Remark

Tag 02V6, permanent. Cite it as [Bourbaki, Tag 02V6].

Théories spectrales, chapter III, § 6. Propriétés spectrales des endomorphismes des espaces de Banach

The assumption $\varrho (u)>0$ cannot in general be omitted in the Krein-Rutman theorem. For example, let V be the endomorphism of the Banach space $\mathscr{C}([0,1])$ defined by

V(f)(x) =\int_0^xf(y)dy

for $f\in \mathscr{C}([0,1])$ . The mapping V is compact and its spectral radius is zero (Exercise 1 of I, p. 187). It preserves the total salient closed convex cone in $\mathscr{C}([0,1])$ formed by the positive functions, and has no eigenvalue (loc. cit.).

The following proposition describes a sufficient condition for an endomorphism preserving a cone to have a strictly positive spectral radius, and then sharpens the Krein-Rutman theorem.

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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