Additional Studies on Displacement Mapping with Restrictions

TL;DR

This paper advances the mathematical understanding of displacement mappings by deriving formulas for inverses and analyzing specific operators, with applications to projections and reflections in linear subspaces.

Contribution

It provides new formulas for set-valued and Moore-Penrose inverses of displacement mappings and analyzes related operators, extending previous theoretical work.

Findings

Formulas for set-valued and Moore-Penrose inverses of displacement mappings.

Analysis of operators ((1/2)Id + T) and its inverse.

Applications to reflected and projection operators in linear subspaces.

Abstract

The theory of monotone operators plays a major role in modern optimization and many areas of nonlinera analysis. The central classes of monotone operators are matrices with a positive semidefinite symmetric part and subsifferential operators. In this paper, we complete our study to the displacement mappings. We derive formulas for set-valued and Moore-Penrose inverses. We also give a comprehensive study of the the operators ($(1/2) {\rm Id} + T$ and its inverse) and provide a formula for $((1/2) {\rm Id} + T)^{-1}$. We illustrate our results by considering the reflected and the projection operators to closed linear subspaces.