Remark

Let E be a real Banach space and let $u$ be an endomorphism of E. Suppose that 0 is an isolated point of the complex spectrum of $u$, that is to say, of the spectrum of the endomorphism $1\otimes u$ of the complexified complete normable space $E_{(\mathbf{C})}$ of E (I, p. 85, No. 13). The subset $\{0\}$ of the complex spectrum of $u$ is open and closed, and invariant under conjugation; let us denote by $e_0(u)\in \mathscr{L}(E)$ the spectral projector associated with it (No. 13 of I, p. 85). Proposition 14 remains valid mutatis mutandis in this setting.