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The role of domination and smoothing conditions in the theory of eventually positive semigroups

We perform an in-depth study of some domination and smoothing properties of linear operators and of their role within the theory of eventually positive operator semigroups. On the one hand we prove that, on many important function spaces, they imply compactness properties. On the other hand, we show that these conditions can be omitted in a number of Perron--Frobenius type spectral theorems. We furthermore prove a Kre\u{\i}n--Rutman type theorem on the existence of positive eigenvectors and eigenfunctionals under certain eventual positivity conditions.


Introduction
The solution of a linear autonomous evolution equation is often described by means of a C 0 -semigroup on a Banach space, usually some kind of functions space. While, in many models, one expects the solution semigroup to be positive, that is, solutions with positive initial conditions remain positive, there are also examples which exhibit a more subtle type of positive behaviour. For example, it was noted in [9] and [10] that the solution semigroup of the bi-harmonic heat equation on R d , while not being positive, behaves in some sense eventually positive. This observation complemented earlier results on the corresponding elliptic problem; see for instance [14,15,16] and the references therein and also the recent paper [22]. A similar phenomenon occurs for the semigroup generated by the Dirichletto-Neumann operator on a two-dimensional disk as shown in [2].
These observations suggest that a general theory of eventually positive C 0semigroups would be useful. While, in finite dimensions, such a theory has been developed during the last decade (see for instance [19,20], [6,Theorem 2.9] and [8]), a systematic study of this phenomenon in infinite dimensions was initiated only recently in [4,3]. Several spectral results for infinite dimensional operators with eventually positive powers were recently proved by the second author in [12], after eventually positive matrix powers had been intensively studied for at least two decades; see the introduction of [12] for references and additional details.
A domination and a smoothing condition In the present note we are mainly concerned with two conditions appearing in various characterisation theorems in [3]. The conditions involve the principal ideal E u generated by some element u of the positive cone E + of a real or complex Banach lattice E. That principal ideal is defined by It is a subspace of E and when equipped with the gauge norm · u given by a Banach lattice in its own right. We will often assume that u ∈ E + is a quasiinterior point of the positive cone, that is, a point such that E u is dense in E. We refer to [18,21] for the general theory of Banach lattices. First condition: Given a linear operator A : E ⊇ D(A) → E we refer to as the domination condition. It plays an important role in the characterisation of eventually positive behaviour of the resolvent of A in [3,Theorem 4.4]. We call this a domination condition since for every v ∈ D(A) it implies the existence of c > 0 such that |v| ≤ cu. Second condition: If A generates a C 0 -semigroup (e tA ) t≥0 on E, then we refer to ∃t 0 ≥ 0 : as the smoothing condition. This condition is an important assumption in [3, Theorem 5.2] which characterises eventual positivity of (e tA ) t≥0 by means of Perron-Frobenius like properties. We call (Smo) a smoothing condition since in general the gauge norm is stronger than the norm induced by on E, and also because (E u , · u ) is isometrically Banach lattice isomorphic to the space of real-or complex-valued continuous functions on some compact Hausdorff space K. The latter follows from the corollary to [ In a typical application we can think of E as an L p -space over a bounded domain Ω ⊆ R d with 1 < p < ∞ and of A as a differential operator, defined on an appropriate Sobolev space. The vector u could, for instance, be the constant function with value 1 in which case E u coincides with L ∞ (Ω). In this case the domination condition (Dom) means that all functions in the domain of A are bounded; it is fulfilled if an appropriate Sobolev embedding theorem holds. The smoothing condition (Smo) means that the semigroup operator e t 0 A maps every function to a bounded function, that is, it "smooths" unbounded initial data in some sense, see also the comment above.
It should be noted that, for analytic semigroups, condition (Dom) implies (Smo); see [13,Remark 9.3.4] or the proof of [3, Corollary 5.3] for details and for a slightly stronger assertion. Given the fact that the assumptions (Dom) and (Smo) are fulfilled in many applications, they were not studied in much detail in [3]; it was merely demonstrated in [3,Example 5.4] that these conditions cannot be dropped in [3,Theorems 4.4 and 5.2] without one of the implications in those theorems failing.
