consequence and thanks to the representation formula (2.13) for any fixed $\lambda > \lambda_1$ which holds thanks to Lemma 2.2-(ii), we have

$$ \langle \mathcal{R}_{\mathcal{L}}(\lambda)g, \phi \rangle = \left\langle \int_{0}^{\infty} e^{-\lambda t} S(t) dt, \phi \right\rangle > 0. $$

Because $\phi \in Y_+ \setminus \{0\}$ is arbitrary, we have established that $\mathcal{R}_{\mathcal{L}}(\lambda)g \in X_{++}$ for any $g \in X_+ \setminus \{0\}$. In other words, when $\lambda > \lambda_1$ and $f \in X_+ \cap D(\mathcal{L})$ satisfy $g := (\lambda - \mathcal{L})f \ge 0$, we deduce that $f = \mathcal{R}_{\mathcal{L}}(\lambda)g \in X_{++}$, what is the strong maximum principle. This one is obviously equivalent to the strong positivity property $\mathcal{R}_{\mathcal{L}}(\lambda) : X_+ \setminus \{0\} \to X_{++}$. On the other way round, writing the above identity as

$$ \int_{0}^{\infty} e^{-\lambda t} \langle S(t)g, \phi \rangle dt = \langle \mathcal{R}_{\mathcal{L}}(\lambda)g, \phi \rangle, $$

we see that the strong maximum principle implies that the RHS term is positive for any $g \in X_+ \setminus \{0\}$, $\phi \in Y_+ \setminus \{0\}$. As a consequence, the LHS term is positive and there exists $\tau > 0$ such that $\langle S(\tau)g, \phi \rangle > 0$, which is nothing but the irreducibility condition (4.11). $\square$

We present two other elementary results about the strong maximum principle.

**Lemma 4.9.** Consider $\mathcal{L}$ satisfying (H1) and $\lambda \in \mathbb{R}$. Then the following assertions are equivalent
(1) $\lambda - \mathcal{L}$ satisfies the strong maximum principle for any $f \in D(\mathcal{L}) \cap X_+$;

(2) $\lambda - \mathcal{L}$ satisfies the strong maximum principle for any $f \in D(\mathcal{L}^k) \cap X_+$ for some $k \ge 1$;

(3) $\lambda - \mathcal{L}^*$ satisfies the strong maximum principle for any $\phi \in D(\mathcal{L}^*) \cap Y_+$;

(4) $\lambda - \mathcal{L}^*$ satisfies the strong maximum principle for any $\phi \in D((\mathcal{L}^*)^\ell) \cap Y_+$ for some $\ell \ge 1$.

**Proof of Lemma 4.9.** Assume that $\lambda - \mathcal{L}$ satisfies the strong maximum principle for some $\lambda \in \mathbb{R}$ and $k \ge 1$ and consider $\phi \in D(\mathcal{L}^*) \cap Y_+ \setminus \{0\}$ such that $(\lambda - \mathcal{L}^*)\phi \ge 0$. For any $\kappa > \max(\lambda, \lambda_1)$ and $g \in D(\mathcal{L}^{k-1}) \cap X_+ \setminus \{0\}$, thanks to (H1) and the strong maximum principle, there exists $f \in D(\mathcal{L}^k) \cap X_{++}$ such that $(\kappa - \mathcal{L})f = g$. As a consequence, we have

$$ \begin{aligned} \langle \phi, g \rangle &= \langle \phi, (\kappa - \mathcal{L})f \rangle \\ &= \langle (\kappa - \mathcal{L}^*)\phi, f \rangle \ge (\kappa - \lambda) \langle \phi, f \rangle > 0. \end{aligned} $$

Since $g \in D(\mathcal{L}^{k-1}) \cap X_+$ is arbitrary and $D(\mathcal{L}^{k-1}) \cap X_+$ is dense in $X_+$, we deduce that $\phi \gg 0$. We have proved that $\lambda - \mathcal{L}^*$ satisfies the strong maximum principle. The other implications can be proved similarly. $\square$

**Remark 4.10.** (1) In many applications, we start proving the strong maximum principle on smooth enough functions (belonging to the iterated domain) for which pointwise arguments may be used.
(2) We may replace the condition (1) by assuming that $\lambda - \mathcal{L}$ satisfies the strong maximum principle for $f \in C \cap X_+$ for a subspace $C \subset D(\mathcal{L})$ such that $(\lambda - \mathcal{L})^{-1} \in \mathcal{B}(C)$ and $C$ is dense in $X$.

The strong maximum principle can be seen as a consequence of the weak maximum principle together with the existence of a family of strictly positive barrier functions. We give now a typical result which can be applied (or modified in order to be applied) in many situations.

**Lemma 4.11.** We assume that

(i) the operator $\lambda - \mathcal{L}$ satisfies the weak maximum principle and Kato's inequalities;

(ii) there exists a subset $\mathcal{G} \subset X_{++} \cap \{g \in D(\mathcal{L}); (\mathcal{L} - \lambda)g \ge 0\}$ such that $\forall f \in D(\mathcal{L}) \cap X_+ \setminus \{0\}$, $\exists g \in \mathcal{G}$ such that $(g - f)_+ \in D(\mathcal{L})$.

Then $\lambda - \mathcal{L}$ satisfies the strong maximum principle.

**Proof of Lemma 4.11.** We consider $f \in D(\mathcal{L}) \cap X_+ \setminus \{0\}$ such that $(\lambda - \mathcal{L})f \ge 0$ and choose $g \in \mathcal{G}$ such that $h := (g - f)_+ \in D(\mathcal{L})$. We remark that from Kato's inequality (4.6), we have

$$ (\mathcal{L} - \lambda)h \ge \operatorname{sign}_+(g - f)(\mathcal{L} - \lambda)(g - f) \ge 0. $$

As a consequence of the weak maximum principle, we have $h \le 0$. That implies $h = 0$, so that $g - f \le 0$ and finally $f \gg 0$. $\square$

The above barrier functions technique is also useful for obtaining the condition (H2) (possibly in a constructive way).
(5) One can observe from the proof below that the conclusion (i) in Theorem 4.13 holds under the sole assumptions (X1)-(i) for X and Y, (C1) and (H4) for $\mathcal{L}$ and $\mathcal{L}^*$. The conclusion (4.20) holds when assuming furthermore that (4.10) holds in X and $\mathcal{L}$ satisfies (H1'). The similar conclusion (4.21) holds when assuming furthermore that (4.10) holds in Y and $\mathcal{L}^*$ satisfies (H1').

(6) We finally recall that under condition (H1), the strong maximum principle (H4) for $\mathcal{L}$ is equivalent to the strong maximum principle (H4) for $\mathcal{L}^*$ (see Lemma 4.9). When furthermore condition (H2) holds and $\lambda_1$ in (C1) is defined by (2.16), the weak maximum principle (4.16) for $\mathcal{L}$ is equivalent to the weak maximum principle (4.16) for $\mathcal{L}^*$ (see the proof of Lemma 2.3).

The proof of Theorem 4.13 is split into the following Lemma 4.15, Lemma 4.17, Lemma 4.18 and Lemma 4.20.

**Lemma 4.15.** Under assumptions (X1)-(i), (C1) and (H4) for both $\mathcal{L}$ and $\mathcal{L}^*$, the solution $(\lambda_1, f_1, \phi_1)$ to the first eig triplet problem satisfies

$$ (4.22) \qquad f_1 \gg 0 \quad \text{and} \quad \phi_1 \gg 0. $$

*Proof of Lemma 4.15.* We just apply the strong maximum principle to the two eigenfunctions $f_1 \in X \setminus \{0\}$ and $\phi_1 \in Y \setminus \{0\}$. $\square$

**Remark 4.16.** It is worth emphasizing that the same conclusion clearly holds when instead of (C1) we only assume that $f_1 \in X_+ \setminus \{0\}$ and $\phi_1 \in Y_+ \setminus \{0\}$ satisfy

$$ (4.23) \qquad \mathcal{L}f_1 = \lambda_1 f_1, \quad \mathcal{L}^*\phi_1 = \lambda_1^*\phi_1. $$

In that case, we deduce that $\lambda_1^* = \lambda_1$ by writing

$$ \lambda_1 \langle f_1, \phi_1 \rangle = \langle \mathcal{L}f_1, \phi_1 \rangle = \langle f_1, \mathcal{L}^*\phi_1 \rangle = \lambda_1^* \langle f_1, \phi_1 \rangle, $$

and observing that $\langle f_1, \phi_1 \rangle \neq 0$.

**Lemma 4.17.** Under assumptions (X1)-(i), (C1) and (H4) for $\mathcal{L}^*$ (resp. $\mathcal{L}$), $\lambda_1$ is the unique eigenvalue associated to a positive eigenvector for $\mathcal{L}$ (resp. for $\mathcal{L}^*$).

*Proof of Lemma 4.17.* Consider $\lambda \in \mathbb{C}$ and $f \in X_+ \setminus \{0\}$ such that $\mathcal{L}f = \lambda f$ and observe that from the proof of Lemma 4.15, we have $\phi_1 \gg 0$. We compute

$$ 0 = \langle (\lambda - \mathcal{L})f, \phi_1 \rangle = \langle f, (\lambda - \mathcal{L}^*)\phi_1 \rangle = (\lambda - \lambda_1)\langle f, \phi_1 \rangle, $$

and thus $\lambda = \lambda_1$ since $\langle f, \phi_1 \rangle > 0$. The same proof applies to the dual problem. $\square$

**Lemma 4.18.** We assume again (X1)-(i), (C1) and (H4) for both $\mathcal{L}$ and $\mathcal{L}^*$. Under the additional condition (H1') for $\mathcal{L}$ and (4.10) in X (resp. (H1') for $\mathcal{L}^*$ and (4.10) in Y), we have $N(\mathcal{L} - \lambda_1) = \text{Span}(f_1)$ (resp. $N(\mathcal{L}^* - \lambda_1) = \text{Span}(\phi_1)$). In particular, $f_1$ (resp. $\phi_1$) is unique (because of the positivity and the normalization conditions).

