The Gâteaux derivative and orthogonality in C_∞

The general problem in this paper is minimizing the C_∞− norm of suitable affine mappings from B(H) to C_∞, 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.