Aim of this note The paper is devoted to an in-depth study of the conditions (Dom) and (Smo). While, on spaces of continuous functions over a compact space, both conditions are always fulfilled, we will show in Section 2 that both conditions are rather strong on other function spaces such as the L p -spaces. When p ∈ [1, ∞), we see in Corollary 2.5 that condition (Smo) forces the semigroup (e tA ) t≥0 to be eventually compact.
In Section 3 we present a short intermezzo on the existence of positive eigenvectors complementing earlier results in [4,Theorem 7.7 Eventual positivity: terminology Several notions of eventual positivity were discussed in [4] and [3], some of which we recall for the convenience of the reader. For a concise formulation we introduce some notation. Let E be a real or complex Banach lattice. As usual we call f ∈ E positive if f ≥ 0, and we write f > 0 if f ≥ 0 but f = 0. If u, f ∈ E + , then we write f u 0 if there exists ε > 0 such that f ≥ εu; in this case we call f strongly positive with respect to u. By L(E) we denote the space of bounded linear operators on E. An operator T ∈ L(E) is called positive, which we denote by T ≥ 0, if T E + ⊆ E + . We call T strongly positive with respect to a vector u ∈ E + if T f u 0 for every 0 < f ∈ E + . Now, let E be a complex Banach lattice with real part E R and let A : The first notion of eventual positivity which we recall relates to the resolvent of A. We recall that the resolvent λ → R(λ, A) := (λI − A) −1 ∈ L(E) is an analytic map on the resolvent set ρ(A). We denote the spectrum of A by σ(A).
Definition. Let A : E ⊇ D(A) → E be a linear operator on a complex Banach lattice E and let u ∈ E be a quasi-interior point of E + . Let λ 0 ∈ R ∩ σ(A) be an isolated spectral value of A.
(a) The resolvent R( · , A) is called individually eventually strongly positive with respect to u at λ 0 if, for every 0 < f ∈ E, there exists a λ 1 > λ 0 with the following properties: (λ 0 , λ 1 ] ⊆ ρ(A) and R(λ, A)f u 0 for all λ ∈ (λ 0 , λ 1 ]. (b) The resolvent R( · , A) is called individually eventually strongly negative with respect to u at λ 0 if, for every 0 < f ∈ E, there exists a λ 1 < λ 0 with the following properties: We speak of individual eventual positivity as λ 1 can depend on f . One can, of course, also define uniform eventual positivity; see [3, Definitions 4.1 and 4.2] for details. Note that if R(·, A) is eventually positive or negative at some λ 0 ∈ σ(A), then A is real, that is, A leaves the real part E R of E invariant.
The above definitions make sense even if λ 0 is not necessarily an isolated point of σ(A), see [3, Definitions 4.1 and 4.2], but the above definition is sufficient for our purposes. In fact we will usually assume that λ 0 is a pole of the resolvent R( · , A) as an analytic map on ρ(A). Such a pole is always an eigenvalue of A as seen in [24, Theorem 2 in Section VIII.8], and the pole is of order one if and only if the geometric and algebraic multplicities of λ 0 as an eigenvalue of A coincide.
We next deal with C 0 -semigroup on E generated by an operator A and denoted by (e tA ) t≥0 .
Definition. Let (e tA ) t≥0 be a C 0 -semigroup on a complex Banach lattice E and let u ∈ E be a quasi-interior point of E + . The semigroup (e tA ) t≥0 is called individually eventually strongly positive with respect to u if, for every 0 < f ∈ E, there exists a time t 0 ≥ 0 such that e tA f u 0 for all t ≥ t 0 .
We talk about uniform eventual positivity if t 0 can be chosen independently of f ∈ E + , see [3, Definition 5.1] for details. It is not difficult to see that A is a real operator if and only if the operator e tA is real for every t ∈ [0, ∞).
To a great extent the long-term behaviour of the semigroup is determined by properties relating to the spectral bound s(A) := sup{Re λ : , then of particular importance is the peripheral spectrum of A given by σ per (A) := {λ ∈ σ(A) : Re λ = s(A)} and the existence of a dominant spectral value, that is, In Section 3 we will also encounter a slightly weaker notion of eventual positivity; see Corollaries 3.2 and 3.3 and the preceeding discussions.