*Proof of Lemma 4.18.* Consider a eigenfunction $f \in X \setminus \{0\}$ associated to the eigenvalue $\lambda_1$. First, we observe from Kato's inequality that

$$ \lambda_1 |f| = \lambda_1 \text{sign}(f) f = \text{sign}(f) \mathcal{L}f \leq \mathcal{L}|f|. $$

That inequality is in fact an equality, otherwise we would have

$$ \lambda_1 \langle |f|, \phi_1 \rangle \neq \langle \mathcal{L}|f|, \phi_1 \rangle = \langle |f|, \mathcal{L}^* \phi_1 \rangle = \lambda_1 \langle |f|, \phi_1 \rangle, $$

and a contradiction. As a consequence, $|f|$ is a solution to the eigenvalue problem $\lambda_1|f| = \mathcal{L}|f|$, so that

$$ \lambda_1 f_{\pm} = \mathcal{L} f_{\pm}, $$

by writing $f_{\pm} = (|f| \pm f)/2$. The strong maximum principle assumption implies $f_{\pm} \gg 0$ or $f_{\pm} = 0$, and thus $f_{+} \gg 0$ or $f_{-} \gg 0$ since $f \neq 0$. Without loss of generality we may assume $f_{+} \gg 0$. From (4.10), we then deduce $f \gg 0$. We introduce the normalized eigenfunctions $\tilde{f} := rf$ and $\tilde{f}_1 = r_1 f_1$ with

$$ (4.24) \qquad \langle \tilde{f}, \phi_1 \rangle = \langle \tilde{f}_1, \phi_1 \rangle = 1. $$

Now, thanks to Kato's inequality again, we write

$$ \lambda_1 (\tilde{f} - \tilde{f}_1)_+ = \text{sign}_+(\tilde{f} - \tilde{f}_1) \mathcal{L} (\tilde{f} - \tilde{f}_1) \leq \mathcal{L} (\tilde{f} - \tilde{f}_1)_+, $$
**Lemma 4.12.** For an operator $\mathcal{L}$, we assume that

(i) the condition (H1) holds with a constant $\kappa_1 \in \mathbb{R}$;

(ii) the hypothesis (ii) in Lemma 4.11 holds with $\lambda = \kappa_1$;

(iii) there exists $h_0 \in X_+ \setminus \{0\}$ such that for any $g \in \mathcal{G}$ there exists $\varepsilon > 0$ such that $g \ge \varepsilon h_0$.

Then the property (H2) holds true.

*Proof of Lemma 4.12.* Thanks to assumption (i), we may define $f_0 \in D(\mathcal{L}) \cap X_+ \setminus \{0\}$ as the solution to the equation $(\kappa_1 - \mathcal{L})f_0 = h_0$. From the proof of Lemma 4.11 and condition (iii), there exists $g \in \mathcal{G}$ and next $\varepsilon > 0$ such that $f_0 \ge g \ge \varepsilon h_0$. Coming back to the equation, we have

$$ \mathcal{L}f_0 = \kappa_1 f_0 - h_0 \ge (\kappa_1 - \varepsilon^{-1})f_0, $$

so that condition (H2) holds true with $\kappa_0 := \kappa_1 - \varepsilon^{-1}$ thanks to Lemma 2.4-(ii). $\square$

<figure><img src="image_2.png" /></figure>

**4.3. The geometry of the first eigenvalue problem.** We come back on and state a result about the geometry of the first eigenvalue. For that purpose, we introduce a general framework for the couple of Banach lattices we will use in the sequel:

(X1) $X_{++} \neq \emptyset$, $Y_{++} \neq \emptyset$ and the signum operator is well defined in both X and Y.

We consider an operator $\mathcal{L}$ on $X$ which satisfies the conclusion (C1) about the existence of solutions $(\lambda_1, f_1)$ and $(\lambda_1^*, \phi_1)$ to the direct and dual first eigenvalue problems. We next assume

(H1') the weak maximum principle

$$ (4.15) \qquad \lambda > \lambda_1, f \in D(\mathcal{L}), (\lambda - \mathcal{L})f \ge 0 \quad \text{imply} \quad f \ge 0 $$

and its Kato's inequalities counterpart

$$ (4.16) \qquad (\operatorname{sign} f) \mathcal{L} f \le \mathcal{L} |f|, \quad (\operatorname{sign}_+ f) \mathcal{L} f \le \mathcal{L} f_+ $$

We recall that these two conditions are automatically satisfied for both $\mathcal{L}$ and $\mathcal{L}^*$ when $\mathcal{L}$ is the generator of a positive semigroup, see Remark 4.1.

Then we define the set of eigenvectors

$$ \mathcal{E} := \{f \in X \setminus \{0\}; \exists \lambda \in \mathbb{C}, \mathcal{L}f = \lambda f\}, \quad \mathcal{E}^* := \{\phi \in Y \setminus \{0\}; \exists \lambda \in \mathbb{C}, \mathcal{L}^*\phi = \lambda \phi\}, $$

and we additionally assume that

$$ (H4) \quad \left\{ \begin{array}{ll} \text{(i)} & \mathcal{L} \text{ and } \mathcal{L}^* \text{ satisfy the strong maximum principle (4.10)}; \\ \text{(ii)} & \text{the property (4.9) holds in both sets } \mathcal{E} \text{ and } \mathcal{E}^*. \end{array} \right. $$

Observe that (H4)-(ii) is verified when $\mathcal{E}$ and $\mathcal{E}^*$ are subsets of $L_{\text{loc}}^1$, due to Lemma 4.6, so in particular when $D(\mathcal{L}^k)$, $D((\mathcal{L}^*)^k) \subset L_{\text{loc}}^1$ for some $k \in \mathbb{N}$.

We may then state our main result in this section, where we recall that $N(A)$ denotes the null space associated to the operator $A$.

**Theorem 4.13.** We assume that $X$ and $Y$ are Banach lattices satisfying (X1). We consider an unbounded operator $\mathcal{L}$ on $X$ which satisfies the conclusion (C1) about the existence part of the first eigenvalue problem. We also assume that $\mathcal{L}$ and $\mathcal{L}^*$ both satisfy the weak maximum principle and Kato's inequalities (H1') as well as the strong positivity condition (H4).

Then the following hold

i) $f_1 \gg 0$, $\phi_1 \gg 0$ and $\lambda_1 = \lambda_1^*$ is the unique eigenvalue associated to a positive eigenvector. We next make the normalization choice

$$ (4.17) \qquad \|\phi_1\| = 1, \quad \langle \phi_1, f_1 \rangle = 1. $$

ii) $\lambda_1$ is algebraically simple:

$$ (4.18) \qquad N((\mathcal{L} - \lambda_1)^k) = \operatorname{Span}(f_1), \quad \forall k \ge 1, $$

$$ (4.19) \qquad N((\mathcal{L}^* - \lambda_1)^k) = \operatorname{Span}(\phi_1), \quad \forall k \ge 1. $$

In particular $f_1$ (resp. $\phi_1$) is the unique positive and normalized eigenvector of $\mathcal{L}$ (resp. of $\mathcal{L}^*$) associated to $\lambda_1$. Finally, the projection on the first eigenspace (associated to $\lambda_1$) is given by

$$ \Pi f := \langle f, \phi_1 \rangle f_1. $$
and for the same reason as above that last inequality is in fact an inequality. The strong maximum principle implies that either $(\tilde{f} - \tilde{f}_1)_+ = 0$, which also reads $\tilde{f} \le \tilde{f}_1$, or $(\tilde{f} - \tilde{f}_1)_+ \gg 0$, which implies that $\tilde{f} \gg \tilde{f}_1$ by using again (4.10). Because of the identity (4.24) and the fact that $\phi_1 \in X'_+ \setminus \{0\}$, the second case in the above alternative is not possible. Repeating the same argument with $(\tilde{f}_1 - \tilde{f})_+$ we get that $\tilde{f}_1 \le \tilde{f}$ and we conclude with $\tilde{f} = \tilde{f}_1$. The same proof applies to the dual problem. $\square$

**Remark 4.19.** Under the same hypotheses as in Lemma 4.18, we have $\psi \in \operatorname{span}(\phi_1)$ if $\psi \in Y_+$ satisfies $\mathcal{L}^*\psi \ge \lambda_1\psi$ and $g \in \operatorname{span}(f_1)$ if $g \in X_+$ satisfies $\mathcal{L}g \ge \lambda_1 g$. In the second case, we indeed cannot have $\mathcal{L}^*g - \lambda_1 g \in X'_+ \setminus \{0\}$, since this would implies

$$ \langle \mathcal{L}g - \lambda_1 g, \phi_1 \rangle > 0, $$

and this would be in contradiction with the fact that

$$ \langle \mathcal{L}g - \lambda_1 g, \phi_1 \rangle = \langle g, \mathcal{L}^* \phi_1 - \lambda_1 \phi_1 \rangle = 0. $$

We thus must have $\mathcal{L}g - \lambda_1 g = 0$ and we conclude thanks to Lemma 4.18. The same proof applies to the dual problem.

**Lemma 4.20.** Under the same assumptions as in Lemma 4.18, $\lambda_1$ is algebraically simple for $\mathcal{L}$ (resp. for $\mathcal{L}^*$).

**Proof of Lemma 4.20.** We use an induction argument. We have already proved that $N((\mathcal{L} - \lambda_1)^k) = \operatorname{Span}(f_1)$ for $k = 1$. Assume then the result proved for any $\ell$, $1 \le \ell \le k$, and consider $f \in N((\mathcal{L} - \lambda_1)^{k+1})$. That means that $(\mathcal{L} - \lambda_1)f \in N((\mathcal{L} - \lambda_1)^k)$, and thus $(\mathcal{L} - \lambda_1)f = rf_1$, with $r \in \mathbb{R}$, thanks to the induction hypothesis. If $r = 0$, then $f \in N(\mathcal{L} - \lambda_1) = \operatorname{Span}(f_1)$. Otherwise, $r \ne 0$, and then

$$ \lambda_1 \langle f, \phi_1 \rangle = \langle f, \mathcal{L}^* \phi_1 \rangle = \langle \mathcal{L}f, \phi_1 \rangle = \langle \lambda_1 f + rf_1, \phi_1 \rangle, $$

which in turn implies $r \langle f_1, \phi_1 \rangle = 0$ and a contradiction. That concludes the proof. $\square$

**4.4. Mean ergodicity.** We deduce from the above analysis a first classical and general but rough information about the long-time behaviour of the trajectories associated to a semigroup. More precisely, assuming the existence and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$ for the generator $\mathcal{L}$ of a semigroup $S$ and introducing the rescaled semigroup $\tilde{S}_t := e^{-\lambda_1 t} S(t)$, we wish to establish the following mean ergodic property

(E1) for any $f \in X$, there holds

$$ (4.25) \qquad \frac{1}{T} \int_0^T \tilde{S}_t f dt \rightarrow \langle f, \phi_1 \rangle f_1, \quad \text{as } T \rightarrow \infty, $$

in a sense to be specified.

We start with a general result, taken from [152, Thm. V.4.5], which states that, under the conclusions of Theorem 4.13, (E1) holds for the strong topology if the semigroup $(\tilde{S}_t)$ is bounded.

**Theorem 4.21.** Consider a positive semigroup $S$ on a Banach lattice $X$ and assume that its generator $\mathcal{L}$ satisfies the conclusions of Theorem 4.13 about the existence and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. Assume furthermore that $(\tilde{S}_t)_{t \ge 0}$ is bounded. Then, the above mean ergodic property (E1) holds for the strong topology.

