The proposed research considers problems in noncommutative harmonic analysis, operator algebras, and interpolation in several variables. The framework of this proposal is mainly the full Fock space, certain noncommutative (resp. commutative) analytic Toeplitz algebras, and the algebra of all bounded linear operators on a Hilbert space. Noncommutative dilation theory, Poisson transforms on $C^*$-algebras generated by isometries, and commutant lifting theorems are considered in order to find noncommutative (resp. commutative) multivariable analogues to some classical results. The main directions of this proposed research are the following: harmonic analysis on Fock spaces; power bounded sequences of operators, structure, and numerical invariants; central intertwining lifting, suboptimization, and analytic interpolation in several variables; dilation theory for tuples of operators (noncontractions) and non-analytic interpolation in several variables.
The motivation of this research is the recent worldwide interest in the noncommutative aspect of harmonic analysis originated from the concept of quantization which links together several branches of mathematics and is closely related to mathematical physics. The objective of this research is to advance the understanding of the relatively new area of multivariable operator theory and apply some of these results to the study of completely positive maps and their invariants, function theory and interpolation in several variables, multivariable linear systems, scattering, control theory, and model theory for tuples of operators.
