QI SEMINAR: (Multi-variable) Operator Theory, (Non-commutative) Function Theory and the Free Hardy space

The classical Hardy space of analytic functions on the open complex unit disk has played a seminal role in the development of operator theory and complex function theory. Recently, the full Fock space over several complex dimensions has been identifi ed with the `Free Hardy space', a canonical several non-commuting variable generalization of Hardy space. Elements of this space are free or non-commutative holomorphic functions on a certain non-commutative multi-variable open unit ball. In this talk we further strengthen the interpretation of the full Fock space as the Free Hardy space by demonstrating that many classical Hardy space function theory results have natural and faithful extensions from one to several (non-commuting) variables. Based on joint work with Michael Jury.