**Proof of Theorem 4.21.** Following the proof of [152, Thm. V.4.5], we consider the subspace

$$ X_0 := \operatorname{Span} f_1 \oplus \operatorname{Span}\{f - \tilde{S}(t)f : f \in X, t \ge 0\} $$

of $X$ and we take $\phi \in Y$ which vanishes on $X_0$. Since $\phi$ vanishes on each element of the form $f - S(t)f$, this implies that $\tilde{S}^*(t)\phi = \phi$ for all $t \ge 0$. We deduce that $\mathcal{L}^*\phi = \lambda_1\phi$, and consequently $\phi \in \operatorname{Span} \phi_1$ due to the point ii) in Theorem 4.13. Since we also have $\langle \phi, f_1 \rangle = 0$, we deduce that $\phi = 0$ and therefore $\overline{X_0} = X$. We observe now

$$ \left( \int_{0}^{T} \tilde{S}(s) ds \right) \left( I - \tilde{S}(t) \right) = \left( I - \tilde{S}(T) \right) \int_{0}^{t} \tilde{S}(s) ds $$

for all $t, T \ge 0$, which is an immediate consequence of the semigroup property. The above relation and the boundedness assumption on $(\tilde{S}_T)_{T \ge 0}$ imply that the convergence (4.25) holds for $f =$
**Remark 4.14.** (1) It is worth emphasizing again that (4.15) and (4.16) for both $\mathcal{L}$ and $\mathcal{L}^*$ are true when $\mathcal{L}$ is the generator of a positive semigroup (see Lemma 2.1 and Remark 4.1) and that (H4)-(i) is true when $S_{\mathcal{L}}$ enjoys additional strong positivity (or irreducibility) condition as formulated in (4.11), (4.12), (4.13) or (4.14). As a consequence, the conclusion of Theorem 4.13 holds true when $\mathcal{L}$ is the generator of a positive semigroup which satisfies the hypotheses of the existence part of the Krein-Rutman Theorem 2.21 and one of the additional above strict positivity conditions.

(2) Theorem 4.13 has to be compared with the seminal Krein and Rutman Theorem 1.2 ([237]), to the many results gathered in [15, Part C-III] (see in particular [15, Prop. C.3.5], [15, Thm. C.3.8] and the original paper [184]) and to the more recent contributions [277, Thm. 5.3], [41, Thm. 14.15] and [230, Thm. 5.1]. Probably many of the conclusions of Theorem 4.13 are very similar (or even included) in the material of [15, Part C-III]. However, our assumptions slightly different since we do not make explicit reference to a positive semigroup but rather refer to the weak and strong maximum principles.

(3) Our proof is quite direct and elementary and uses similar arguments as those used during the proof of [277, Thm. 4.3] and [230, Thm. 5.1]. We learnt this kind of technique in the (less abstract and general) proof of the uniqueness part of [312, Lem. 2.1].

(4) From ii), we deduce that $\mathcal{L}$ decomposes according to $X = X_0 \oplus X_1$ with $X_1 := \text{Span } f_1$ and $X_0 := (\text{Span } \phi_1)^\perp = \{f \in X; \langle f, \phi_1 \rangle = 0\}$ in the sense of [228, § III.5.6]. More precisely, $X = X_0 \oplus X_1$ is a topological direct sum, $\mathcal{L} : X_0 \cap D(\mathcal{L}) \to X_0$ and $\mathcal{L} : X_1 \to X_1$.

(5) One can observe from the proof below that the conclusion (i) in Theorem 4.13 holds under the sole assumptions (X1) for X and Y, (C1) and (H4)-(i) for $\mathcal{L}$ and $\mathcal{L}^*$. The conclusion (4.18) holds when assuming furthermore that (4.9) holds in $\mathcal{E}$ and $\mathcal{L}$ satisfies (H1'). The similar conclusion (4.19) holds when assuming furthermore that (4.9) holds in $\mathcal{E}^*$ and $\mathcal{L}^*$ satisfies (H1').

(6) We finally recall that under condition (H1), the strong maximum principle (4.10) for $\mathcal{L}$ is equivalent to the strong maximum principle (4.10) for $\mathcal{L}^*$ (see Lemma 4.9). When furthermore condition (H2) holds and $\lambda_1$ in (C1) is defined by (2.16), the weak maximum principle (4.15) for $\mathcal{L}$ is equivalent to the weak maximum principle (4.15) for $\mathcal{L}^*$ (see the proof of Lemma 2.3).

The proof of Theorem 4.13 is split into the following Lemma 4.15, Lemma 4.16, Lemma 4.17 and Lemma 4.19.

**Lemma 4.15.** Under assumptions (X1), (C1) and (H4)-(i), we have

$$ (4.20) \qquad f_1 \gg 0, \quad \phi_1 \gg 0 \quad \text{and} \quad \lambda_1 = \lambda_1^*. $$

*Proof of Lemma 4.15.* Applying the strong maximum principle to the two eigenfunctions $f_1 \in X \setminus \{0\}$ and $\phi_1 \in Y \setminus \{0\}$, we get the strict positivity of $f_1$ and $\phi_1$. In particular $\langle f_1, \phi_1 \rangle \neq 0$. Next, we write

$$ \lambda_1 \langle f_1, \phi_1 \rangle = \langle \mathcal{L} f_1, \phi_1 \rangle = \langle f_1, \mathcal{L}^* \phi_1 \rangle = \lambda_1^* \langle f_1, \phi_1 \rangle, $$

which yields $\lambda_1 = \lambda_1^*$. $\square$

<figure><img src="image_3.png" /></figure>

**Lemma 4.16.** Under assumptions (X1), (C1) and (H4)-(i) for $\mathcal{L}^*$ (resp. $\mathcal{L}$), $\lambda_1$ is the unique eigenvalue associated to a positive eigenvector for $\mathcal{L}$ (resp. for $\mathcal{L}^*$).

*Proof of Lemma 4.16.* Consider $\lambda \in \mathbb{C}$ and $f \in X_+ \setminus \{0\}$ such that $\mathcal{L}f = \lambda f$ and observe that from Lemma 4.15, we have $\phi_1 \gg 0$. We compute

$$ 0 = \langle (\lambda - \mathcal{L})f, \phi_1 \rangle = \langle f, (\lambda - \mathcal{L}^*)\phi_1 \rangle = (\lambda - \lambda_1)\langle f, \phi_1 \rangle, $$

and thus $\lambda = \lambda_1$ since $\langle f, \phi_1 \rangle > 0$. The same proof applies to the dual problem. $\square$

<figure><img src="image_4.png" /></figure>

**Lemma 4.17.** We assume again (X1), (C1) and (H4)-(i). Under the additional condition (H1') for $\mathcal{L}$ and (4.9) in $\mathcal{E}$ (resp. (H1') for $\mathcal{L}^*$ and (4.9) in $\mathcal{E}^*$), we have $N(\mathcal{L} - \lambda_1) = \text{Span}(f_1)$ (resp. $N(\mathcal{L}^* - \lambda_1) = \text{Span}(\phi_1)$). In particular, $f_1$ (resp. $\phi_1$) is unique (because of the positivity and the normalization conditions).

*Proof of Lemma 4.17.* Consider a eigenfunction $f \in X \setminus \{0\}$ associated to the eigenvalue $\lambda_1$. First, we observe from Kato's inequality that

$$ \lambda_1 |f| = \lambda_1 \text{sign}(f) f = \text{sign}(f) \mathcal{L} f \le \mathcal{L} |f|. $$
$g - \tilde{S}(t)g$ with $g \in X$, $t \ge 0$, and thus for any $f \in X_0$. Finally, since $X_0$ is dense in $X$ and using again the fact that $(\tilde{S}_t)_{t \ge 0}$ is bounded, we can readily extend the validity of (4.25) to any $f \in X$. $\square$

We will now give weaker versions of Theorem 4.21 with proofs which are based on compactness arguments. The motivation for providing such alternative proofs that require stronger assumptions is that, unlike the proof of Theorem 4.21, the methods can be adapted to derive stronger ergodicity results, namely without averaging in time, see Section 5.5.

**Theorem 4.22.** Consider a positive semigroup $S$ on a Banach lattice $X$ and assume that its generator $\mathcal{L}$ satisfies the conclusions of Theorem 4.13 about the existence and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. With the above notations, we assume furthermore that

(1) $(\tilde{S}_t)$ is bounded;

(2) $B_X$ is weakly compact for a topology which makes $f \mapsto \langle f, \phi_1 \rangle$ continuous. Then, the above mean ergodic property (E1) holds for the topology introduced in (2).

*Proof of Theorem 4.22.* Fix $f \in X$ and define

$$u_T := \frac{1}{T} \int_0^T \tilde{S}_t f \, dt.$$

From (1), we have

$$\|u_T\| \le \frac{1}{T} \int_0^T \|\tilde{S}_t f\| \, dt \le M \|f\|, \quad \forall T > 0.$$

We also compute

$$\langle u_T, \phi_1 \rangle = \frac{1}{T} \int_0^T \langle \tilde{S}_t f, \phi_1 \rangle \, dt = \langle f, \phi_1 \rangle, \quad \forall T > 0.$$

Thanks to assumption (2), we deduce that there exists $f^* \in X$ and a sequence $(T_k)$ such that

$$u_{T_k} \to f^* \quad \text{and} \quad \langle f^*, \phi_1 \rangle = \langle f, \phi_1 \rangle.$$

Because $(\tilde{S}_t f)$ is bounded, we may use the usual ergodicity trick as in the second proof of Theorem 3.1 and for any $t > 0$, we have

$$\tilde{S}_t f^* - f^* = \lim_{k \to \infty} \frac{1}{T_k} \left\{ \int_{T_k}^{T_k+t} \tilde{S}_s f \, ds - \int_0^t \tilde{S}_s f \, ds \right\} = 0.$$

We have established $(\mathcal{L} - \lambda_1)f^* = 0$, so that $f^* \in \text{Span}(f_1)$ and more precisely $f^* = \langle f, \phi_1 \rangle f_1$. By uniqueness of the limit, it is the whole family $(u_T)$ which converges to $f^*$. $\square$

We present a variant of the previous result in which we see that in a very general framework (including all the applications we present in the second part of this work) the above hypotheses (1) and (2) are not needed (or more precisely are automatically satisfied).

**Theorem 4.23.** (1) Consider a Banach lattice $X \subset L_{\text{loc}}^1(E, \mathcal{E}, \mu)$ and $Y \subset L_{\text{loc}}^1(E, \mathcal{E}, \mu)$ (so that in particular $\phi_1 \in L_{\text{loc}}^1$ and $L_{\phi_1}^1$ is well-defined) and a positive semigroup $S$ on $X$ such that its generator $\mathcal{L}$ satisfies the conclusions of Theorem 4.13 about the existence, positivity and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. Then the mean ergodic convergence (E1) holds for the weak topology of $L_{\phi_1}^1$.

(2) Assuming additionally that $S$ is strongly continuous and that

$$\chi^k := (D(\mathcal{L}^k), \| \cdot \|_{\chi^k}) \subset L_{\text{loc}}^1 \text{ with strong compact embedding for some } k \ge 1,$$

where

$$\|f\|_{\chi^k} := \|f\|_{L_{\phi_1}^1} + \cdots + \|\mathcal{L}^k f\|_{L_{\phi_1}^1}, \quad \forall f \in D(\mathcal{L}^k),$$

then the mean ergodic convergence (E1) holds for the strong topology of $L_{\phi_1}^1$.

