Mixing Douglas' and weak majorization and factorization theorems

TL;DR

This paper introduces a combined version of Douglas' majorization and factorization theorems for operators into product spaces, with applications to control theory of PDE systems.

Contribution

It extends Douglas' theorems to operators with codomain in product spaces, linking operator range inclusion with kernel and range relations, and applies this to control theory.

Findings

Extended Douglas' theorems to product space operators

Established new operator inclusion relations

Applied results to control of PDE systems

Abstract

The Douglas' majorization and factorization theorem characterizes the inclusion of operator ranges in Hilbert spaces. Notably, it reinforces the well-established connections between the inclusion of kernels of operators in Hilbert spaces and the (inverse) inclusion of the closures of the ranges of their adjoints. This note aims to present a ''mixed'' version of these concepts for operators with a codomain in a product space. Additionally, an application in control theory of coupled systems of linear partial differential equations is presented.