Departmental Colloquium Series: Dr. Robert T.W. Martin

Name: Dr. Robert T.W. Martin 

Affiliation: University of Manitoba

Talk title: Non-commutative measure theory and function theory

Abstract: Measure theory on the complex unit circle and analytic function theory in the unit disk, in
particular the theory of Hardy spaces, are fundamentally connected. Several celebrated theorems
due to P. Fatou, G. Herglotz, F. and M. Riesz and G. Szeg¨o describe the relationship between these
theories.
We will show that many of these classical results have natural extensions to the multivariate
and non-commutative settings of the full Fock space, or free Hardy space of square–summable
power series in several non-commuting variables and positive non-commutative (NC) measures.
Here a positive NC measure is any positive linear functional on the free disk system, the selfadjoint
subspace of operators generated by all NC monomials in the left free shifts, which act as
left multiplication by the independent NC variables on the free Hardy space. Our generalizations
of the single–variable theory typically require new operator–theoretic techniques and recover the
classical results, with new proofs, as a special case.