We complete this section by clarifying some notation we will use throughout. The dual space of a real or complex Banach lattice E is denoted by E ; it is also a Banach lattice and its positive cone E + is called the dual cone of E + . A vector ϕ ∈ E is positive if and only if ϕ, f ≥ 0 for all 0 ≤ f ∈ E. Since E is a Banach lattice, all the notation introduced above implies to the elements of this space, too; in particular, we write ϕ > 0 if a functional ϕ ∈ E fulfils ϕ ≥ 0 but ϕ = 0. We call the functional ϕ ∈ E strictly positive if ϕ, f > 0 for all 0 < f ∈ E. Note that every quasi-interior point of E + is a strictly positive functional, but the converse is not in genera true. If A : E ⊇ D(A) → E is a densely defined linear operator, then its dual operator is denoted by A : Acknowledgement Some results presented here appear in the PhD Thesis of the second author [13], in particular Corollary 3.3, Theorem 4.1 and a weaker version of Theorem 2.2. We would like to thank James Kennedy for useful discussions on some of the material.

Domination, smoothing and compactness
In this section we show that, on certain types of Banach lattices, the conditions (Dom) and (Smo) have rather strong consequences. Let E be a complex Banach lattice, let u ∈ E + . The fact that the gauge norm on E u is stronger than the induced norm from E has severe consequences on every operator T ∈ L(E) which maps E to E u as we shall see in the main theorems of this section.
To state the theorems we need to recall that a complex Banach lattice E is said to have order continuous norm if its real part E R has order continuous norm. We refer to [18,Definition 2.4.1] for a precise definition. We recall that every L pspace with 1 ≤ p < ∞ has order continuous norm, as has the space c 0 of all realor complex-valued sequences which converge to 0 endowed with the supremum norm. The space of continuous functions on a compact Hausdorff space K has never order continuous norm unless K is finite. We start with a lemma.
Lemma 2.1. Let E be a real or complex Banach lattice with order continuous norm and let u ∈ E + .
Proof. It suffices to prove the lemma in case that the scalar field is real. Since E has order continuous norm every order interval in E is weakly compact; see [18,Theorem 2.4.2]. By definition of the gauge norm (1.1) every bounded set in E u is contained in an order interval in E. Hence the natural injection j : E u → E given by j(x) = x is weakly compact. If we are precise, then T ∈ L(E) is the composition j • T .
(i) As T : E → E u is bounded and j : E u → E is weakly compact we conclude that T : E → E is weakly compact.
(ii) Because E u is a Dunford-Pettis space and j ∈ L(E u , E) is weakly compact, j is a Dunford-Pettis operator, that is, x n 0 weakly in E u implies that x n → 0 in E; see Definition 3.7.6, Proposition 3.7.9 and Proposition 1.2.13 in [18]. Let now (x n ) be a bounded sequence in E. Then by the weak compactness of T and the Eberlein-Šmulian theorem [5, Theorem V.6.1], we can find a subsequence (x n k ) such that T x n k y weakly in E u for some y ∈ E u . Using that j ∈ L(E u , E) is a Dunford-Pettis operator we conclude that (j • T )x n k → T y in E. This proves that T : E → E is a compact operator.
Theorem 2.2. Let E be a real or complex Banach lattice with order continuous norm and let u ∈ E + . If T k ∈ L(E) and Proof. First note that due to the closed graph theorem, T k ∈ L(E, E u ) for k = 1, 2, where E u is as usual endowed with the gauge norm. By Lemma 2.1(i) T 1 ∈ L(E) is weakly compact. As T 2 : E → E u is continuous it is also weakly continuous and hence the composition As a special case we can consider one operator T 1 = T 2 = T . If we assume that E is reflexive, then we obtain an even stronger result. Examples for reflexive Banach lattices are the L p -spaces with 1 < p < ∞ on an arbitrary measure space. Theorem 2.3. Let E be a real or complex Banach lattice and let u ∈ E + . Suppose that T ∈ L(E) and that T E ⊆ E u . Then the following assertions are true.
