Shift invariant subspaces of slice $L^2$ functions

TL;DR

This paper characterizes invariant subspaces of slice L^2 functions in quaternionic spaces, leading to an inner-outer factorization theorem and a new characterization of outer functions based on cyclicity.

Contribution

It introduces a complete description of invariant subspaces for quaternionic slice L^2 functions and establishes an inner-outer factorization framework.

Findings

Characterization of closed invariant subspaces for quaternionic multiplier operators

Inner-outer factorization theorem for quaternionic Hardy space

New characterization of quaternionic outer functions via cyclicity

Abstract

In this paper we characterize the closed invariant subspaces for the ($*$-)multiplier operator of the quaternionic space of slice $L^2$ functions. As a consequence, we obtain the inner-outer factorization theorem for the quaternionic Hardy space on the unit ball and we provide a characterization of quaternionic outer functions in terms of cyclicity.