In the last two decades, a free analogue of Sz.-Nagy-Foias theory on the unit ball in the space of n-tuples of Hilbert space operators has been pursued by the investigator and others. This theory has already had remarkable applications in n-dimensional complex interpolation, multivariable prediction and entropy optimization, control theory, systems theory, scattering theory, and wavelet theory. The investigator will continue the ongoing program to develop a free analogue of the Nagy-Foias theory of contractions for more general noncommutative domains and varieties in the space of n-tuples of operators, in a uniform framework that includes both aspects: noncommutative and commutative. The investigator will continue to work on function theory on noncommutative balls, with the emphasis on the geometric aspects of the theory of free holomorphic functions and the connection with the hyperbolic geometry. The main directions of this proposed research are the following: (i) Noncommutative domains, universal models, operator algebras, and classification; (ii) Unitary invariants on noncommutative domains; (iii) Noncommutative hyperbolic geometry; (iv) Free holomorphic functions on noncommutative balls. Each noncommutative domain (resp. variety) to be studied is associated with a universal model consisting on operators on Fock spaces, a domain (resp. Hardy, C*-) algebra, and Reinhardt (resp. circular) domains in n-dimensional complex space. The proposer will work on the classification of certain classes of noncommutative domain algebras, the classification of the corresponding noncommutative domains up to free biholomorphic maps, and the connections with the classification of Reinhardt domains. The investigator will try to develop a model theory on noncommutative polydomains and formulate a theory of curvature invariant in this setting, in the attempt to extend and unify the existent results (commutative and noncommutative). Significant progress towards the classification of the elements of a noncommutative domain up to unitary equivalence is expected; an important part of the proposed research concerns unitary invariants: characteristic function, curvature, and entropy. The investigator will continue to study the hyperbolic geometry of the unit ball of n-tuples of operators in close connection with the theory free holomorphic functions, and will try to extend the theory to more general noncommutative domains.
Originated from the concept of quantization, operator theory links together several branches of mathematics, is closely related to mathematical physics, and has numerous applications in engineering. The motivation of the proposed research is the recent worldwide interest in the noncommutative aspects of multivariable operator theory and function theory, and their interplay with the theory of holomorphic functions in several complex variables, the representation theory of operator algebras, and the harmonic analysis on Fock spaces. The objective is to advance the understanding of these relatively new areas of research and make new connections with hyperbolic complex analysis and algebraic geometry. The expected results have potential applications in interpolation and biholomorphic classification in several complex variables, systems theory, and scattering theory. Potential impact in fields such as control theory, entropy optimization, wavelet theory, and image processing is also expected. The results will be disseminated in graduate seminars, conferences and workshops.