*Proof of Theorem 4.23. Step 1.* We first recall a very classical result about conservative semigroups. Denoting $\tilde{S}_t := e^{-\lambda_1 t} S(t)$, we observe that this rescaled semigroup satisfies

(i) $\tilde{S}_t \ge 0$;

(ii) $\tilde{S}_t f_1 = f_1$ for any $t \ge 0$;
That inequality is in fact an equality, otherwise we would have

$$ \lambda_1 \langle |f|, \phi_1 \rangle \neq \langle \mathcal{L} |f|, \phi_1 \rangle = \langle |f|, \mathcal{L}^* \phi_1 \rangle = \lambda_1 \langle |f|, \phi_1 \rangle, $$

and a contradiction. As a consequence, $|f|$ is a solution to the eigenvalue problem $\lambda_1|f| = \mathcal{L}|f|$, so that

$$ \lambda_1 f_{\pm} = \mathcal{L} f_{\pm}, $$

by writing $f_{\pm} = (|f| \pm f)/2$. The strong maximum principle assumption implies $f_{\pm} \gg 0$ or $f_{\pm} = 0$, and thus $0 \ll f_{+} \in \mathcal{E}$ or $0 \ll f_{-} \in \mathcal{E}$ since $f \neq 0$. Without loss of generality we may assume $f_{+} \gg 0$. From (4.9) in $\mathcal{E}$, we then deduce $f \gg 0$. We introduce the normalized eigenfunctions $\tilde{f} := rf$ and $\tilde{f}_1 = r_1 f_1$ with

$$ (4.21) \qquad \langle \tilde{f}, \phi_1 \rangle = \langle \tilde{f}_1, \phi_1 \rangle = 1. $$

Now, thanks to Kato's inequality again, we write

$$ \lambda_1 (\tilde{f} - \tilde{f}_1)_+ = \text{sign}_+(\tilde{f} - \tilde{f}_1) \mathcal{L} (\tilde{f} - \tilde{f}_1) \le \mathcal{L} (\tilde{f} - \tilde{f}_1)_+, $$

and for the same reason as above that last inequality is in fact an inequality. The strong maximum principle implies that either $(\tilde{f} - \tilde{f}_1)_+ = 0$, which also reads $\tilde{f} \le \tilde{f}_1$, or $(\tilde{f} - \tilde{f}_1)_+ \gg 0$, which implies that $\tilde{f} \gg \tilde{f}_1$ by using again (4.9) in $\mathcal{E}$. Because of the identity (4.21) and the fact that $\phi_1 \in X'_+ \setminus \{0\}$, the second case in the above alternative is not possible. Repeating the same argument with $(\tilde{f}_1 - \tilde{f})_+$ we get that $\tilde{f}_1 \le \tilde{f}$ and we conclude with $\tilde{f} = \tilde{f}_1$. The same proof applies to the dual problem. $\square$

**Remark 4.18.** Under the same hypotheses as in Lemma 4.17, we have $\psi \in \text{span}(\phi_1)$ if $\psi \in Y_+$ satisfies $\mathcal{L}^*\psi \ge \lambda_1\psi$ and $g \in \text{span}(f_1)$ if $g \in X_+$ satisfies $\mathcal{L}g \ge \lambda_1 g$. In the second case, we indeed cannot have $\mathcal{L}^*g - \lambda_1 g \in X'_+ \setminus \{0\}$, since this would implies

$$ \langle \mathcal{L}g - \lambda_1 g, \phi_1 \rangle > 0, $$

and this would be in contradiction with the fact that

$$ \langle \mathcal{L}g - \lambda_1 g, \phi_1 \rangle = \langle g, \mathcal{L}^* \phi_1 - \lambda_1 \phi_1 \rangle = 0. $$

We thus must have $\mathcal{L}g - \lambda_1 g = 0$ and we conclude thanks to Lemma 4.17. The same proof applies to the dual problem.

**Lemma 4.19.** Under the same assumptions as in Lemma 4.17, $\lambda_1$ is algebraically simple for $\mathcal{L}$ (resp. for $\mathcal{L}^*$).

*Proof of Lemma 4.19.* We use an induction argument. We have already proved that $N((\mathcal{L}-\lambda_1)^k) = \text{Span}(f_1)$ for $k=1$. Assume then the result proved for any $\ell$, $1 \le \ell \le k$, and consider $f \in N((\mathcal{L}-\lambda_1)^{k+1})$. That means that $(\mathcal{L}-\lambda_1)f \in N((\mathcal{L}-\lambda_1)^k)$, and thus $(\mathcal{L}-\lambda_1)f = rf_1$, with $r \in \mathbb{R}$, thanks to the induction hypothesis. If $r=0$, then $f \in N(\mathcal{L}-\lambda_1) = \text{Span}(f_1)$. Otherwise, $r \neq 0$, and then

$$ \lambda_1 \langle f, \phi_1 \rangle = \langle f, \mathcal{L}^* \phi_1 \rangle = \langle \mathcal{L}f, \phi_1 \rangle = \langle \lambda_1 f + rf_1, \phi_1 \rangle, $$

which in turn implies $r\langle f_1, \phi_1 \rangle = 0$ and a contradiction. That concludes the proof. $\square$

**4.4. Mean ergodicity.** We deduce from the above analysis a first classical and general but rough information about the long-time behaviour of the trajectories associated to a semigroup.

More precisely, assuming the existence and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$ for the generator $\mathcal{L}$ of a semigroup $S$ and introducing the rescaled semigroup $\tilde{S}_t := e^{-\lambda_1 t} S(t)$, we wish to establish the following mean ergodic property

(E1) for any $f \in X$, there holds

$$ (4.22) \qquad \frac{1}{T} \int_0^T \tilde{S}_t f dt \to \langle f, \phi_1 \rangle f_1, \quad \text{as } T \to \infty, $$

in a sense to be specified.

We start with a general result, taken from [151, Thm. V.4.5], which states that, under the conclusions of Theorem 4.13, (E1) holds for the strong topology if the semigroup $(\tilde{S}_t)$ is bounded.
(iii) $\langle \tilde{S}_t g, \phi_1 \rangle = \langle g, \phi_1 \rangle$ for any $g \in X$ and $t \ge 0$.

We denote $[f]_1 := \langle |f|, \phi_1 \rangle$ which is a norm on $X$ (we use here that $\phi_1 \gg 0$) and $\tilde{S}_t$ is obviously a contraction for this one. Indeed, for any $f \in X$, there holds

$$|\tilde{S}_t f| = |\tilde{S}_t f_+ - \tilde{S}_t f_-| \le \tilde{S}_t f_+ + \tilde{S}_t f_- = \tilde{S}_t |f|,$$

using (i) in the inequality, and next

$$(4.27) \qquad [\tilde{S}_t f]_1 = \langle |\tilde{S}_t f|, \phi_1 \rangle \le \langle \tilde{S}_t |f|, \phi_1 \rangle = [f]_1,$$

using (iii) in the last equality. Abusing notations, we also denote by $\mathcal{X}$ the completion of $X$ for the $L_{\phi_1}^1$ norm (so that we may identify $\mathcal{X}$ to a closed subspace of $L_{\phi_1}^1$). We may then extend $\tilde{S}_t$ to $\mathcal{X}$ by uniform continuity and this extension still satisfies the properties (i)-(ii)-(iii) on $\mathcal{X}$. Consider now $f \in X$ such that $H(f/f_1)f_1 \in X$ for some convex function $H: \mathbb{R} \to \mathbb{R}$, where we use here that $X \subset L_{\text{loc}}^1$, and thus in particular $f_1 > 0$ a.e. on $E$, in order to give a sense to the term $H(f/f_1)f_1$. From (ii), we have

$$\ell[(\tilde{S}_t f)/f_1]f_1 = \tilde{S}_t[\ell(f/f_1)f_1],$$

for any real affine function $\ell$. Next from (i) and (2.7), we have

$$H[(\tilde{S}_t f)/f_1]f_1 \le \tilde{S}_t[H(f/f_1)f_1],$$

because of $H = \sup_{\ell \le H} \ell$ and the supremum can be taken on a numerable set of affine functions. Thanks to (iii), we conclude that

$$(4.28) \qquad \langle H[(\tilde{S}_t f)/f_1]f_1, \phi_1 \rangle \le \langle H[f/f_1]f_1, \phi_1 \rangle, \quad \forall t \ge 0.$$

Step 2. We normalize $\langle f_1, \phi_1 \rangle = 1$. For $f \in \mathcal{X} \subset L_{\phi_1}^1$ so that $f\phi_1 = (f/f_1)f_1\phi_1 \in L^1$, the de la Vallée Poussin theorem tells us that there exists an even and convex function $H: \mathbb{R} \to \mathbb{R}_+$ such that $H(s)/s \to +\infty$ as $s \to \infty$ and $H(f/f_1)f_1\phi_1 \in L^1$. Using the notations of the proof of Theorem 4.22, the Jensen inequality and the above estimate (4.28), we deduce

$$\int_E H(u_T/f_1)f_1\phi_1 d\mu \le \frac{1}{T} \int_0^T \int_E H[(\tilde{S}_t f)/f_1] f_1\phi_1 d\mu dt \le \int_E H(f/f_1)f_1\phi_1 d\mu,$$

for any $T > 0$. Now, for any $A \in \mathcal{E}$ and $T, K > 0$, we have

$$\begin{aligned} \int_E u_T \mathbf{1}_A \phi_1 d\mu &= \int_E \frac{u_T}{f_1} \mathbf{1}_{\frac{|u_T|}{f_1} > K} \mathbf{1}_A f_1 \phi_1 d\mu + \int_E \frac{u_T}{f_1} \mathbf{1}_{\frac{|u_T|}{f_1} \le K} \mathbf{1}_A f_1 \phi_1 d\mu \\ &\le \frac{K}{H(K)} \int_E H(u_T/f_1) f_1 \phi_1 d\mu + K \int_E \mathbf{1}_A f_1 \phi_1 d\mu \\ &\le \frac{K}{H(K)} \int_E H(f/f_1) f_1 \phi_1 d\mu + K \int_E \mathbf{1}_A f_1 \phi_1 d\mu, \end{aligned}$$

from what we immediately deduce that $(u_T)$ belongs to a weak compact set of $L_{\phi_1}^1$. We conclude that (4.25) holds for the weak convergence in $L_{\phi_1}^1$ as in the proof of Theorem 4.22.

