This research project deals with five areas of multivariable operator theory: (i) algebraic conditions for existence, uniqueness, and localization of the support of representing measures for truncated moment problems; (ii) model theory for 2-hyponormal operators; (iii) structure and spectral theory for polynomially hyponormal operators; (iv) operator models over Reinhardt and matrix Reinhardt domains; and (v) a multivariable analog of Apostol's Lemma on hyperinvariant subspaces for contractions. Concerning the first area, Curto plans to extend his recent work (joint with L. Fialkow) on flat extensions of positive moment matrices, which has led to a general framework for the study of truncated complex moment problems. He plans to apply these methods to study the case of singular moment matrices, and to continue to analyze the question of localization of the support of representing measures. Curto expects to settle a conjecture on the solubility index of a truncated moment sequence, and to characterize quadrature domains in terms of their moment matrices. As part of (ii), a model theory for 2-hyponormal operators will be sought, along the lines of the existing theory for subnormal operators, and using as test ground recent results on unilateral weighted shifts. In joint work with W.Y. Lee, Curto has introduced the class of weakly subnormal operators and obtained preliminary results on its position relative to subnormality and 2-hyponormality, including a proof that contractive 2-hyponormal operators with closed range self-commutator are extremal for the family of 2-hyponormal contractions. The third area is closely related to the first two, in that both originate in Curto's work on quadratic and joint hyponormality, which eventually led to the solution of the subnormal completion problem for weighted shifts and to the existence of non-subnormal polynomially hyponormal operators. Curto will seek a characterization of polynomially hyponormal weighted shifts, a structure theorem for quadratically hyponormal shifts, and the detection of non-subnormal polynomially hyponormal operators through the Pincus principal function. The fourth area deals with a Sz. Nagy-Foias dilation theory in several variables, by extending existing results to functional Hilbert spaces over Reinhardt domains. The suitability of multi-shifts as standard models and the validity of von Neumann's inequality for special n-tuples will be considered. Curto plans to extend the description of the spectral picture of multiplication operators associated with Reinhardt measures to functional Hilbert spaces over matrix Reinhardt domains. Finally, the fifth area deals with the invariant subspace structure of commuting contractions on Hilbert space. Two main goals will be pursued: (a) an extension to the Taylor spectrum of results on spectral dominance currently available only for the Harte spectrum, and (b) an analog of Apostol's Theorem in several variables.
Multivariable operator theory is a rapidly evolving area of mathematics, with deep and significant connections with areas of differential geometry, topology, complex analysis, and algebraic geometry, and with exciting applications to engineering, quantum and relativistic mechanics, and computational mathematics. The theory of truncated moment problems provide easily accessible formulas for the evaluation of areas and volumes of complex regions, of moments of inertia and centers of gravity. Dilation theory and invariant subspace theory are essential tools in the description of algebraic properties of elaborate physical or engineering systems, and the study of transformations on function spaces has often led to the solution of problems in control theory, intimately tied to systems theory and electrical engineering. Our research project 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.
