Lemma 1

Tag 02SO, permanent. Cite it as [Bourbaki, Tag 02SO].

Théories spectrales, chapter III, § 4. Perturbations dans les espaces de Banach

Let E be a Banach space. Let $p$ and $q$ be continuous projectors of E such that $\|q-p\|<1$ . Then $p$ induces an isomorphism of Im( $q$ ) onto Im( $p$ ), and $1_E-p$ induces an isomorphism of Ker( $q$ ) onto Ker( $p$ ).

Consider the continuous linear mappings $u:x\mapsto p(x)$ of Im( $q$ ) into Im( $p$ ) and $v:y\mapsto q(y)$ of Im( $p$ ) into Im( $q$ ). For every $x\in$ Im( $q$ ), we have $x=q(x)$ , whence

(q-p)^2(x) =q^2(x)-q(p(x))-p(q(x)) +p^2(x)

=x-q(p(x)) =x-v(u(x))

Hence $\|1_E-v\circ u\|\leqslant \|q-p\|^2<1$ . By Corollary 1 of I, p. 22 $,v\circ u$ is an automorphism of Im( $q$ ). One proves analogously that $u\circ v$ is an automorphism of Im( $p$ ). This implies that $u$ is an isomorphism of Im( $q$ ) onto Im( $p$ ), whence the first assertion of the lemma. The second follows by replacing $p$ and $q$ by $1_E-p$ and $1_E-q$ respectively.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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