Analysis Seminar

Wednesday, November 3, 2021 4:00 pm - 4:00 pm EDT (GMT -04:00)

Robert Martin, University of Manitoba

"Non-commutative rational multipliers of the Fock space"

The theory of rational functions in several non-commuting (NC) variables has experienced rapid growth in recent years; deep connections have been developed with established branches of pure and applied mathematics including Free Algebraic Geometry, Non-commutative Algebra, Free Probability and Control Theory. In this talk we extend classical results on rational multipliers of the Hardy space of square--summable power series in the complex unit disk to the NC setting of the Fock space, or free Hardy space, of square--summable power series in several NC variables.  

Our results include: an NC rational Fejer--Riesz Theorem, proof that a contractive NC rational multiplier of Fock space is either inner, i.e. isometric or not an extreme point, a characterization of the NC Clark measures of both inner and contractive NC rational multipliers, as well as an explicit description of their minimal realizations and an Aronszajn--Donoghue theorem on the mutual singularity of the singular parts of the NC Clark measures of a contractive NC rational multiplier. We will show that each of these results is a natural analogue of classical fact.

This seminar will be held jointly online and in person: