Global minimum and orthogonality in C1-classes ✩

In this paper we characterize the global minimum of an arbitrary function defined on a Banach space, in terms of a new concept of derivatives adapted for our case from a recent work due to D.J. Keckic (J. Operator Theory, submitted for publication). Using these results we establish several new characterizations of the global minimum of the map Fψ :U →R+ defined by Fψ(X)= ψ(X) 1, where ψ :U →C1 is a map defined by ψ(X) = S + φ(X) and φ :B(H)→B(H) is a linear map, S ∈ C1, and U = {X ∈ B(H): φ(X) ∈ C1}. Further, we apply these results to characterize the operators which are orthogonal to the range of elementary operators.  2003 Elsevier Inc. All rights reserved.