Topics in Multivariable Operator Theory and Interpolation

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The main directions of this proposed research are the following: (1) Harmonic analysis on Fock spaces, (2) Entropy and multivariable interpolation, (3) Numerical in-variants for Hilbert modules over free semigroup algebras. A new notion of entropy for operators on Fock spaces is proposed in connection with factorizations of multi-Toeplitz and multi-analytic operators, multivariable interpolation, and numerical invariants for completely positive maps. Under the first heading are also found a number of problems pertaining to harmonic analysis on Fock spaces, the geometry of the unit ball of non-commutative analytic Toeplitz algebras generated by the left creation operators, interpo-lation sequences, and Fejer type inequalities in several variables. These results have po-tential applications to function theory in several complex variables, prediction and multi-variate stochastic processes. In recent years, there has been exciting progress in multi-variable interpolation. The PI will continue his work in this area of research and expects to find the maximal entropy solutions of several multivariable interpolation problems (Sarason, Caratheodory-Schur, Nevanlinna-Pick) including the abstract noncommutative commu-tant lifting theorem for row contractions. This proposed research is expected to play a role in multivariable control theory and systems theory. A new invariant, entropy, is pro-posed for n-tuples of operators, that seems to complement the curvature invariant. The goal is to make significant progress towards a complete set of numerical invariants that classify large classes of completely positive maps.Originated from the concept of quantization, operator theory links together several branches of mathematics and is closely related to mathematical physics. The motivation of this research is the recent worldwide interest in the noncommutative aspect of har-monic analysis and multivariable operator theory. The objective is to advance the under-standing of these relatively new areas of research and apply the results to the study of completely positive maps and their invariants, function theory and interpolation in several variables, multivariable linear systems and control theory, and prediction and stochastic processes. Potential applications in fields such as geophysics and image processing are also expected.