Complex interpolation of a Banach space with its dual - Watbled, Frédérique (MATHEMATICA SCANDINAVICA, 2000)
Let X be a Banach space compatible with its antidual X^*, where X^* stands for the vector space X^* where the multiplication by a scalar is replaced by the multiplication λ x^* = λ x^*. Let H be a Hilbert space intermediate between X and X^* with a scalar product compatible with the duality (X,X^*), and such that X X^* is dense in H. Let F denote the closure of X X^* in X^* and suppose X X^* is dense in X. Let K denote the natural map which sends H into the dual of X F and for every Banach space A which contains X F densely let A’ be the realization of the dual space of A inside the dual of X F. We show that if K^-1a, K^-1b _H ≤ a _X’ b _F’ whenever a and b are both in X’ F’ then (X, X^*)_12 = H with equality of norms. In particular this equality holds true if X embeds in H or H embeds densely in X. As other particular cases we mention spaces X with a 1-unconditional basis and Köthe function spaces on Ω intermediate between L^1(Ω) and L^∞(Ω).