Multivariable Operator Theory

University Of Iowa, Iowa City IA

Investigators

Abstract

Abstract Curto 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.