Minimization of the norm of a nonlinear mapping

Abstract

In this paper, we minimize the map $F_{\psi }: U\rightarrow {\mathcal {R}}^{+}$ defined by $F_{\psi }(X) = ||\psi (X)||$ where $\psi : U \rightarrow B(H)$ is a map defined by $\psi (X) = ||S+\phi (X)||$, with $\phi : B(H)\rightarrow B(H)$ a linear or a nonlinear map, $S \in B(H)$, and $U = \{X \in B(H): \phi (X) \in B(H)\}$, using convex and differential analysis (Gâteaux derivative) as well as input from operator theory. The mappings considered generalize the so-called elementary operators and in particular the generalized derivations, which are of great interest by themselves. The main results obtained characterize global minima in terms of (Banach space) orthogonality, and constitute an interesting combination of infinite-dimensional differential analysis, operator theory and duality. It is interesting to point out that our results are the first (at least in our knowledge) in the (linear or nonlinear) case.