Step 3. We now additionally assume that (4.26) holds with strong compact embedding for some $k \ge 1$. Taking $f \in D(\mathcal{L}^k)$, we compute

$$\langle |\mathcal{L}^j(\tilde{S}_t f)|, \phi_1 \rangle = \langle |\tilde{S}_t(\mathcal{L}^j f)|, \phi_1 \rangle \le \langle |\mathcal{L}^j f|, \phi_1 \rangle,$$

for any $j \le k$ and any $t \ge 0$, and thus the same bound holds for $(u_T)$. From (4.26), we deduce that up to the extraction of a subsequence, $(u_T)$ converges a.e. on $E$. Together with the weak convergence in $L_{\phi_1}^1$ yet established, we classically deduce that the whole family $(u_T)$ converges for the strong topology in $L_{\phi_1}^1$. We conclude that the same holds for any $f \in X$ by taking advantage of the fact that $D(\mathcal{L}^k)$ is dense in $X$ for the strong topology of $X$, and thus for the strong topology of $\mathcal{X}$, and of the estimate of contraction (4.27). $\square$

**Remark 4.24.** (1) A similar conclusion holds as in Theorem 4.23 when we assume $X \subset M_{\text{loc}}^1$, $D(\mathcal{L}^k) \subset L_{\text{loc}}^1$ and $D(\mathcal{L}^{*k}) \subset L_{\text{loc}}^1$ for some $k \ge 1$ instead of $X, Y \subset L_{\text{loc}}^1$. For $f \in D(\mathcal{L}^k) \subset L_{\text{loc}}^1$, we may indeed repeat the proof of Theorem 4.23 and we obtain the same conclusion. We next define $\mathcal{X}$ as the closure of $D(\mathcal{L}^k)$ for the norm $[\cdot]_1$. We conclude that (4.25) holds weakly in $L_{\phi_1}^1$ for any $f \in \mathcal{X}$ by a density argument.
**Theorem 4.20.** Consider a positive semigroup $S$ on a Banach lattice $X$ and assume that its generator $\mathcal{L}$ satisfies the conclusions of Theorem 4.13 about the existence and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. Assume furthermore that $(\tilde{S}_t)_{t \ge 0}$ is bounded. Then, the above mean ergodic property (E1) holds for the strong topology.

*Proof of Theorem 4.20.* Following the proof of [151, Thm. V.4.5], we consider the subspace

$$X_0 := \operatorname{Span} f_1 \oplus \operatorname{Span}\{f - \tilde{S}(t)f : f \in X, t \ge 0\}$$

of $X$ and we take $\phi \in Y$ which vanishes on $X_0$. Since $\phi$ vanishes on each element of the form $f - \tilde{S}(t)f$, this implies that $\tilde{S}^*(t)\phi = \phi$ for all $t \ge 0$. We deduce that $\mathcal{L}^*\phi = \lambda_1\phi$, and consequently $\phi \in \operatorname{Span} \phi_1$ due to the point ii) in Theorem 4.13. Since we also have $\langle \phi, f_1 \rangle = 0$, we deduce that $\phi = 0$ and therefore $\overline{X_0} = X$. We observe now

$$\left( \int_{0}^{T} \tilde{S}(s)ds \right) (I - \tilde{S}(t)) = (I - \tilde{S}(T)) \int_{0}^{t} \tilde{S}(s)ds$$

for all $t, T \ge 0$, which is an immediate consequence of the semigroup property. The above relation and the boundedness assumption on $(\tilde{S}_T)_{T \ge 0}$ imply that the convergence (4.22) holds for $f = g - \tilde{S}(t)g$ with $g \in X, t \ge 0$, and thus for any $f \in X_0$. Finally, since $X_0$ is dense in $X$ and using again the fact that $(\tilde{S}_t)_{t \ge 0}$ is bounded, we can readily extend the validity of (4.22) to any $f \in X$. $\square$

We will now give weaker versions of Theorem 4.20 with proofs which are based on compactness arguments. The motivation for providing such alternative proofs that require stronger assumptions is that, unlike the proof of Theorem 4.20, the methods can be adapted to derive stronger ergodicity results, namely without averaging in time, see Section 5.5.

**Theorem 4.21.** Consider a positive semigroup $S$ on a Banach lattice $X$ and assume that its generator $\mathcal{L}$ satisfies the conclusions of Theorem 4.13 about the existence and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. With the above notations, we assume furthermore that

(1) $(\tilde{S}_t)$ is bounded;

(2) $B_X$ is weakly compact for a topology which makes $f \mapsto \langle f, \phi_1 \rangle$ continuous.

Then, the above mean ergodic property (E1) holds for the topology introduced in (2).

*Proof of Theorem 4.21.* Fix $f \in X$ and define

$$u_T := \frac{1}{T} \int_{0}^{T} \tilde{S}_t f \, dt.$$

From (1) and denoting by $M$ the uniform bound of $(\|\tilde{S}_t\|)$, we have

$$\|u_T\| \le \frac{1}{T} \int_0^T \|\tilde{S}_t f\| \, dt \le M \|f\|, \quad \forall T > 0.$$

We also compute

$$\langle u_T, \phi_1 \rangle = \frac{1}{T} \int_0^T \langle \tilde{S}_t f, \phi_1 \rangle \, dt = \langle f, \phi_1 \rangle, \quad \forall T > 0.$$

Thanks to assumption (2), we deduce that there exists $f^* \in X$ and a sequence $(T_k)$ such that

$$u_{T_k} \to f^* \quad \text{and} \quad \langle f^*, \phi_1 \rangle = \langle f, \phi_1 \rangle.$$

Because $(\tilde{S}_t f)$ is bounded, we may use the usual ergodicity trick as in the second proof of Theorem 3.1 and for any $t > 0$, we have

$$\tilde{S}_t f^* - f^* = \lim_{k \to \infty} \frac{1}{T_k} \left\{ \int_{T_k}^{T_k+t} \tilde{S}_s f \, ds - \int_{0}^{t} \tilde{S}_s f \, ds \right\} = 0.$$

We have established $(\mathcal{L} - \lambda_1)f^* = 0$, so that $f^* \in \operatorname{Span}(f_1)$ and more precisely $f^* = \langle f, \phi_1 \rangle f_1$. By uniqueness of the limit, it is the whole family $(u_T)$ which converges to $f^*$. $\square$

We present a variant of the previous result in which we see that in a very general framework (including all the applications we present in the second part of this work) the above hypotheses (1) and (2) are not needed (or more precisely are automatically satisfied).
(2) The proof of Theorem 4.23 is based on so-called General Relative Entropy (GRE) techniques as developed for instance in [258], [269] and [49]. These ones are known to be useful for some classes of PDEs and for stochastic semigroups in order to establish uniform in time estimates and longtime convergence results.

The main interest of the two previous results is that they do not ask any new information on the semigroup but they are just based on the eigentriplet stationary problem. The shortcoming is that they are formulated in terms of the norm $[\cdot]_1$ instead of the norm of $X$. We present a second variant of Theorem 4.22 which is well adapted to the splitting framework developed in Sections 2 and 3 and is precisely formulated in a weak or strong topology of a space $X_0 \supset X$.

**Theorem 4.25.** Consider a positive semigroup $S = S_\mathcal{L}$ such that $\mathcal{L}$ satisfies the conclusions of Theorem 4.13 about the existence and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. Assume furthermore that $S$ satisfies the splitting structure introduced in (HS2) in section 3.2 or (HS3) in Section 3.2, or more precisely, there exist two families of operators $(V(t))$ and $(W(t))$ such that

$$S = V + W * S,$$

a real number $\kappa \le \lambda_1$ and some decaying functions $\Theta_i : \mathbb{R}_+ \to \mathbb{R}_+$ with $\Theta_1(t) \to 0$ as $t \to \infty$, $\Theta_2 \in L^1(\mathbb{R}_+)$ such that the following estimates hold

$$(4.29) \qquad \|V(t)e^{-\kappa t}\|_{\mathcal{B}(X)} = \mathcal{O}(1), \quad \|V(t)e^{-\kappa t}\|_{\mathcal{B}(X, X_0)} = \mathcal{O}(\Theta_1),$$

$$(4.30) \qquad \|W(t)e^{-\kappa t}\|_{\mathcal{B}(X_0, X_1)} = \mathcal{O}(\Theta_2),$$

with $X_1 \subset X_0 \subset X_0$, where $X_0$ is the space $X$ endowed with the norm $[g]_1 := \langle |g|, \phi_1 \rangle$.

(1) Assume furthermore that $X_1 \subset X_0$ with compact embedding for the weak or the strong topology in $X_0$ and this topology makes $f \mapsto \langle f, \phi_1 \rangle$ continuous. Then the mean ergodic convergence (E1) holds true for the above strong or weak topology.

(2) Assume furthermore that $X \subset L_{\text{loc}}^1$, $S$ is strongly continuous, and that the space $X^k$ defined by (4.26) is strongly compact embedded in $L_{\text{loc}}^1$ for some $k \ge 1$. Then the mean ergodic convergence (E1) holds true for the strong topology of $X_0$.

**Proof of Theorem 4.25.** We define

$$\tilde{V}(t) := V(t)e^{-\lambda_1 t}, \quad \tilde{W}(t) := W(t)e^{-\lambda_1 t},$$

so that

$$\tilde{S} = \tilde{V} + \tilde{W} * \tilde{S},$$

and

$$(4.31) \qquad M := \sup_{t \ge 0} \|\tilde{V}(t)\|_{\mathcal{B}(X)} < \infty, \quad \|\tilde{V}\|_{\mathcal{B}(X, X_0)} \lesssim \tilde{\Theta}_1 \in C_0(\mathbb{R}_+), \\ \qquad \tilde{\Theta}_2(t) := \|\tilde{W}(t)\|_{\mathcal{B}(X_0, X_1)} \in L^1(\mathbb{R}_+).$$

**Step 1.** We furthermore assume (1) and that the weak topology of $X_0$ makes $f \mapsto \langle f, \phi_1 \rangle$ continuous. We denote by $\mathcal{T}$ the weak or the strong topology $X_0$ (depending of the assumption made on the embedding $X_1 \subset X_0$). For $f_0 \in X$, we split

$$f(t) := \tilde{S}_t f_0 = v(t) + k(t), \quad v(t) := \tilde{V}(t) f_0, \quad k(t) := (\tilde{W} * \tilde{S})(t) f_0,$$

and we observe that $\|v(t)\|_{X_0} \to 0$ as $t \to \infty$ from the second estimate in (4.31). On the other hand, we have

$$\sup_{t \ge 0} \|k(t)\|_{X_1} \le \|\tilde{W}\|_{L^1} \sup_{t \ge 0} \|\tilde{S}_t f_0\|_{X_0} \le \|\tilde{W}\|_{L^1} \|f_0\|_{X_0},$$

from (4.27). In particular, $k(t)$ belongs to a compact set of $\mathcal{T}$, so that $(f(t))_{t \ge 0}$ also belongs to a compact set for the same topology $\mathcal{T}$. The same argument used on the Cesàro function $(u_T)$ defined during the proof of Theorem 4.22 implies that there exist $f^* \in X$ and a sequence $(T_k)$ such that

$$u_{T_k} \to f^* \text{ in the sense of } \mathcal{T} \text{ and } \langle f^*, \phi_1 \rangle = \langle f, \phi_1 \rangle,$$

the last identity following from the assumption that $f \mapsto \langle f, \phi_1 \rangle$ continuous for $\mathcal{T}$. We may then conclude as in the proof of Theorem 4.22.
**Theorem 4.22.** (1) Consider a Banach lattice $X \subset L_{\text{loc}}^{1}(E, \mathcal{E}, \mu)$ and $Y \subset L_{\text{loc}}^{1}(E, \mathcal{E}, \mu)$ (so that in particular $\phi_1 \in L_{\text{loc}}^{1}$ and $L_{\phi_1}^{1}$ is well-defined) and a positive semigroup $S$ on $X$ such that its generator $\mathcal{L}$ satisfies the conclusions of Theorem 4.13 about the existence, positivity and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. Then the mean ergodic convergence (E1) holds for the weak topology of $L_{\phi_1}^{1}$.

