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

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

Daniel Daners School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
daniel.daners@sydney.edu.au
Jochen Glück Partially supported by a scholarship within the scope of the LGFG Baden-Württemberg, Germany. Institut für Angewandte Analysis, Universität Ulm, D-89069 Ulm, Germany
jochen.glueck@uni-ulm.de
July 15, 2019
Abstract

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 Kreĭn–Rutman type theorem on the existence of positive eigenvectors and eigenfunctionals under certain eventual positivity conditions.

footnotetext: Mathematics Subject Classification (2010): 47D06, 47B65, 34G10, 47A10footnotetext: Keywords: One-parameter semigroups of linear operators; eventually positive semigroup; domination condition; smoothing condition; Perron-Frobenius theory;

1 Introduction

The solution of a linear autonomous evolution equation is often described by means of a -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 [Ferrero2008] and [Gazzola2008] that the solution semigroup of the bi-harmonic heat equation on , while not being positive, behaves in some sense eventually positive. This observation complemented earlier results on the corresponding elliptic problem; see for instance [Grunau1997, Grunau1998, Grunau1999] and the references therein and also the recent paper [Sweers2016]. A similar phenomenon occurs for the semigroup generated by the Dirichlet-to-Neumann operator on a two-dimensional disk as shown in [Daners2014].

These observations suggest that a general theory of eventually positive -semigroups would be useful. While, in finite dimensions, such a theory has been developed during the last decade (see for instance [Noutsos2008, Olesky2009], [Ellison2009, Theorem 2.9] and [Erickson2015]), a systematic study of this phenomenon in infinite dimensions was initiated only recently in [Daners2016, Daners2016a]. Several spectral results for infinite dimensional operators with eventually positive powers were recently proved by the second author in [GlueckEPO], after eventually positive matrix powers had been intensively studied for at least two decades; see the introduction of [GlueckEPO] 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 [Daners2016a]. The conditions involve the principal ideal generated by some element of the positive cone of a real or complex Banach lattice . That principal ideal is defined by

It is a subspace of and when equipped with the gauge norm given by

(1.1)

a Banach lattice in its own right. We will often assume that is a quasi-interior point of the positive cone, that is, a point such that is dense in . We refer to [Meyer-Nieberg1991, Schaefer1974] for the general theory of Banach lattices.

First condition: Given a linear operator we refer to

(Dom)

as the domination condition. It plays an important role in the characterisation of eventually positive behaviour of the resolvent of in [Daners2016a, Theorem 4.4]. We call this a domination condition since for every it implies the existence of such that .

Second condition: If generates a -semigroup on , then we refer to

(Smo)

as the smoothing condition. This condition is an important assumption in [Daners2016a, Theorem 5.2] which characterises eventual positivity of 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 , and also because is isometrically Banach lattice isomorphic to the space of real- or complex-valued continuous functions on some compact Hausdorff space . The latter follows from the corollary to [Schaefer1974, Proposition II.7.2] and from Kakutani’s representation theorem for AM-spaces [Meyer-Nieberg1991, Theorem 2.1.3].

If is the space of real- or complex-valued continuous functions on a compact Hausdorff space , endowed with the supremum norm, then we always have . Hence, conditions (Dom) and (Smo) are automatically fullfilled on such spaces. On many other Banach lattices, however, both conditions are quite strong. In a typical application we can think of as an -space over a bounded domain with and of as a differential operator, defined on an appropriate Sobolev space. The vector could, for instance, be the constant function with value in which case coincides with . In this case the domination condition (Dom) means that all functions in the domain of are bounded; it is fulfilled if an appropriate Sobolev embedding theorem holds. The smoothing condition (Smo) means that the semigroup operator 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 [GlueckDISS, Remark 9.3.4] or the proof of [Daners2016a, 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 [Daners2016a]; it was merely demonstrated in [Daners2016a, Example 5.4] that these conditions cannot be dropped in [Daners2016a, 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 -spaces. When , we see in Corollary 2.5 that condition (Smo) forces the semigroup to be eventually compact.

In Section 3 we present a short intermezzo on the existence of positive eigenvectors complementing earlier results in [Daners2016, Theorem 7.7.(i)]. In Sections 4 and 5 we show that some of the implications in the characterisation results in [Daners2016a, Theorems 4.4 and 5.2] remain true without the conditions (Dom) and (Smo).

Eventual positivity: terminology

Several notions of eventual positivity were discussed in [Daners2016] and [Daners2016a], some of which we recall for the convenience of the reader. For a concise formulation we introduce some notation. Let be a real or complex Banach lattice. As usual we call positive if , and we write if but . If , then we write if there exists such that ; in this case we call strongly positive with respect to . By we denote the space of bounded linear operators on . An operator is called positive, which we denote by , if . We call strongly positive with respect to a vector if for every .

Now, let be a complex Banach lattice with real part and let be a linear operator. The operator is called real if and if maps to . The first notion of eventual positivity which we recall relates to the resolvent of . We recall that the resolvent is an analytic map on the resolvent set . We denote the spectrum of by .

Definition.

Let be a linear operator on a complex Banach lattice and let be a quasi-interior point of . Let be an isolated spectral value of .

  1. The resolvent is called individually eventually strongly positive with respect to at if, for every , there exists a with the following properties: and for all .

  2. The resolvent is called individually eventually strongly negative with respect to at if, for every , there exists a with the following properties: and for all

We speak of individual eventual positivity as can depend on . One can, of course, also define uniform eventual positivity; see [Daners2016a, Definitions 4.1 and 4.2] for details. Note that if is eventually positive or negative at some , then is real, that is, leaves the real part of invariant.

The above definitions make sense even if is not necessarily an isolated point of , see [Daners2016a, Definitions 4.1 and 4.2], but the above definition is sufficient for our purposes. In fact we will usually assume that is a pole of the resolvent as an analytic map on . Such a pole is always an eigenvalue of as seen in [Yosida1995, Theorem 2 in Section VIII.8], and the pole is of order one if and only if the geometric and algebraic multplicities of as an eigenvalue of coincide.

We next deal with -semigroup on generated by an operator and denoted by .

Definition.

Let be a -semigroup on a complex Banach lattice and let be a quasi-interior point of . The semigroup is called individually eventually strongly positive with respect to if, for every , there exists a time such that for all .

We talk about uniform eventual positivity if can be chosen independently of , see [Daners2016a, Definition 5.1] for details. It is not difficult to see that is a real operator if and only if the operator is real for every .

To a great extent the long-term behaviour of the semigroup is determined by properties relating to the spectral bound of . If , then of particular importance is the peripheral spectrum of given by and the existence of a dominant spectral value, that is, such that .

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 is denoted by ; it is also a Banach lattice and its positive cone is called the dual cone of . A vector is positive if and only if for all . Since is a Banach lattice, all the notation introduced above implies to the elements of this space, too; in particular, we write if a functional fulfils but . We call the functional strictly positive if for all . Note that every quasi-interior point of is a strictly positive functional, but the converse is not in genera true. If is a densely defined linear operator, then its dual operator is denoted by .

Acknowledgement

Some results presented here appear in the PhD Thesis of the second author [GlueckDISS], 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.

2 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 be a complex Banach lattice, let . The fact that the gauge norm on is stronger than the induced norm from has severe consequences on every operator which maps to as we shall see in the main theorems of this section.

To state the theorems we need to recall that a complex Banach lattice is said to have order continuous norm if its real part has order continuous norm. We refer to [Meyer-Nieberg1991, Definition 2.4.1] for a precise definition. We recall that every -space with has order continuous norm, as has the space of all real- or complex-valued sequences which converge to endowed with the supremum norm. The space of continuous functions on a compact Hausdorff space has never order continuous norm unless is finite. We start with a lemma.

Lemma 2.1.

Let be a real or complex Banach lattice with order continuous norm and let .

  1. If , then is weakly compact.

  2. If is weakly compact, then is compact.

Proof.

It suffices to prove the lemma in case that the scalar field is real. Since has order continuous norm every order interval in is weakly compact; see [Meyer-Nieberg1991, Theorem 2.4.2]. By definition of the gauge norm (1.1) every bounded set in is contained in an order interval in . Hence the natural injection given by is weakly compact. If we are precise, then is the composition .

(i) As is bounded and is weakly compact we conclude that is weakly compact.

(ii) Because is a Dunford-Pettis space and is weakly compact, is a Dunford–Pettis operator, that is, weakly in implies that in ; see Definition 3.7.6, Proposition 3.7.9 and Proposition 1.2.13 in [Meyer-Nieberg1991]. Let now be a bounded sequence in . Then by the weak compactness of and the Eberlein–Šmulian theorem [dunford1958, Theorem V.6.1], we can find a subsequence such that weakly in for some . Using that is a Dunford–Pettis operator we conclude that in . This proves that is a compact operator. ∎

Theorem 2.2.