(i) If E has order continuous norm, then T 2 ∈ L(E) is compact.
Proof. (i) This is an obvious consequence of Theorem 2.2 taking T 1 = T 2 = T .
(ii) First note that due to the closed graph theorem, T ∈ L(E, E u ). If E is reflexive, then by the Banach-Alaoglu theorem every bounded set in E is contained in a weakly compact set. As T ∈ L(E, E u ) is continuous and thus weakly continuous it follows that T ∈ L(E, E u ) is weakly compact. Since it follows from [18,Theorem 2.4.2(v)] that every reflexive Banach lattice has order continuous norm, we can now apply Lemma 2.1(ii) which shows that T ∈ L(E) is compact.
In [3, Theorems 4.4 and 5.2] it was always assumed that certain spectral values of A be poles of the resolvent. In the corollaries below we will show that the above results imply that such assumptions are automatically satisfied if E has order continuous norm and if one of the conditions (Dom) or (Smo) is fulfilled. It is worthwhile pointing out the the assumption of the first corollary is a bit more general than condition (Dom).
Corollary 2.4. Let E be a complex Banach lattice, u ∈ E + and let A : E ⊇ D(A) → E be a linear operator with non-empty resolvent set. Suppose that D(A n ) ⊆ E u for some n ∈ N. Then the following assertions are true.
(i) If E has order continuous norm, then R(λ, A) 2n is compact for every λ ∈ ρ(A).
(ii) If E is reflexive, then R(λ, A) n is compact for every λ ∈ ρ(A).
In either case, all spectral values of A are poles of the resolvent R( · , A) and have finite algebraic multiplicity. Corollary 2.5. Let E be a complex Banach lattice with order continuous norm, u ∈ E + and let (e tA ) t≥0 be a C 0 -semigroup on E. Suppose that e t 0 A E ⊆ E u for some t 0 ≥ 0.
Then the semigroup (e tA ) t≥0 is eventually compact. In particular, all spectral values of A are poles of the resolvent R( · , A) and have finite algebraic multiplicity. Moreover, the peripheral spectrum of A is finite.
Proof. The semigroup is eventually compact since Theorem 2.3 implies that the operator e 2t 0 A is compact. Hence, according to [ As similar comment as given after Corollary 2.4 also applies here. In order to show that a given semigroup is eventually positive, one can combine results from [3, Section 5] with Corollary 2.5.

The existence of positive eigenvectors
as λ ↓ λ 0 . Then the following assertions hold: (i) The number λ 0 is an eigenvalue of A and the corresponding eigenspace ker(λ 0 I − A) contains a positive, non-zero vector.
(ii) If A is densely defined, then λ 0 is an eigenvalue of the dual operator A and the corresponding eigenspace ker(λ 0 I − A ) contains a positive, non-zero vector.
We note in passing that in [3] the condition (3.1) is referred to as R( · , A) being individually asymptotically positive at λ 0 if λ 0 is a first-order pole of R( · , A). (ii) Now assume that A is densely defined so that it has a well-defined dual operator A . Then R(λ, A ) = R(λ, A) for all λ ∈ ρ(A) = ρ(A ), so it follows from (3.2) that the Laurent expansion of R( · , A ) about λ 0 is given by In particular, as Q −m = 0, the point λ 0 ∈ σ(A ) is an m-th order pole of R( · , A ). As Q −m is positive, so is Q −m and hence, im(Q −m ) contains a positive non-zero vector. As im(Q −m ) ⊆ ker(λ 0 I − A ), this proves the assertion as in (ii).
Let us formulate two corollaries where Theorem 3.1 is applied to eventually positive resolvents and to eventually positive semigroups.
