By D.M. Gallo, R.M. Porter

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A very important contribution in the investigation of this question was made by Lyubich and Ph` ong [82], and independently, by Arendt and Batty [11]. We follow the Lyubich–Ph` ong approach to this result and its proof. We begin with the following lemma (see [82]). 10. Let (Ut )t≥0 be a C0 -semigroup of isometries in a Banach space X with the generator S. 43) for all x ∈ D(S). Moreover, if σ(S) ∩ iR = iR then σ(S) ⊆ iR. Proof. Let x ∈ D(S). We consider the X-valued function u(t) = e−λt Ut x, t ≥ 0, then u(t) = exp (|Re (λ)| t) · x .

Let G be the generator of a bounded C0 -semigroup (Tt )t≥0 in a Banach space X. If the intersection of the spectrum of G with the imaginary axis is at most countable and the adjoint operator G∗ has no imaginary eigenvalues, then a solution uf of the ACP du = Gu dt (t ≥ 0), u(0) = f, satisﬁes lim uf (t) = 0 for all f ∈ D(G). t→∞ Proof. We assume without loss of generality that the semigroup (Tt )t≥0 consists of contractions. Then the function Tt x : R+ → R+ is non-increasing for each ﬁxed x, and hence the following limit exists: l(x) = lim Tt x t→∞ (x ∈ X).

This inequality shows that there exists an open neighborhood W of λ0 such that Zλ1 = X for every λ1 ∈ W . Hence Zλ = X for all λ in the left half-plane {Reλ < 0}, and λI − S is invertible for all λ, Reλ < 0. 2, λI − S is already invertible for all λ, Reλ > 0, then σ(S) ⊆ iR. 11 (Lyubich–Ph` ong). Let G be the generator of a bounded C0 -semigroup (Tt )t≥0 in a Banach space X. If the intersection of the spectrum of G with the imaginary axis is at most countable and the adjoint operator G∗ has no imaginary eigenvalues, then a solution uf of the ACP du = Gu dt (t ≥ 0), u(0) = f, satisﬁes lim uf (t) = 0 for all f ∈ D(G).