To: Dr. Joe Jenkins >From: Raul Curto Date: January 7, 1998 Re: Abstract of Project, Multivariable Operator Theory This research project deals with four areas of multivariable operator theory: (i) existence, uniqueness, and localization of the support of representing measures for truncated moment problems; (ii) structure and spectral theory for polynomially hyponormal operators; (iii) standard operator models over Reinhardt domains; and (iv) a multivariable analog of Apostol's Lemma on hyperinvariant subspaces for contractions. Concerning the first area, special emphasis will be given to the study of moment problems associated with singular matrices, building on recent work (joint with L.A. Fialkow) on flat extensions of positive moment matrices, which has led to a general framework for the study of truncated complex moment problems. With the aid of a new moment matrix associated to a semi-algebraic set, progress is expected in the localization-of-support problem. A characterization of quadrature domains in terms of their moment matrices, and further applications to quadrature and cubature formulas will also be sought. As part of (ii), 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 and through Putinar's 2-subscalar model, will be sought. The research aim in the third area (which deals with a Sz. Nagy- Foias dilation theory in several variables and builds on recent work of A. Athavale, V. Muller, F.-H. Vasilescu and others), is to extend the existing theory to functional Hilbert spaces over Reinhardt domains. The suitability of multi-shifts as standard models, the validity of von Neumann's inequality for special n-tuples, and the structure of C*-algebras generated by spherical isometries, will be considered. The fourth and final area deals with the invariant subspace structure of commuting contractions on Hilbert space. Using recent results of J. Eschmeier, M. Kosiek, A. Octavio, M. Ptak, and others, 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 to describe the 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.