(2) Assuming additionally that $S$ is strongly continuous and that

$$ (4.23) \qquad \mathcal{X}^k := (D(\mathcal{L}^k), \|\cdot\|_{\mathcal{X}^k}) \subset L_{\text{loc}}^1 \text{ with strong compact embedding for some } k \ge 1, $$

where

$$ \|f\|_{\mathcal{X}^k} := \|f\|_{L_{\phi_1}^1} + \dots + \|\mathcal{L}^k f\|_{L_{\phi_1}^1}, \quad \forall f \in D(\mathcal{L}^k), $$

then the mean ergodic convergence (E1) holds for the strong topology of $L_{\phi_1}^1$.

*Proof of Theorem 4.22. Step 1.* We first recall a very classical result about conservative semi-groups. Denoting $\tilde{S}_t := e^{-\lambda_1 t} S(t)$, we observe that this rescaled semigroup satisfies

(i) $\tilde{S}_t \ge 0$;

(ii) $\tilde{S}_t f_1 = f_1$ for any $t \ge 0$;

(iii) $\langle \tilde{S}_t g, \phi_1 \rangle = \langle g, \phi_1 \rangle$ for any $g \in X$ and $t \ge 0$.

We denote $[f]_1 := \langle |f|, \phi_1 \rangle$ which is a norm on $X$ (we use here that $\phi_1 \gg 0$) and $\tilde{S}_t$ is obviously a contraction for this one. Indeed, for any $f \in X$, there holds

$$ |\tilde{S}_t f| = |\tilde{S}_t f_+ - \tilde{S}_t f_-| \le \tilde{S}_t f_+ + \tilde{S}_t f_- = \tilde{S}_t |f|, $$

using (i) in the inequality, and next

$$ (4.24) \qquad [\tilde{S}_t f]_1 = \langle |\tilde{S}_t f|, \phi_1 \rangle \le \langle \tilde{S}_t |f|, \phi_1 \rangle = [f]_1, $$

using (iii) in the last equality. Abusing notations, we also denote by $\mathcal{X}$ the completion of $X$ for the $L_{\phi_1}^1$ norm (so that we may identify $\mathcal{X}$ to a closed subspace of $L_{\phi_1}^1$). We may then extend $\tilde{S}_t$ to $\mathcal{X}$ by uniform continuity and this extension still satisfies the properties (i)-(ii)-(iii) on $\mathcal{X}$. Consider now $f \in X$ such that $H(f/f_1)f_1 \in X$ for some convex function $H: \mathbb{R} \to \mathbb{R}$, where we use here that $X \subset L_{\text{loc}}^1$, and thus in particular $f_1 > 0$ a.e. on $E$, in order to give a sense to the term $H(f/f_1)f_1$. From (ii), we have

$$ \ell([\tilde{S}_t f]/f_1) f_1 = \tilde{S}_t [\ell(f/f_1)f_1], $$

for any real affine function $\ell$. Next from (i) and (2.7), we have

$$ H([\tilde{S}_t f]/f_1) f_1 \le \tilde{S}_t [H(f/f_1)f_1], $$

because of $H = \sup_{\ell \le H} \ell$ and the supremum can be taken on a numerable set of affine functions. Thanks to (iii), we conclude that

$$ (4.25) \qquad \langle H([\tilde{S}_t f]/f_1) f_1, \phi_1 \rangle \le \langle H[f/f_1] f_1, \phi_1 \rangle, \quad \forall t \ge 0. $$

*Step 2.* We normalize $\langle f_1, \phi_1 \rangle = 1$. For $f \in \mathcal{X} \subset L_{\phi_1}^1$ so that $f\phi_1 = (f/f_1)f_1\phi_1 \in L^1$, the de la Vallée Poussin theorem tells us that there exists an even and convex function $H: \mathbb{R} \to \mathbb{R}_+$ such that $H(s)/s \to +\infty$ as $s \to \infty$ and $H(f/f_1)f_1\phi_1 \in L^1$. Using the notations of the proof of Theorem 4.21, the Jensen inequality and the above estimate (4.25), we deduce

$$ \int_E H(u_T/f_1)f_1\phi_1 d\mu \le \frac{1}{T} \int_0^T \int_E H([\tilde{S}_t f]/f_1) f_1\phi_1 d\mu dt \le \int_E H(f/f_1)f_1\phi_1 d\mu, $$

for any $T > 0$. Now, for any $A \in \mathcal{E}$ and $T, K > 0$, we have

$$ \begin{align*} \int_E u_T \mathbf{1}_A \phi_1 d\mu &= \int_E \frac{u_T}{f_1} \mathbf{1}_{\frac{|u_T|}{f_1} > K} \mathbf{1}_A f_1 \phi_1 d\mu + \int_E \frac{u_T}{f_1} \mathbf{1}_{\frac{|u_T|}{f_1} \le K} \mathbf{1}_A f_1 \phi_1 d\mu \\ &\le \frac{K}{H(K)} \int_E H(u_T/f_1) f_1 \phi_1 d\mu + K \int_E \mathbf{1}_A f_1 \phi_1 d\mu \\ &\le \frac{K}{H(K)} \int_E H(f/f_1) f_1 \phi_1 d\mu + K \int_E \mathbf{1}_A f_1 \phi_1 d\mu, \end{align*} $$
Step 2. We furthermore assume (2), and by linearity we may assume $f_0 \in X$, $\langle f_0, \phi_1 \rangle = 0$. We recall that (4.25) holds for the strong topology of $L_{\phi_1}^1$ from Theorem 4.23 and that $\|v(t)\|_{X_0} \to 0$ as $t \to \infty$ from Step 1. Arguing as in Step 3 of the proof of Theorem 3.4, we have

$$ 
\begin{aligned}
K(T) &:= \frac{1}{T} \int_0^T (W * \tilde{S})(t) dt = \frac{1}{T} \int_0^T W(s) \int_0^{T-s} \tilde{S}(u) du ds \\
&= \int_0^T W(s) \frac{T-s}{T} \tilde{U}(T-s) ds,
\end{aligned}
$$

where $\tilde{U}_T := U_T e^{-\lambda_1 T}$, $U_t$ is defined by (3.44), so that $u_T = \tilde{U}_T f_0$ and $[u_T]_1 \to 0$ as $T \to \infty$ from Theorem 4.23. As a consequence, we have

$$ 
\begin{aligned}
\|K(T)f_0\|_{X_1} &\le \int_0^{T/2} \Theta_2(s)[\tilde{U}(T-s)]_0 ds + \int_{T/2}^T \Theta(s)[\tilde{U}(T-s)]_0 ds \\
&\le \|\tilde{\Theta}_2\|_{L^1} \sup_{t \ge T/2} [\tilde{U}(t)]_0 + \int_{T/2}^\infty \tilde{\Theta}_2(s) ds \sup_{t \ge 0} [\tilde{U}(t)]_0 \to 0,
\end{aligned}
$$

as $T \to \infty$. All together, we have established that $\|u_T\|_{X_0} \to 0$ as $T \to \infty$. $\square$

## 5. THE GEOMETRY OF THE BOUNDARY POINT SPECTRUM

We summarize the results established up to now by assuming that the main conclusions in the previous sections are achieved, namely

(C2) the first eigentriplet problem (4.1) has a unique solution $(\lambda_1, f_1, \phi_1)$, and furthermore, $f_1 \gg 0$ and $\phi_1 \gg 0$. In that situation, we make the usual normalization (4.19).

In this section, we aim to describe one step further the geometry of the spectrum and more precisely to get some information on the boundary point spectrum

$$ \Sigma_P^+(\mathcal{L}) := \Sigma_P(\mathcal{L}) \cap \bar{\Delta}_{\lambda_1} = \Sigma_P(\mathcal{L}) \cap \Sigma_+(\mathcal{L}). $$

That will be possible by introducing first a suitable and usual complexification framework and next by assuming a stronger positivity property on $\mathcal{L}$ or on the associated semigroup. Here and for further references below, we recall that we define the sets

$$ \Sigma_d(\mathcal{L}) \subset \Sigma_P(\mathcal{L}) \subset \Sigma(\mathcal{L}), $$

where the point spectrum set $\Sigma_P(\mathcal{L})$ is the set of eigenvalues, namely $\lambda \in \Sigma_P(\mathcal{L})$ if $N(\mathcal{L}-\lambda) \ne \{0\}$, and the discrete spectrum set $\Sigma_d(\mathcal{L})$ is the set of eigenvalues which are isolated and have finite algebraic multiplicity.

### 5.1. Complexification and the sign operator.

We present some materials, most of them being very classical, about the sign operator in a complex Banach lattice and we refer to [337, 15] for more details.

**Complexification.** The complexification space $X_C$ associated to a real Banach lattice $X$ is defined by $X_C := X + iX$ so that $f \in X_C$ if $f = g + ih, g, h \in X$. In general, we just write $X$ without mentioning the field, although when we need to specify it, we write $X_C$ or $X_R$. We extend on $X_C$ the order defined on $X_R$ by writing

$$ f = g + ih \ge 0 \quad \text{if} \quad g \ge 0 \text{ and } h = 0. $$

The conjugate $\bar{f}$ of a complex vector $f = g + ih$ is classically defined by $\bar{f} = g - ih$. We then define the modulus

$$ (5.1) \qquad |f| := \sup_{\theta \in [0, 2\pi]} (g \cos \theta + h \sin \theta), $$

which indeed exists for such a family of vectors. One checks the usual modulus properties:

$$ |f| \ge 0, \quad |f| = 0 \quad \text{iff} \quad f = 0, \quad |\lambda f| = |\lambda| |f|, \quad |f + g| \le |f| + |g|, $$

for any $f, g \in X$ and $\lambda \in \mathbb{C}$. We finally define the norm on $X_C$ by

$$ \|f\| := \|g + ih\|_{X_C}, $$
from what we immediately deduce, by using the Dunford-Pettis theorem, that $(u_T)$ belongs to a weak compact set of $L_{\phi_1}^1$. We conclude that (4.22) holds for the weak convergence in $L_{\phi_1}^1$ as in the proof of Theorem 4.21.