First we recall the definition of an eventually positive resolvent from [4, Section 8]. Let A : E ⊇ D(A) → E be a linear operator on a complex Banach lattice E and let λ 0 ∈ σ(A)∩R. We call the resolvent R( · , A) of A individually eventually positive at λ 0 if for every f ∈ E + there exists λ 1 > λ 0 such that (λ 0 , λ 1 ] ⊆ ρ(A) and R(λ, A)f ≥ 0 for all λ ∈ (λ 0 , λ 1 ]. The following corollary is an immediate consequence of Theorem 3.1. To formulate the second corollary, we recall the definition of an eventually positive semigroup from [4,Section 7]. Let (e tA ) t≥0 be a C 0 -semigroup on a complex Banach lattice E. We call this C 0 -semigroup individually eventually positive if, for every f ∈ E + , there exists a time t 0 ≥ 0 such that e tA f ≥ 0 for all t ≥ t 0 . We recall from [4,Theorem 7.6] that the spectral bound s(A) of the generator of an individually eventually positive C 0 -semigroup (e tA ) t≥0 is always contained in the spectrum unless s(A) = −∞.
For individually eventually positive C 0 -semigroups we obtain the following corollary of Theorem 3.1 which is a generalisation of [4, Theorem 7.7(a)] in that it also yields the existence of a positive eigenvector for the dual operator.

A Perron-Frobenius theorem for resolvents
In this section we prove a Perron-Frobenius type theorem for eventually positive resolvents. In contrast to the results of Section 3 we prove not only the existence, but also the uniqueness of positive eigenvectors. Let us start by recalling that a certain Perron-Frobenius type property can be characterised by considering the spectral projection of the eigenvalue under consideration. More precisely, let A : E ⊇ D(A) → E be a real densely defined linear operator A : E ⊇ D(A) → E on a complex Banach lattice E, λ 0 ∈ σ(A) ∩ R a pole of R( · , A) and u a quasiinterior point of E + . A typical conclusion of such a Perron-Frobenius type theorem is: The eigenvalue λ 0 of A is geometrically simple and the corresponding eigenspace ker(λ 0 I − A) contains a vector v u 0. Moreover, the eigenspace ker(λ 0 I −A ) of the dual operator contains a strictly positive functional. (4.1) It was shown in [3, Corollary 3.3] that a very concise way of stating this conclusion is to say that the spectral projection P associated with λ 0 fulfills P u 0. Assertion (4.1) also implies that λ 0 is algebraically simple and the only eigenvalue with a positive eigenfunction.
It was further proved in [3,Theorem 4.4] that, under appropriate spectral assumptions combined with the domination condition (Dom), P u 0 is equivalent to a certain eventual positivity property of the resolvent R( · , A). On the other hand, it was demonstrated in [3,Example 5.4] that such an equivalence is no longer true if one drops the condition (Dom). However, we prove in the next theorem that some implications in [3,Theorem 4.4], namely "(ii) or (iii) ⇒ (i)", remain true without (Dom).
Theorem 4.1. Let A : E ⊇ D(A) → E be a densely defined and real linear operator on a complex Banach lattice E and let u ∈ E + be a quasi-interior point. Assume that λ 0 ∈ σ(A) ∩ R is a pole of the resolvent R( · , A) and denote the corresponding spectral projection by P . If R( · , A) is individually eventually strongly positive or negative with respect to u at λ 0 , then P u 0.
The proof of the implications "(ii) or (iii) ⇒ (i)" in [3,Theorem 4.4] cannot simply be adapted to work in our more general setting here. The major obstacle is that [3, Lemma 4.8] relies on the domination condition (Dom). Here, we use a different approach which has been inspired by the proof of [1, Proposition B-III. 3.5]. We also need a simple auxiliary result which was implicitly contained in the proof of [4,Lemma 7.4].
Lemma 4.2. Let E be a complex Banach lattice and let (T j ) j∈J ⊆ L(E) an individually eventually positive net of operators, in the sense that for all f ∈ E + there exists j 0 ∈ J such that T j f ≥ 0 for all j ≥ j 0 . Then, for every f in the real part E R of E, there exists j 1 ∈ J such that |T j f | ≤ T j |f | for all j ≥ j 1 .
