Multivariable Operator Theory

University Of Iowa, Iowa City IA

Investigators

Abstract

Many questions in physics, mathematics, and engineering can be 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 (think infinite-size) 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. A main goal of this project involves finding such models for operators. Once that is done, many basic structural questions 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. The aim is to resolve several outstanding questions in so-called multivariable operator theory. The project also involves working with students and creating recruitment and retention opportunities particularly for women and minorities to pursue careers in mathematics and other STEM fields. The main idea/thrust of this project will be to utilize recently established connections between analytic geometry and analysis to study questions in multivariable operator theory. Attention will be focused on two principal areas: (i) truncated moment problems (TMP); and (ii) (joint) hyponormality and subnormality for commuting families of operators on Hilbert space. In the first area, algebraic conditions will be determined for the existence, uniqueness, and localization of the support of representing measures for TMP. Development of new solubility criteria in the case of moment matrices with column relations is to be tied to irreducible algebraic curves and cubic column relations associated with finite algebraic varieties. The second direction concerns operator theory over Reinhardt domains, with special emphasis on spectral and structural properties of multivariable weighted shifts. Planned are both the study of a new bridge connecting 2-variable weighted shifts to the theory of weighted shifts on directed trees, and the characterization of moment infinitely divisible weighted shifts, using the theory of completely alternating sequences and completely monotone functions, Bernstein functions, and Laplace and Fourier transforms. The planned methodology will include multivariable techniques in the study of block Toeplitz operators. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.