The spectra of closed interpolated operators (Q5952395)

Let \((E_0,E_1)\) be a Banach couple, \(E_\lambda\) (\(0\leq \text{Re }\lambda\leq 1\)) the interpolated spaces by the complex method, \(T\) a closed linear operator in \(E_0+E_1\), \(T_\lambda\) its restriction to \(E_\lambda\), with domain \(\{x\in E_\lambda\); \(Tx\in E_\lambda\}\). Then \(T_\lambda\) is closed and the main result of the paper states that \(\lambda \mapsto \widetilde{\sigma}(T_\lambda)\) is an analytic multifunction from the strip \(\{\lambda\); \(0< \text{Re }\lambda<1\}\) to \({\mathbb C}\cup\{\infty\}\). Here \(\widetilde{\sigma}(T_\lambda)\) stands for the extended spectrum of \(T_\lambda\). Some applications to the spectral theory of semigroups are given.