Stability in interpolation of families of Banach spaces

Abstract:

Let $D$ be a simply connected domain in the complex plane whose boundary $\Gamma$ is a rectifiable simple closed curve. Let $\left \{ {A(\gamma )/\gamma \in \Gamma } \right \}$ and $\left \{ {B(\gamma )/\gamma \in \Gamma } \right \}$ be interpolation families of Banach spaces. Let $T$ be a linear operator mapping $A(\gamma )$ continuously into $B(\gamma )$. For $z \in D$ let ${T_z}$ be the restriction of $T$ to the interpolation space ${A_z}$. Then $\{ z \in D/\operatorname {cod}(T_z) = d < \infty$ and $\dim \operatorname {Ker}({T_z}) = 0 \}$ and $\{ z \in D/\dim \operatorname {Ker}(T_z) = d < \infty$ and $T_z$ is onto $B_z\}$ are open sets.