Exercise 7, § 4

Let E and F be Banach spaces and let $u$ be an element of $\mathscr{Q}\mathscr{M}(E; F)$ (resp. of $\mathscr{Q}\mathscr{E}(E; F)$). Let $h$ be an element of $\mathscr{L}(E; F)$. Suppose that there exist positive real numbers $a$ and $b$ such that $a+b((u))^{-1}<1$ so that

for all $x\in E$.

Prove that $u+h\in \mathscr{Q}\mathscr{M}(E; F)$ (resp. $u+h\in \mathscr{Q}\mathscr{E}(E; F)$) and

dim Ker($u+h$)$\leqslant$ dim Ker($u$),

dim Coker($u+h$)$\leqslant$ dim Coker($u$),

ind($u+h$) $=$ ind($u$).

(Reduce to the case where $a$ and $b$ are $>0$ and let $c >0$ be such that $a+b/c <1$; consider the norm $p$ on E defined by

$p(x) =a\|u(x)\|+b\|x\|$ for $x\in E$,

and $E_1$ the Banach space E endowed with the norm $p$; one has $\mathscr{Q}\mathscr{M}(E_1; F) =\mathscr{Q}\mathscr{M}(E; F)$, $\mathscr{Q}\mathscr{E}(E_1; F) =\mathscr{Q}\mathscr{E}(E; F)$ and $\mathscr{L}(E_1; F) =\mathscr{L}(E; F)$. Apply then lemma 2 of III, p. 62.)