Topics in multivariable operator theory

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This project is concerned with the further development of certain aspects of multivariable operator theory, especially its connection with function theory in several complex variables. The basic point of view of the theory is that one can profitably study analytic functions by considering their action on matrix or operator inputs, rather than just scalar inputs. Over the last half-century this viewpoint has been extensively developed in the one-variable setting, establishing deep connections between complex function theory, operator theory, and functional analysis; von Neumann's inequality and the Sz.-Nagy Dilation theorem are fundamental results in this area. Particular problems that will be investigated in this project include the failure of von Neumann's inequality in several variables, function-theoretic aspects of analytic functions realized as transfer functions generated by operator tuples, and the analysis of functions of positive real part in terms of "noncommutative spectral measures," that is, positive functionals on operator systems. A unifying theme will be the theory of composition operators in several variables, especially compactness questions.The project belongs to the branch of mathematics known as Operator Theory. Originally developed as the mathematical language of quantum mechanics, over the last century it has expanded to influence many other areas of science and mathematics. It has found many applications in engineering, for example in the design of robust control systems for aircraft and spacecraft; the analysis of acoustical scattering data; and more recently in the study of linear matrix inequalities, which lie behind many optimization problems. Conversely, problems in engineering and mathematical physics have raised new questions in operator theory; one related to the project is the description of the electrostatic properties of composite materials.