Let be a real or complex Banach lattice with order continuous norm and let . If and for , then is compact.

Proof.

First note that due to the closed graph theorem, for , where is as usual endowed with the gauge norm. By Lemma 2.1(i) is weakly compact. As is continuous it is also weakly continuous and hence the composition is weakly compact. Now Lemma 2.1(ii) implies that is compact. ∎

As a special case we can consider one operator . If we assume that is reflexive, then we obtain an even stronger result. Examples for reflexive Banach lattices are the -spaces with on an arbitrary measure space.

Theorem 2.3.

Let be a real or complex Banach lattice and let . Suppose that and that . Then the following assertions are true.

  1. If has order continuous norm, then is compact.

  2. If is reflexive, then is compact.

Proof.

(i) This is an obvious consequence of Theorem 2.2 taking .

(ii) First note that due to the closed graph theorem, . If is reflexive, then by the Banach-Alaoglu theorem every bounded set in is contained in a weakly compact set. As is continuous and thus weakly continuous it follows that is weakly compact. Since it follows from [Meyer-Nieberg1991, 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 is compact. ∎

In [Daners2016a, Theorems 4.4 and 5.2] it was always assumed that certain spectral values of be poles of the resolvent. In the corollaries below we will show that the above results imply that such assumptions are automatically satisfied if 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 be a complex Banach lattice, and let be a linear operator with non-empty resolvent set. Suppose that for some . Then the following assertions are true.

  1. If has order continuous norm, then is compact for every .

  2. If is reflexive, then is compact for every .

In either case, all spectral values of are poles of the resolvent and have finite algebraic multiplicity.

Proof.

Let . Then . Now Theorem 2.3(i) and (ii) yield (i) and (ii) respectively. In either case [Taylor1958, Theorem 5.8-F] implies that all spectral values of are poles of and have finite algebraic multiplicity. ∎

Corollary 2.4 is useful to prove that an operator in a concrete application has an eventually positive resolvent. This can often be done by using [Daners2016a, Theorem 4.4]. As it turns out, if has order continuous norm and the domination condition (Dom) is fulfilled, then, as a consequence of Corollary 2.4, some of the spectral theoretic assumptions in [Daners2016a, Theorem 4.4] are automatically satisfied.

Corollary 2.5.

Let be a complex Banach lattice with order continuous norm, and let be a -semigroup on . Suppose that for some .

Then the semigroup is eventually compact. In particular, all spectral values of are poles of the resolvent and have finite algebraic multiplicity. Moreover, the peripheral spectrum of is finite.

Proof.

The semigroup is eventually compact since Theorem 2.3 implies that the operator is compact. Hence, according to [Engel2000, Corollary V.3.2], all spectral values of are poles of and have finite algebraic multiplicity. It now follows from [Engel2000, Theorem II.4.18] that the peripheral spectrum of is finite. ∎

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 [Daners2016a, Section 5] with Corollary 2.5.

3 The existence of positive eigenvectors

This section is devoted to a Kreĭn–Rutman type theorem about the existence of positive eigenvectors. For eventually positive semigroup, a related result was given in [Daners2016, Theorem 7.7(i)]. Similar results for eventually and asymptotically positive operators can be found in [GlueckEPO, Section 6]. The latter results also contain existence results about positive eigenvectors of the dual operator. The following theorem and its corollaries are in the spirit of this latter result. The proof of Theorem 3.1 is inspired by the proofs of [Daners2016, Theorem 7.7(i)] and [GlueckEPO, Theorem 6.1].

Theorem 3.1.

Let be a linear operator on a complex Banach lattice and let be a pole of the resolvent . Suppose that we have, for every ,

(3.1)

as . Then the following assertions hold:

  1. The number is an eigenvalue of and the corresponding eigenspace contains a positive, non-zero vector.

  2. If is densely defined, then is an eigenvalue of the dual operator and the corresponding eigenspace contains a positive, non-zero vector.

We note in passing that in [Daners2016a] the condition (3.1) is referred to as being individually asymptotically positive at if is a first-order pole of .

Proof of Theorem 3.1.

(i) Let , denote the order of as a pole of and let

(3.2)

be the Laurent series expansion of about , where . Then and ; see [Yosida1995, Theorem 2 in Section VIII.8]. In particular, is an eigenvalue of and with respect to the operator norm as . Hence Assumption (3.1) implies that is a positive operator. Since is non-zero its range contains a positive non-zero vector and this vector is an eigenvector of corresponding to the eigenvalue .

