Spectrum of interpolated operators

Let $(X_0,X_1)$ be a compatible pair of Banach spaces and let $T$ be an operator that acts boundedly on both $X_0$ and $X_1$. Let $T_{[\theta]} \quad(0\le\theta\le 1)$ be the corresponding operator on the complex interpolation space $(X_0,X_1)_{[\theta]}$.

The aim of this paper is to study the spectral properties of $T_{[\theta]}$. We show that in general the set-valued function $\theta\mapsto \sigma(T_{[\theta]})$ is discontinuous even in inner points $\theta\in(0,1)$ and show that each operator satisfies the local uniqueness-of-resolvent condition of Ransford. Further we study connections with the real interpolation method.