Additional Studies on Displacement Mapping with Restrictions

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.