(ii) Now assume that is densely defined so that it has a well-defined dual operator . Then for all , so it follows from (3.2) that the Laurent expansion of about is given by

In particular, as , the point is an -th order pole of . As is positive, so is and hence, contains a positive non-zero vector. As , 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 [Daners2016, Section 8]. Let be a linear operator on a complex Banach lattice and let . We call the resolvent of individually eventually positive at if for every there exists such that and for all . The following corollary is an immediate consequence of Theorem 3.1.

Corollary 3.2.

Let be a linear operator on a complex Banach lattice and let be a pole of the resolvent . Suppose that the resolvent of is individually eventually positive at .

Then the assertions (i) and (ii) of Theorem 3.1 are fulfilled.

To formulate the second corollary, we recall the definition of an eventually positive semigroup from [Daners2016, Section 7]. Let be a -semigroup on a complex Banach lattice . We call this -semigroup individually eventually positive if, for every , there exists a time such that for all . We recall from [Daners2016, Theorem 7.6] that the spectral bound of the generator of an individually eventually positive -semigroup is always contained in the spectrum unless .

For individually eventually positive -semigroups we obtain the following corollary of Theorem 3.1 which is a generalisation of [Daners2016, Theorem 7.7(a)] in that it also yields the existence of a positive eigenvector for the dual operator.

Corollary 3.3.

Let be an individually eventually positive -semigroup on a complex Banach lattice . Suppose that is a pole of the resolvent .

Then the assertions (i) and (ii) of Theorem 3.1 are fulfilled for .

Proof.

Since the semigroup is individually eventually positive and since it follows from [Daners2016, Corollary 7.3] that the resolvent of fulfils property (3.1) in Theorem 3.1 for . Hence, the assertion follows from that theorem. ∎

4 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 be a real densely defined linear operator on a complex Banach lattice , a pole of and a quasi-interior point of . A typical conclusion of such a Perron-Frobenius type theorem is:

The eigenvalue of is geometrically simple and the corresponding eigenspace contains a vector . Moreover, the eigenspace of the dual operator contains a strictly positive functional. (4.1)

It was shown in [Daners2016a, Corollary 3.3] that a very concise way of stating this conclusion is to say that the spectral projection associated with fulfills . Assertion (4.1) also implies that is algebraically simple and the only eigenvalue with a positive eigenfunction.

It was further proved in [Daners2016a, Theorem 4.4] that, under appropriate spectral assumptions combined with the domination condition (Dom), is equivalent to a certain eventual positivity property of the resolvent . On the other hand, it was demonstrated in [Daners2016a, 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 [Daners2016a, Theorem 4.4], namely “(ii) or (iii) (i)”, remain true without (Dom).

Theorem 4.1.

Let be a densely defined and real linear operator on a complex Banach lattice and let be a quasi-interior point. Assume that is a pole of the resolvent and denote the corresponding spectral projection by . If is individually eventually strongly positive or negative with respect to at , then .

The proof of the implications “(ii) or (iii) (i)” in [Daners2016a, Theorem 4.4] cannot simply be adapted to work in our more general setting here. The major obstacle is that [Daners2016a, Lemma 4.8] relies on the domination condition (Dom). Here, we use a different approach which has been inspired by the proof of [arendt1986, Proposition B-III.3.5]. We also need a simple auxiliary result which was implicitly contained in the proof of [Daners2016, Lemma 7.4].

Lemma 4.2.

Let be a complex Banach lattice and let an individually eventually positive net of operators, in the sense that for all there exists such that for all . Then, for every in the real part of , there exists such that for all .

Proof.

Choose such that and for all . For all those we then obtain and . Hence, and and thus for all . ∎

Proof of Theorem 4.1.

We may assume throughout the proof that . Suppose that is individually eventually strongly positive with respect to at . We are going to show that (4.1) is fulfilled.

According to Corollary 3.2 we can find vectors and . We observe that every element fulfils . Indeed, by assumption, for each such we can find a number for which and . Therefore, .

Next we show that the functional is strictly positive. For every we can find a number such that ; in particular, is a quasi-interior point of . Hence,

Thus, is indeed strictly positive.

It remains to show that is one-dimensional. To this end, we first prove that is a sublattice of the real part of . Fix . According to Lemma 4.2 we can find a number such that . By testing the positive vector against the strictly positive functional we obtain

and thus, . This proves that , so is indeed a sublattice of .