Step 3. We now additionally assume that (4.23) holds with strong compact embedding for some $k \ge 1$. Taking $f \in D(\mathcal{L}^k)$, we compute

$$ \langle |\mathcal{L}^j(\tilde{S}_t f)|, \phi_1 \rangle = \langle |\tilde{S}_t(\mathcal{L}^j f)|, \phi_1 \rangle \le \langle |\mathcal{L}^j f|, \phi_1 \rangle, $$

for any $j \le k$ and any $t \ge 0$, and thus the same bound holds for $(u_T)$. From (4.23), we deduce that up to the extraction of a subsequence, $(u_T)$ converges a.e. on $E$. Together with the weak convergence in $L_{\phi_1}^1$ yet established, we classically deduce that the whole family $(u_T)$ converges for the strong topology in $L_{\phi_1}^1$. We conclude that the same holds for any $f \in X$ by taking advantage of the fact that $D(\mathcal{L}^k)$ is dense in $X$ for the strong topology of $X$, and thus for the strong topology of $X$, and of the estimate of contraction (4.24). $\square$

**Remark 4.23.** (1) A similar conclusion holds as in Theorem 4.22 when we assume $X \subset M_{\text{loc}}^1$, $D(\mathcal{L}^k) \subset L_{\text{loc}}^1$ and $D(\mathcal{L}^{*k}) \subset L_{\text{loc}}^1$ for some $k \ge 1$ instead of $X, Y \subset L_{\text{loc}}^1$. For $f \in D(\mathcal{L}^k) \subset L_{\text{loc}}^1$, we may indeed repeat the proof of Theorem 4.22 and we obtain the same conclusion. We next define $\mathcal{X}$ as the closure of $D(\mathcal{L}^k)$ for the norm $[\cdot]_{\phi_1}$. We conclude that (4.22) holds weakly in $L_{\phi_1}^1$ for any $f \in \mathcal{X}$ by a density argument.

(2) The proof of Theorem 4.22 is based on so-called General Relative Entropy (GRE) techniques as developed for instance in [257], [268] and [49]. These ones are known to be useful for some classes of PDE and for stochastic semigroups in order to establish uniform in time estimates and longtime convergence results.

The main interest of the two previous results stated in Theorem 4.22 is that they do not ask any new information on the rescaled semigroup $\tilde{S}$ but they are just based on the eigentriplet stationary problem. The shortcoming is that they are formulated in terms of the norm $[\cdot]_{\phi_1}$ instead of the norm of $X$. We point out that under the same conditions as in Theorem 4.22-(1), we may directly apply Theorem 4.21 and obtain a slightly stronger conclusion. We present a second variant of Theorem 4.21 which is well adapted to the splitting framework developed in Sections 2 and 3 and is precisely formulated in a weak or strong topology of a space $X_0 \supset X$.

**Theorem 4.24.** Consider a positive semigroup $S = S_\mathcal{L}$ such that $\mathcal{L}$ satisfies the conclusions of Theorem 4.13 about the existence and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. Assume furthermore that $S$ satisfies the splitting structure introduced in (HS2) in Section 3.1 or (HS3) in Section 3.3, or more precisely, there exist two families of operators $(V(t))$ and $(W(t))$ such that

$$ S = V + W * S, $$

a real number $\kappa \le \lambda_1$ and some decaying functions $\Theta_i : \mathbb{R}_+ \to \mathbb{R}_+$ with $\Theta_1(t) \to 0$ as $t \to \infty$, $\Theta_2 \in L^1(\mathbb{R}_+)$ such that the following estimates hold

$$ \begin{aligned} \|V(t)e^{-\kappa t}\|_{\mathcal{B}(X)} &= \mathcal{O}(1), & \|V(t)e^{-\kappa t}\|_{\mathcal{B}(X, X_0)} &= \mathcal{O}(\Theta_1), \\ \|W(t)e^{-\kappa t}\|_{\mathcal{B}(X_0, X_1)} &= \mathcal{O}(\Theta_2), \end{aligned} $$

with $\mathcal{X}_1 \subset X_0 \subset \mathcal{X}_0$, where $\mathcal{X}_0$ is the space $X$ endowed with the norm $[g]_{\phi_1} := \langle |g|, \phi_1 \rangle$.

(1) Assume furthermore that $\mathcal{X}_1 \subset X_0$ with compact embedding for the weak or the strong topology in $X_0$ and this topology makes $f \mapsto \langle f, \phi_1 \rangle$ continuous. Then the mean ergodic convergence (E1) holds true for the above strong or weak topology.

(2) Assume furthermore that $X \subset L_{\text{loc}}^1$, $S$ is strongly continuous, and that the space $\mathcal{X}^k$ defined by (4.23) is strongly compact embedded in $L_{\text{loc}}^1$ for some $k \ge 1$. Then the mean ergodic convergence (E1) holds true for the strong topology of $X_0$.

**Proof of Theorem 4.24.** We define

$$ \tilde{V}(t) := V(t)e^{-\lambda_1 t}, \quad \tilde{W}(t) := W(t)e^{-\lambda_1 t}, $$

so that

$$ \tilde{S} = \tilde{V} + \tilde{W} * \tilde{S}, $$
and we observe that $X_{\mathbb{C}}$ has a complex Banach lattice structure. We extend the definition of $A \in \mathcal{B}(X_{\mathbb{R}})$ to $X_{\mathbb{C}}$ by setting

$$A(g + ih) = Ag + iAh, \quad \forall g + ih \in X_{\mathbb{C}}.$$

**The operator sign.** We classically extend the sign operator defined in Section 4.1 to the present complex Banach lattice framework. Instead of dealing with the most general case, we will use some regularity assumption on the Banach lattice $X$ which is suitable for our purpose and that we present below. Similarly as in Remark 4.5, for $f \in X$, we define

$$X_f := \bigcup_n \{g \in X; |g| \le n|f|\},$$

and next, similarly as in Theorem 2.24, we define

$$A_f[g] := \inf\{C > 0; |g| \le C|f|\}, \quad \forall g \in X_f.$$

We summarize the regularity conditions we need on the Banach lattice $X$ by assuming:

(X2) For any $f \in X$ such that $|f| \in X_{++}$, there exists a sign operator sign $f \in \mathcal{B}(X)$, with the following properties

(5.2) $\quad$ sign $f \circ \text{sign} \bar{f} = I$, (sign $f$) $f = |f|$,

(5.3) $\quad$ (sign $f$) $g = (\text{sign } uf)(ug)$, $|(\text{sign } f)g| \le |g|$, $\forall g \in X, \forall u \in \mathbb{S}^1$.

and furthermore

(X3) for any $f \in X$ such that $|f| \in X_{++}$, the inclusion $X_f \subset X$ is dense for the strong, the weak, or the weak-* topology, and for all $f \in X$ and $g \in X_f$

(5.4) $\quad (g \in X_{\mathbb{R}} \text{ and } |g| \le C|f|) \Leftrightarrow A_f[g - ir|f|] \le \sqrt{C^2 + r^2}, \forall r \in \mathbb{R}.$

For a space of functions, the sign operator is defined as the multiplication by (abusing notations)

(5.5) $\quad$ sign $f := \bar{f}/|f|$, $\forall f \in X$, $|f| \in X_{++}$.

**Lemma 5.1.** With (5.5), the properties (X2) and (X3) hold when $X = L^p(E, \mathcal{E}, \mu)$ or $X = C_0(E)$.

*Proof of Lemma 5.1.* For $f \in X$, $|f| \in X_{++}$, we just indicate the proof of $\bar{X}_f = X$, the other algebraic properties being clear from the definition (5.5). When $f \in L^p$ such that $|f| > 0$ $\mu$-a.e. and $0 \le g \in L^p$, we set $g_n := g \wedge (n|f|)$. We have $0 \le g_n \le g$ and $g_n \to g$ strongly $L^p$ if $p < \infty$ and weakly-* $L^\infty$ if $p = \infty$. The general case $g \in L^p$ is dealt in the usual way by introducing positive and negative parts and next real and imaginary part. That concludes the proof of $\bar{X}_f = L^p$. The proof of $\bar{X}_f = C_0(E)$ is similar. $\square$

A sign operator satisfying (X2) and (X3) can actually be built by using Kakutani's theorem in general Banach lattices whenever $|f|$ is a quasi-interior point, see for instance [41, Chapter 14.3]. In $X = L^\infty(E, \mathcal{E}, \mu)$, being a quasi-interior point is more demanding than belonging to $X_{++}$, and our framework is thus more general in that case. In $X = M^1(E)$, the situation is even worst since there is no quasi-interior point, so the approach via Kakutani's theorem does not provide any sign operator. However, we can associate to $f \in M^1(E)$ such that $|f| \gg 0$ a sign operator by means of the Radon-Nikodym theorem. Denoting $\alpha : E \to \mathbb{S}^1$ the measurable function such that $f = \alpha|f|$, the multiplication by $\bar{\alpha}/|\alpha|$ defines a sign operator sign $f \in \mathcal{B}(X)$, or in other words (abusing notations)

(5.6) $\quad$ sign $f := \bar{\alpha}/|\alpha|$, $\forall f = \alpha|f| \in M^1$, $|f| \in M_{++}^1$.

**Lemma 5.2.** With the definition (5.6), $X = M^1(E)$ enjoys the properties (X2) and (X3).

*Proof of Lemma 5.2.* As for Lemma 5.1, we only sketch the proof of the density property $\bar{X}_f = X$, which holds here for the weak-* topology, the other algebraic properties being clear from the definition (5.6). Without loss of generality, we may take $f \in X_{++}$, meaning that $f(\mathcal{O}) > 0$ for any open set $\mathcal{O} \subset E$. For $\varepsilon > 0$ and $\varphi \in C_0(E)$, we can find a partition $E_1, \dots, E_n$ of $E$ and some elements $x_1, \dots, x_n$ of $E$ such that for any $i \in \{1, \dots, n\}$:

$$f(E_i) > 0, \quad x_i \in E_i \quad \text{and} \quad \sup_{x \in E_i} |\varphi(x) - \varphi(x_i)| < \varepsilon.$$
and

$$
(4.26) \qquad
\begin{aligned}
M &:= \sup_{t \ge 0} \|\tilde{V}(t)\|_{\mathcal{B}(X)} < \infty, \quad \|\tilde{V}\|_{\mathcal{B}(X, X_0)} \lesssim \tilde{\Theta}_1 \in C_0(\mathbb{R}_+), \\
\tilde{\Theta}_2(t) &:= \|\tilde{W}(t)\|_{\mathcal{B}(X_0, X_1)} \in L^1(\mathbb{R}_+).
\end{aligned}
$$

Step 1. We furthermore assume (1) and that the weak topology of $X_0$ makes $f \mapsto \langle f, \phi_1 \rangle$ continuous. We denote by $\mathcal{T}$ the weak or the strong topology $X_0$ (depending of the assumption made on the embedding $\mathcal{X}_1 \subset X_0$). For $f_0 \in X$, we split

$$f(t) := \tilde{S}_t f_0 = v(t) + k(t), \quad v(t) := \tilde{V}(t)f_0, \quad k(t) := (\tilde{W} * \tilde{S})(t)f_0,$$

and we observe that $\|v(t)\|_{X_0} \to 0$ as $t \to \infty$ from the second estimate in (4.26). On the other hand, we have

$$\sup_{t \ge 0} \|k(t)\|_{X_1} \le \|\tilde{\Theta}_2\|_{L^1} \sup_{t \ge 0} \|\tilde{S}_t f_0\|_{X_0} \le \|\tilde{\Theta}_2\|_{L^1} \|f_0\|_{X_0},$$

from (4.24). In particular, $k(t)$ belongs to a compact set of $\mathcal{T}$, so that $(f(t))_{t \ge 0}$ also belongs to a compact set for the same topology $\mathcal{T}$. The same argument used on the Cesàro function $(u_T)$ defined during the proof of Theorem 4.21 implies that there exist $f^* \in X$ and a sequence $(T_k)$ such that

$$u_{T_k} \to f^* \text{ in the sense of } \mathcal{T} \quad \text{and} \quad \langle f^*, \phi_1 \rangle = \langle f, \phi_1 \rangle,$$

the last identity following from the assumption that $f \mapsto \langle f, \phi_1 \rangle$ continuous for $\mathcal{T}$. We may then conclude as in the proof of Theorem 4.21.