Proof. Choose j 1 ∈ J such that T j f + ≥ 0 and T j f − ≥ 0 for all j ≥ j 1 . For all those j we then obtain Proof of Theorem 4.1. We may assume throughout the proof that λ 0 = 0. Suppose that R( · , A) is individually eventually strongly positive with respect to u at λ 0 = 0. We are going to show that (4.1) is fulfilled.
According to Corollary 3.2 we can find vectors 0 < v ∈ ker A and 0 < ϕ ∈ ker A . We observe that every element 0 < w ∈ ker A fulfils w u 0. Indeed, by assumption, for each such w we can find a number λ > 0 for which λ ∈ ρ(A) and w = λ R(λ, A)w u 0. Therefore, v u 0.
Next we show that the functional ϕ is strictly positive. For every 0 < f ∈ E we can find a number 0 < λ ∈ ρ(A) such that λ R(λ, A)f u 0; in particular, λ R(λ, A)f is a quasi-interior point of E + . Hence, Thus, ϕ is indeed strictly positive.
It remains to show that ker A is one-dimensional. To this end, we first prove that E R ∩ ker A is a sublattice of the real part E R of E. Fix w ∈ E R ∩ ker A. According to Lemma 4.2 we can find a number 0 < λ ∈ ρ(A) such that |w| = λ| R(λ, A)w| ≤ λ R(λ, A)|w|. By testing the positive vector λ R(λ, A)|w| − |w| against the strictly positive functional ϕ ∈ ker A we obtain ϕ, λ R(λ, A)|w| − |w| = λ R(λ, A )ϕ, |w| − ϕ, |w| = 0 and thus, λ R(λ, A)|w| = |w|. This proves that |w| ∈ E R ∩ ker A, so E R ∩ ker A is indeed a sublattice of E R .
We have seen above that every non-zero positive vector in w ∈ ker A fulfils w u 0 and is thus a quasi-interior point of E R . Hence, according to [21, Corollary 2 to Theorem II. 6.3], v is also a quasi-interior point of the positive cone of the Banach lattice E R ∩ ker A (when endowed with the norm inherited from E R ). We have thus shown that every positive non-zero element of the real Banach lattice E R ∩ker A is a quasi-interior point of its positive cone. This implies that E R ∩ker A is one-dimensional; see [17,Lemma 5.1] or [11,Remark 5.9]. Since A is real, we have ker A = E R ∩ker A+iE R ∩ker A, so we conclude that ker A is one-dimensional over the complex field. This proves (4.1). Now assume instead that R( · , A) is individually eventually strongly negative with respect to u at λ 0 = 0. Then the resolvent of −A, which is given by , is individually eventually strongly positive with respect to u at 0. Hence, by what we have just seen, the spectral projection of −A associated with 0 is strongly positive with respect to u. This spectral projection coincides with P , which proves the assertion by what we have shown above.

A Perron-Frobenius theorem for semigroups
In this final section we pursue a similar goal as in Section 4, but this time for eventually positive semigroups instead of resolvents. In [3,Theorem 5.2] it was shown that, under some assumptions which include the smoothing condition (Smo), individual eventual strong positivity with respect to u of a semigroup (e tA ) t≥0 is equivalent to a certain spectral condition that includes the Perron-Frobenius properties discussed at the start of the previous section. As demonstrated in [3,Example 5.4] this results fails in general if the smoothing condition (Smo) is dropped. However, we are now going to prove that at least a certain part of [3, Theorem 5.2] remains true without the condition (Smo).
Theorem 5.1. Let (e tA ) t≥0 be a real C 0 -semigroup on a complex Banach lattice E and let u ∈ E + be a quasi-interior point. Assume that s(A) is not equal to −∞ and a pole of the resolvent R( · , A). If (e tA ) t≥0 is individually eventually strongly positive with respect to u, then the spectral projection P corresponding to s(A) fulfils P u 0. Proof of Theorem 5.1. We may assume throughout that s(A) = 0. According to Corollary 3.3 there exists a vector 0 < v ∈ ker A and a functional 0 < ϕ ∈ ker A . To prove (4.1) we now proceed similarly as in the proof of Theorem 4.1 with λ 0 = s(A) = 0.
