# Codes and Expansions (CodEx) Seminar

## Marcin Bownik (University of Oregon)

Multiplication-invariant operators and the classification of abelian group frames

In this talk we discuss the properties of multiplication invariant (MI) operators acting on subspaces of the vector-valued space \(L^2(X;\mathcal H)\). We show that there is a natural isomorphism between the category of MI spaces (with MI operators as morphisms) and the category of measurable range functions whose morphisms are measurable range operators. We investigate how global properties of an MI operator are reflected by local pointwise properties of its corresponding range operator. We present several results about frames generated by multiplications in \(L^2(X;\mathcal H)\). This includes the classification of frames of multiplications with respect to unitary equivalence by measurable fields of Gramians. The talk is based on a joint work with Joey Iverson.