Corollary

Let E be a complex Banach space and $u$ a closed partial operator on E. Let $\varepsilon >0$.

a) For every $v\in \mathscr{L}(E)$, one has PSp$_{\varepsilon}(u)\subset$ PSp$_{\varepsilon+\|v\|}(u+v)$;

b) Let $\delta >0$ and let $D_{\delta}$ be the open disk with center 0 and radius $\delta$ in $\mathbf{C}$. One has PSp$_{\varepsilon}(u) + D_{\delta}\subset$ PSp$_{\varepsilon+\delta}(u)$.

Let $\lambda \in$ PSp$_{\varepsilon}(u)$. There exists an endomorphism $w\in \mathscr{L}(E)$ such that $\|w\|< \varepsilon$ and $\lambda \in$ Sp($u+w$) (Prop. 20). Since $u+w= (u+v)+(w-v)$ and $\|w-v\|< \varepsilon +\|v\|$, one has $\lambda \in$ PSp$_{\varepsilon+\|v\|}(u+v) ($loc. cit.).

Let $\mu\in D_{\delta}$; one has $\lambda +\mu\in$ Sp($u+ (w+\mu1_E)$) and $\|w+\mu1_E\|< \varepsilon +\delta$, hence $\lambda +\mu\in$ PSp$_{\varepsilon+\delta}(u) ($loc. cit.).