Lemma 5

Let $u'$ be an element of $\mathscr{L}(E)$. Put $v'= 1_F+p\circ u'\circ q$. If $u'$ is an inverse of $u$, then $v'$ is an inverse of $v$. If $u'$ is a quasi-inverse of $u$, then $v'$ is a quasi-inverse of $v$. If $u'$ commutes with $u$, then $v'$ commutes with $v$.

Using formulas (6), we calculate

whence

which proves the lemma.

Let us prove Prop. 10. Taking Lemma 4 into account, it is enough to prove assertion a) for $n= 1$. By formulas (6), the mappings $p$ and $q$ define by passing to the subspaces continuous linear mappings $p':$ Ker($u$)$\rightarrow$ Ker($v$) and $q':$ Ker($v$)$\rightarrow$ Ker($u$), and by passing to the quotients continuous linear mappings $p'':$ Coker($u$)$\rightarrow$ Coker($v$) and $q'':$ Coker($v$)$\rightarrow$ Coker($u$).

Since $1_E-u=q\circ p$ and $1_F-v=p\circ q$, the mappings $p'$ and $q'$ on the one hand, $p''$ and $q''$ on the other hand, are isomorphisms reciprocal to one another. This proves a).

For the image of a continuous linear mapping to be closed, it is necessary and sufficient that the cokernel of this mapping be a separated space. Thus b) results from a).

Let us denote by $i: Ker(u)\rightarrow E$ and $j:$ Ker($v$)$\rightarrow F$ the canonical injections. Suppose that Ker($u$) possesses a topological complement in E, and let $r: E\rightarrow$ Ker($u$) be a continuous linear retraction of $i$. Put $r'=p'\circ r\circ q$. We have $r'\in \mathscr{L}(F$; Ker($v$)) and

therefore $r'$ is a continuous linear retraction of $j$ and Ker($v$) possesses a topological complement in F. This proves the first assertion of c). The second is proved similarly, by remarking that if $s:$ Coker($u$)$\rightarrow E$ is a continuous linear section of the canonical surjection of E onto Coker($u$), then $p\circ s\circ q''$ is one of the canonical surjection of F onto Coker($v$).

Let us denote by $\overline{u}: E/$ Ker($u$)$\rightarrow$ Im($u$) and $\overline{v}: F/$ Ker($v$)$\rightarrow$ Im($v$) the bijective continuous linear mappings induced by $u$ and $v$. By formulas (6), the mapping $p$ defines, by passing to quotients, a continuous linear mapping $p_1: E/$ Ker($u$)$\rightarrow F/$ Ker($v$), and $p,q$ define, by passing to subspaces, continuous linear mappings $p_0:$ Im($u$)$\rightarrow$ Im($v$) and $q_0:$ Im($v$)$\rightarrow$ Im($u$).

Let $t:$ Im($v$)$\rightarrow F/$ Ker($v$) denote the composed mapping of the canonical injection Im($v$)$\rightarrow F$ and the canonical surjection $F\rightarrow F/$ Ker($v$); it is continuous. We have $\overline{v}\circ t= 1_{Im(v)}-p_0\circ q_0$ and $\overline{v}\circ p_1=p_0\circ \overline{u}$, whence

This proves that $t+p_1\circ \overline{u}^{-1}\circ q_0$ is the inverse bijection of $\overline{v}$. If $u$ is strict, the mapping $\overline{u}^{-1}$ is continuous, and the same is true of $\overline{v}^{-1}$, which proves that $v$ is strict. This proves d).

By lemma 5, if $u$ is an automorphism of E, then $v$ is one of F. Analogously, if $u$ is a Fredholm endomorphism of E, then $v$ is one of F, and one then has ind($u$) $=$ ind($v$) by a). This proves e) and f). Finally, if E and F are separated and if $u$ is a Riesz endomorphism of E, then $v$ is one of F by lemma 5 and condition (ii) of prop. 8; this is assertion g).