A Two-dimensional Hahn-banach Theorem

Let T̃ = ∑n i=1 ũi⊗ vi : V → V = [v1, ..., vn] ⊂ X, where ũi ∈ V ∗ and X is a Banach space. Let T = ∑n i=1 ui ⊗ vi : X → V be an extension of T̃ to all of X (i.e., ui ∈ X∗) such that T has minimal (operator) norm. In this paper we show in particular that, in the case n = 2 and the field is R, there exists a rank-n T̃ such that ‖T‖ = ‖T̃‖ for all X if and only if the unit ball of V is either not smooth or not strictly convex. In this case we show, furthermore, that, for some ‖T‖ = ‖T̃‖, there exists a choice of basis v = v1, v2 such that ‖ui‖ = ‖ũi‖, i = 1, 2; i.e., each ui is a Hahn-Banach extension of ũi.