Proposition 19

Let E be a complex Banach space and $u$ a closed partial operator on E. One has PSp$_{\varepsilon}(u)\subset$ PSp$_{\delta}(u)$ if $0< \varepsilon < \delta$ and

$\bigcap_{\varepsilon\in\mathbf{R}^*_+}$ PSp$_{\varepsilon}(u) =$ Sp($u$).

The first assertion results from the definition. For the second, the spectrum of $u$ is contained in PSp$_{\varepsilon}(u)$ for every $\varepsilon >0$ by definition, and if $\lambda \notin$ Sp($u$), then $\lambda \notin$ PSp$_{\varepsilon}(u)$ when $\varepsilon <\|R(u, \lambda )\|^{-1}$.