A note on the spectrum of a bounded operator on a complex interpolation space

Abstract

Let $(X_0,X_1)$ be a compatible couple of Banach spaces and $T_j$ a bounded operator from $X_j$ into itself $(j = 0, 1)$ satisfying $T_0x = T_1x$ for all $x \in X_0 \cap X_1$. On an additional assumption concerning a boundedness of $T_0|_{X_0\cap X_1} (=T_1|_{X_0\cap X_1})$, the next relations of spectra are proved: $$\sigma(T_\theta) \subset \sigma(T_0)\cup \sigma (T_1) = \sigma(T\Delta)\cup \sigma (T_\Sigma)\quad (\theta \in (0,1)),$$ where $T_\theta $, $T_\Delta$ and $T_\Sigma$ are the bounded operators induced by $T_0$ and $T_1$ on the complex interpolation space $(X_0,X_1)_{[\theta ]}$, the intersection $X_0 \cap X_1$ and the sum $X_0 + X_1$, respectively.