Shift invariant subspaces of the quaternionic space of slice $L^2$ functions

I will give a characterization of the closed shift invariant subspaces of the quaternionic space of slice $L^2$ functions. As a consequence, the inner-outer factorization theorem for the quaternionic Hardy space $H^2$ on the unit ball is obtained. Therefore, I will present some properties of inner and outer functions in the quaternionic setting, providing in particular sufficient conditions as well as necessary ones for functions to be inner or outer.

This talk is based on joint works with Giulia Sarfatti and Daniel Seco.