Complex Interpolation of Operators and Optimal Domains

Let X ₀ and X ₁ be two order continuous Banach function spaces on a finite measure space, (E ₀, E ₁) a Banach space interpolation pair, and \({T: X_0 + X_1 \to E_0 + E_1}\) an admissible operator between the pairs (X ₀,X ₁) and (E ₀,E ₁). If \({T_{\theta} : X_0, X_1]_{\theta ]} \to E_0, E_1]_{\theta]}}\) is the interpolated operator by the first complex method of Calderón and m ₀, m ₁ and m _θ are the vector measures coming from \({{T\vert}_{X_0}}\) and \({{T\vert}_{X_1}}\) and T _θ, respectively, then we study the relationship between the optimal domain \({L^1(m_{\theta})}\) of T _θ and the complex interpolation space \({L^1(m_0),L^1(m_1)]_{\theta]}}\) of the optimal domains of \({{T\vert}_{X_0}}\) and \({{T\vert}_{X_1}}\) . Then, we apply the obtained result to study interpolation of p-th power factorable and bidual (p,q)-power-concave operators.