Abstract (EN)

We correct the article [S. Mischler, J. Scher, Spectral analysis of semigroups and growth-fragmentation equations. Ann. Inst. H. Poincar Anal. Non Linaire 33 (2016), no. 3, 849898]. In the article [2], there was an error in Step 1 in the proof of Theorem 2.1 that we were not able to fix, with fatal repercussions on most of the abstract spectral analysis results (Corollary 2.4, Theorem 3.1, Theorem 3.3 and Theorem 3.5 in [2]). However, we claim that Corollary 2.5 is correct and that we may slightly modify the statements of the other above mentioned abstract results in order to make them correct and then to use these variants in order to repair the proof of [2, Theorem 1.1]. We employ the notation and assumptions of [2], except that we replace assumption (H2) by the assumptions (h2) or (h2 ′) below. More precisely, we consider a Banach space X and the generator Λ of a strongly continuous semigroup S Λ on X. We assume that Λ splits as Λ = A + B where B is the generator of a strongly continuous semigroup S B and A is B-bounded. We also assume that the operators A and S B satisfy one of the two following regularizing properties of an iterated enough convolution product: (h2) there exist ζ ∈ (0, 1] and ζ ′ ∈ [0, ζ) such that A ∈ B(X ζ ′ , X) and there exists an integer n ≥ 1 such that for any a > a * , there holds ∀ t ≥ 0, (AS B) (* n) (t) B(X,X ζ) ≤ C a,n,ζ e at for a constant C a,n,ζ ∈ (0, ∞), or (h2 ′) there exist ζ ∈ [−1, 0) and ζ ′ ∈ (ζ, 0] such that A ∈ B(X, X ζ ′) and there exists an integer n ≥ 1 such that for any a > a * , there holds ∀ t ≥ 0, (S B A) (* n) (t) B(X ζ ,X) ≤ C a,n,ζ e at for a constant C a,n,ζ ∈ (0, ∞).