First note that every vector 0 < w ∈ ker A fulfils w u 0. Indeed, for each such vector we can find a time t ≥ 0 for which we have w = e tA w u 0. In particular we have v u 0. Next we prove that the functional ϕ is strictly positive. To this end, let 0 < x ∈ E. We can find a time t ≥ 0 such that e tA x u 0, so e tA x is a quasi-interior point of E + . Hence, as ϕ is non-zero we obtain ϕ, x = (e tA ) ϕ, x = ϕ, e tA x > 0, which shows that ϕ is indeed strictly positive.
To conclude the proof, we still have to show that ker A is one-dimensional. As in the proof of Theorem 4.1, let us first show that E R ∩ ker A is a sublattice of E R . So, take w ∈ E R ∩ ker A and choose a time t 1 ≥ 0 such that |w| = |e tA w| ≤ e tA |w| for all t ≥ t 1 ; such a time t 1 exists according to Lemma 4.2. For t ≥ t 1 we test the positive vector e tA |w| − |w| against the strictly positive functional ϕ ∈ ker A , thus obtaining ϕ, e tA |w| − |w| = (e tA ) ϕ, |w| − ϕ, |w| = 0 and hence e tA |w| = |w|. For every t ≥ 0 this implies e tA |w| = e tA e t 1 A |w| = e (t+t 1 )A |w| = |w|. Therefore, |w| ∈ ker A, so E R ∩ ker A is indeed a sublattice of E R . Now the same arguments as in the proof of Theorem 4.1 show that ker A is indeed one-dimensional.
If the peripheral spectrum of A is finite and consists of poles of the resolvent and if the smoothing condition (Smo) is fulfilled, then [3, Theorem 5.2] asserts, among other things, that individual eventual strong positivity of (e tA ) t≥0 with respect to u implies that the semigroup (e t(A−s(A)) ) t≥0 is bounded. It is an interesting question whether this result remains true without the condition (Smo). This does not even seem to be clear if the semigroup is strongly positive with respect to u, that is, if e tA u 0 for all t > 0. If, however, the semigroup under consideration is eventually norm continuous, then the situation is much simpler. In this case we obtain the following corollary which shows that the implication "(i) ⇒ (ii)" in [3, Corollary 5.3] is true under weaker assumptions than stated there.
Corollary 5.2. Let (e tA ) t≥0 be a real and eventually norm-continuous C 0 -semigroup on a complex Banach lattice E and let u ∈ E + be a quasi-interior point. Assume that s(A) > −∞ and that the peripheral spectrum of A is finite and consists of poles of the resolvent.
If (e tA ) t≥0 is individually eventually strongly positive with respect to u, then the rescaled semigroup (e t(A−s(A)) ) t≥0 is bounded, the spectral bound s(A) is a dominant spectral value of A and the corresponding spectral projection P fulfils P u 0.
Proof. We may assume that s(A) = 0. First recall from [4, Theorem 7.6] that s(A) is a spectral value of A. It follows from Theorem 5.1 that P u 0. Hence, s(A) is a first order pole of R( · , A) according to [3,Corollary 3.3], and this in turn implies that σ per (A) conists of first order poles of the resolvent; see [4,Theorem 7.7(ii)]. Now, let σ per (A) = {iβ 1 , ..., iβ n } and denote by Q ∈ L(E) the spectral projection of A associated with σ per (A). Since σ per (A) consists of first order poles of the resolvent, we have QE = ⊕ n k=1 ker(iβ k I − A). Hence, the semigroup (e tA ) t≥0 is bounded on the range of Q. On the other hand, since the semigroup is eventually norm continuous and since σ per (A) is isolated from the rest of spectrum by assumption, it follows from [7,Theorem II.4.18] that the spectral bound of A| ker Q fulfils s(A| ker Q ) < 0. Using again that our semigroup is eventually norm continuous, we conclude from [7,Corollary IV.3.11] that e tA → 0 on ker Q with respect to the operator norm as t → ∞. Hence, (e tA ) t≥0 is indeed bounded as claimed.
Finally, the boundedness and the individual eventual positivity of (e tA ) t≥0 imply immediately that this semigroup is individually asymptotically positive, see