Gateaux derivative and Orthogonality in B(H)

Abstract

The general problem in this paper is minimizing themap F Y: B(H) ! R+ defined by FY(X) = k (X)k, where : B(H) ! B(H) is a map defined by (X) = S + O(X) and O : B(H) ! B(H) is a linear map, S 2 B(H). using convex and differential analysis (Gateaux 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 dimentional differentiable analysis, operator theory and duality. Note that the results obtained generalize all results in the leterature concerning operator which are orthogonal to the range of a derivation and the technics have not been done by other authors.