and arise naturally in statistics, spectral analysis, geophysics, image recognition, and economics. Our research is aimed at resolving some outstanding problems in multivariable operator theory, while creating recruitment and retention opportunities for women and minorities to pursue careers in mathematics, by engaging their participation in projects related to the interaction of mathematics with other sciences. The results on truncated moment problems have been used by S. McCullough to obtain a structure theorem in Fejér-Riesz factorization theory; by J. Lasserre in the study of semi-algebraic subset of the plane; and by J. Lasserre and M. Laurent to convert polynomial optimization into an instance of semidefinite programming. We anticipate that such connections with areas outside of operator theory will continue to arise. Several open problems in this proposal are written to generate research projects accessible to undergraduate and graduate students, especially those related to cubatures, low-degree moment problems, their connections with algebraic geometry, and multivariable weighted shifts.
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 organize the information, and codify it in layers according to its relevance for the mathematical problem under consideration. Matrices can also be made part of algebraic systems; that is, they can be added and multiplied as if they were numbers. When those operations are properly defined, several properties of addition and multiplication are preserved (e.g., associativity and distributivity), while other properties (e.g., commutativity) do not hold. Matrices also serve as concrete representations of spatial transformations which preserve addition and scalar multiplication. Such transformations are called linear, and they map elements of the underlying space, called vectors, into other vectors. For instance, planar transformations such as rotations, translations, dilations and contractions are best described using matrices. In short, a matrix provides a graphical representation of linear transformation acting on a vector space. Thus, determining the structure of the matrix reveals important properties of the linear transformation. 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 our research program involves finding such models for operators. Once the models are obtained, many basic structural questions about the operators become natural. Beginning in the 1950’s, 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. Our research has focused, among other topics, on the study of subnormality for two classes of Hilbert space operators: unilateral weighted shifts and Toeplitz operators. For the so-called 2-variable weighted shifts, we have been able to define and describe in great detail a number of criteria that classify these families of operators according to how close or how far they satisfy the subnormality property. We have done this with tests and algorithms that produce concrete answers for 2-variable weighted shifts with specific collections of weights. We have also been able to calculate the spectral picture of a 2-variable weighted shift in a number of concrete situations, in a manner completely analogous to the well known description which exists for classical subnormal unilateral weighted shifts. Similarly, the classes of scalar and block Toeplitz operators arise in a variety of areas of mathematics and physics. For block Toeplitz operators, we have been able to generalize and expand on the existing theory of scalar Toeplitz operators. In the process, we have provided new insights into the so-called Halmos' Problem 5, which asks for a characterization of subnormal Toeplitz operators. A separate part of our research has dealt with inverse problems, esp. moment problems for probability density functions, which occur naturally in statistics, spectral analysis, geophysics, image recognition, global positioning tools, signal detection theory, and economics. Our work on truncated moment problems has been applied by a number of researchers in optimization theory, real algebraic geometry, numerical analysis, semidefinite programming and sensor network localization. Of special significance has been the study of cubic column relations for the matrix associated to a truncated moment problem, and of truncated moment problems whose associated matrices are recursively determinate. In our research we have aimed at resolving some outstanding problems in multivariable operator theory, while creating recruitment and retention opportunities for women and minorities to pursue careers in mathematics, and its interaction with other sciences. Several questions in our research projects were written to generate research problems accessible to undergraduate and graduate students. One of the graduate students completed his degree in December 2010, with a dissertation on truncated moment problems associated with cubic column relations. A postdoctoral fellow, in residence at the University of Iowa during the calendar year 2012, worked on invariant subspace problems arising in single variable operator theory. Finally, the NSF funds helped support the Iowa-Nebraska Functional Analysis Seminar (INFAS), a regional meeting held twice a year in Des Moines, Iowa, and which gathers researchers from a number of nearby institutions, including Iowa, Iowa State, Nebraska-Lincoln, Creighton, and Nebraska-Omaha. With a typical attendance of 20-to-25 mathematicians, INFAS provides an opportunity for graduate students and junior mathematicians to listen to state-of-the-art presentations given by operator theorists and functional analysis from the upper Midwest. INFAS also provides a seminar-like atmosphere conducive to the exchanges of ideas amongst researchers from the above mentioned institutions.
