Complex Interpolation of Operators and Optimal Domains

Abstract

Let X0 and X1 be two order continuous Banach function spaces on a finite measure space, (E0, E1) a Banach space interpolation pair, and T : X0 + X1 → E0 + E1 an admissible operator between the pairs (X0, X1) and (E0, E1). If Tθ : [X0, X1][θ] → [E0, E1][θ] is the inter polated operator by the first complex method of Calder´on and m0, m1 and mθ are the vector measures coming from T| X0 and T| X1 and Tθ, respectively, then we study the relationship between the optimal domain L1(mθ) of Tθ and the complex interpolation space [L1(m0), L1(m1)][θ] of the optimal domains of T| X0 and T| X1 . Then, we apply the obtained result to study interpolation of p-th power factorable and bidual (p, q)- power-concave operators.