Complex Interpolation of Operators and Optimal Domains

Abstract

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.