We have seen above that every non-zero positive vector in fulfils and is thus a quasi-interior point of . Hence, according to [Schaefer1974, Corollary 2 to Theorem II.6.3], is also a quasi-interior point of the positive cone of the Banach lattice (when endowed with the norm inherited from ). We have thus shown that every positive non-zero element of the real Banach lattice is a quasi-interior point of its positive cone. This implies that is one-dimensional; see [Lotz1968, Lemma 5.1] or [GlueckGR, Remark 5.9]. Since is real, we have , so we conclude that is one-dimensional over the complex field. This proves (4.1).

Now assume instead that is individually eventually strongly negative with respect to at . Then the resolvent of , which is given by for all , is individually eventually strongly positive with respect to at . Hence, by what we have just seen, the spectral projection of associated with is strongly positive with respect to . This spectral projection coincides with , which proves the assertion by what we have shown above. ∎

5 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 [Daners2016a, Theorem 5.2] it was shown that, under some assumptions which include the smoothing condition (Smo), individual eventual strong positivity with respect to of a semigroup is equivalent to a certain spectral condition that includes the Perron–Frobenius properties discussed at the start of the previous section. As demonstrated in [Daners2016a, 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 [Daners2016a, Theorem 5.2] remains true without the condition (Smo).

Theorem 5.1.

Let be a real -semigroup on a complex Banach lattice and let be a quasi-interior point. Assume that is not equal to and a pole of the resolvent . If is individually eventually strongly positive with respect to , then the spectral projection corresponding to fulfils .

For a similar reason as in Section 4 we cannot simply modify the relevant part of the proof of [Daners2016a, Theorem 5.2]. Instead we adapt the argument in the proof of Theorem 4.1 for semigroups.

Proof of Theorem 5.1.

We may assume throughout that . According to Corollary 3.3 there exists a vector and a functional . To prove (4.1) we now proceed similarly as in the proof of Theorem 4.1 with .

First note that every vector fulfils . Indeed, for each such vector we can find a time for which we have . In particular we have . Next we prove that the functional is strictly positive. To this end, let . We can find a time such that , so is a quasi-interior point of . Hence, as is non-zero we obtain

which shows that is indeed strictly positive.

To conclude the proof, we still have to show that is one-dimensional. As in the proof of Theorem 4.1, let us first show that is a sublattice of . So, take and choose a time such that for all ; such a time exists according to Lemma 4.2. For we test the positive vector against the strictly positive functional , thus obtaining

and hence . For every this implies . Therefore, , so is indeed a sublattice of . Now the same arguments as in the proof of Theorem 4.1 show that is indeed one-dimensional. ∎

If the peripheral spectrum of is finite and consists of poles of the resolvent and if the smoothing condition (Smo) is fulfilled, then [Daners2016a, Theorem 5.2] asserts, among other things, that individual eventual strong positivity of with respect to implies that the semigroup 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 , that is, if for all .

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 [Daners2016a, Corollary 5.3] is true under weaker assumptions than stated there.

Corollary 5.2.

Let be a real and eventually norm-continuous -semigroup on a complex Banach lattice and let be a quasi-interior point. Assume that and that the peripheral spectrum of is finite and consists of poles of the resolvent.

If is individually eventually strongly positive with respect to , then the rescaled semigroup is bounded, the spectral bound is a dominant spectral value of and the corresponding spectral projection fulfils .

Proof.

We may assume that . First recall from [Daners2016, Theorem 7.6] that is a spectral value of . It follows from Theorem 5.1 that . Hence, is a first order pole of according to [Daners2016a, Corollary 3.3], and this in turn implies that conists of first order poles of the resolvent; see [Daners2016, Theorem 7.7(ii)].

Now, let and denote by the spectral projection of associated with . Since consists of first order poles of the resolvent, we have . Hence, the semigroup is bounded on the range of . On the other hand, since the semigroup is eventually norm continuous and since is isolated from the rest of spectrum by assumption, it follows from [Engel2000, Theorem II.4.18] that the spectral bound of fulfils . Using again that our semigroup is eventually norm continuous, we conclude from [Engel2000, Corollary IV.3.11] that on with respect to the operator norm as . Hence, is indeed bounded as claimed.

Finally, the boundedness and the individual eventual positivity of imply immediately that this semigroup is individually asymptotically positive, see [Daners2016a, Definition 8.1]. Hence, it follows from [Daners2016a, Theorem 8.3] that is a dominant spectral value of . ∎

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