Step 2. We furthermore assume (2), and by linearity we may assume $f_0 \in X$, $\langle f_0, \phi_1 \rangle = 0$. We recall that (4.22) holds for the strong topology of $L^1_{\phi_1}$ from Theorem 4.22 and that $\|v(t)\|_{X_0} \to 0$ as $t \to \infty$ from Step 1. Arguing as in Step 3 of the proof of Theorem 3.4, we have

$$K(T) := \frac{1}{T} \int_0^T (\tilde{W} * \tilde{S})(t) dt = \frac{1}{T} \int_0^T \tilde{W}(s) \int_0^{T-s} \tilde{S}(u) du ds \\ = \int_0^T \tilde{W}(s) \frac{T-s}{T} \tilde{U}(T-s) ds,$$

where $\tilde{U}_T := U_T e^{-\lambda_1 T}$, $U_t$ is defined by (3.43), so that $u_T = \tilde{U}_T f_0$ and $[u_T]_{\phi_1} \to 0$ as $T \to \infty$ from Theorem 4.22. As a consequence, we have

$$\|K(T)f_0\|_{X_1} \le \int_0^{T/2} \tilde{\Theta}_2(s)[\tilde{U}(T-s)f_0]_{\phi_1} ds + \int_{T/2}^T \tilde{\Theta}_2(s)[\tilde{U}(T-s)f_0]_{\phi_1} ds \\ \le \|\tilde{\Theta}_2\|_{L^1} \sup_{t \ge T/2} [\tilde{U}(t)f_0]_{\phi_1} + \int_{T/2}^{\infty} \tilde{\Theta}_2(s) ds \sup_{t \ge 0} [\tilde{U}(t)f_0]_{\phi_1} \to 0,$$

as $T \to \infty$. Altogether, we have established that $\|u_T\|_{X_0} \to 0$ as $T \to \infty$. $\square$

Let us finally summarize the results obtained in this section concerning our spectral analysis program. For this purpose, we emphasize that the spectrum set of $\mathcal{L}$ may be sensitive to the norm considered for defining it. We still denote by $\Sigma(\mathcal{L})$ the spectrum set defined in $X$ (and its norm) in Section 2.1. We may similarly define the spectrum of $\mathcal{L}$ in the space $L^1(\phi_1) \supset X$ and we denote it by $\Sigma'(\mathcal{L})$ in the following discussion.

On the one hand, Theorem 4.20, Theorem 4.21 or Theorem 4.24 (when $\mathcal{X}_1 = X$ and $\mathcal{X}_0 = L^1(\phi_1)$) implies that $\Sigma(\mathcal{L}) \subset \{z \in \mathbb{C}; \Re z \le \lambda_1\}$ in $X$, because $(\|\tilde{S}(t)\|_{\mathcal{B}(X)})$ is bounded in these cases and we may use the resolvent identity (2.13). For a similar reason, Theorems 4.22 implies that $\Sigma'(\mathcal{L}) \subset \{z \in \mathbb{C}; \Re z \le \lambda_1\}$ in $L^1(\phi_1)$, because $(\|\tilde{S}(t)\|_{\mathcal{B}(L^1(\phi_1))})$ is bounded in this case. As a consequence, all these results imply that (S1) holds in either $X$ or $L^1(\phi_1)$.

On the other hand, the conclusion (C1) together with Theorem 4.13 provide the first answer (S2) about the geometry of the spectrum problem. The mean ergodicity convergence (E1) is next deduced in the many variants Theorem 4.20, Theorem 4.21, Theorem 4.22, Theorem 4.24. Altogether, these results thus provide a proof for Theorem 1.1-(2). In the weak dissipativity framework, we may also observe that (C1) and the above mentioned theorems imply that (S1), (S2) and (E1) hold at least in $L^1(\phi_1)$, and for the weak convergence.
For $g \in X$ and $\varepsilon > 0$, defining $g_\varepsilon$ by

$$g_\varepsilon := \sum_{i=1}^{n} \frac{g(E_i)}{f(E_i)} f_{|E_i} \in X_f,$$

we have

$$\left|\langle g_{\varepsilon}, \varphi \rangle - \sum_{i=1}^{n} \varphi(x_i) g(E_i)\right| \leq \sum_{i=1}^{n} |g(E_i)| \int_{E_i} |\varphi(x) - \varphi(x_i)| \frac{f(dx)}{f(E_i)} \leq \varepsilon \|g\|_X,$$

as well as

$$\left|\langle g, \varphi \rangle - \sum_{i=1}^{n} \varphi(x_i) g(E_i)\right| \leq \sum_{i=1}^{n} \int_{E_i} |\varphi(x) - \varphi(x_i)| |g|(dx) \leq \varepsilon \|g\|_X.$$

We have established that $|\langle g_\varepsilon - g, \varphi \rangle| \leq 2\varepsilon \|g\|$ for any $\varepsilon > 0$, from what we deduce that $g$ belongs to the weak-* closure of $X_f$. $\square$

**Lemma 5.3.** Assume (X2)-(X3), and $f \in X_{++}$. Consider a linear operator $Q: X_f \to X_f$ such that $Qf = f$ and $A_f(Qg) \leq A_f(g)$ for any $g \in X_f$. Then $Q \geq 0$.

*Proof of Lemma 5.3.* Take $0 \leq g \in X_f$ such that $g \leq 2Cf$, $C > 0$, and observe that

$$-Cf \leq g - Cf \leq Cf.$$

For any $r \in \mathbb{R}$, we compute

$$\begin{aligned} A_f[(Qg) - Cf - irf] &= A_f[Q(g - Cf - irf)] \\ &\leq A_f[g - Cf - irf] \\ &\leq \sqrt{C^2 + r^2}, \end{aligned}$$

by using the non expansion property of $Q$ and the claim (5.4). Using again (5.4), we deduce $-Cf \leq (Qg) - Cf \leq Cf$ and the conclusion. $\square$

We generalize Kato's inequality (4.5) to the present complex framework by saying that an operator $\mathcal{L}$ on $X$ satisfies (the complex) Kato's inequality if

$$(5.7) \qquad \forall f \in D(\mathcal{L}), \quad \Re(\operatorname{sign} f) \mathcal{L} f \leq \mathcal{L} |f|,$$

possibly in a dual sense as in (4.6). As for the real Kato's inequality, when $\mathcal{L}$ is the generator of a semigroup, Kato's inequality (5.7) is a consequence of the positivity of the semigroup, and we refer to Remark 4.1 for references about this claim.

## 5.2. On the subgroup and discrete structure of the boundary point spectrum.

In this section, we establish that the boundary point spectrum enjoys a subgroup structure under the same kind of hypotheses as considered in the previous sections.

**Lemma 5.4.** Under assumptions (C2), (X2) and the complex Kato's inequality (5.7), for any $\lambda \in \Sigma_P^+(\mathcal{L}) \setminus \{0\}$ the associated normalized eigenfunction $f$ satisfies $|f| = f_1$.

*Proof of Lemma 5.4.* By definition $\mathcal{L}f = \lambda f$ and $f \in D(\mathcal{L})$. By linearity of the operator sign and thanks to (5.2) and Kato's inequality (5.7), we have

$$\lambda_1 |f| = \Re[e[\lambda(\operatorname{sign} f) f]] = \Re(e(\operatorname{sign} f)(\lambda f)) = \Re(e(\operatorname{sign} f) \mathcal{L} f) \leq \mathcal{L} |f|.$$

By the duality argument introduced during the proof of Lemma 4.18, we must have $\lambda_1 |f| = \mathcal{L} |f|$ and the conclusion. $\square$

**Theorem 5.5.** Assume (C2), (X2), (X3) and that the complex Kato's inequality (5.7) holds true. Denoting $\tilde{\mathcal{L}} = \mathcal{L} - \lambda_1$, the set $\mathcal{S} := \Sigma_P(\tilde{\mathcal{L}}) \cap i\mathbb{R}$ is an additive subgroup and $\dim N(\tilde{\mathcal{L}} - i\alpha)^k = 1$ for any $i\alpha \in \mathcal{S}$ and $k \geq 1$.

Theorem 5.5 is similar but more general than [15, C-III, Cor. 2.12] and [41, Prop. 14.15]. Our proof is also very similar to the proof of [41, Prop. 14.15]. However, it is more direct and avoid the use of the $C(K)$ algebra and Kakutani's Theorem [227] (see also [264, Thm. 2.1.3]).

*Proof of Theorem 5.5.* We split the proof into three steps.
## 5. THE GEOMETRY OF THE BOUNDARY POINT SPECTRUM

We establish the sharper geometric parts (S31) and (S32) as well as the evolution parts (E2) and (E31) of the Krein-Rutman theorem. For that purpose, we introduce in particular the partial conclusion (C2) of (S1)-(S2) established in the previous section, the structure conditions (X2) and (X3) on the Banach lattice X and the aperiodicity like conditions (H5) and (H5').

We summarize the results established up to now by assuming that the main conclusions in the previous sections are achieved, namely

(C2) the first eigentriplet problem (1.1)-(1.2) has a unique solution ($\lambda_1, f_1, \phi_1$), and furthermore, $f_1 \gg 0$ and $\phi_1 \gg 0$. In that situation, we make the usual normalization (4.17).

Compared to (S1)-(S2), the (geometric and algebraic) simplicity of $f_1$ and $\phi_1$ is not required in (C2) nor the localization of the spectrum $\Sigma(\mathcal{L}) \subset \{z \in \mathbb{C}, \Re e(z) \le \lambda_1\}$.

In this section, we aim to describe one step further the geometry of the spectrum and more precisely to get some information about the principal point spectrum

$$ \Sigma_P^1(\mathcal{L}) := \Sigma_P(\mathcal{L}) \cap \bar{\Delta}_{\lambda_1}. $$

Here and for further references below, we recall that we define the sets

$$ \Sigma_d(\mathcal{L}) \subset \Sigma_P(\mathcal{L}) \subset \Sigma(\mathcal{L}), $$

where the point spectrum set $\Sigma_P(\mathcal{L})$ is the set of eigenvalues, namely $\lambda \in \Sigma_P(\mathcal{L})$ if $N(\mathcal{L}-\lambda) \ne \{0\}$, and the discrete spectrum set $\Sigma_d(\mathcal{L})$ is the set of eigenvalues which are isolated and have finite algebraic multiplicity. It is worth pointing out that in general the principal point spectrum $\Sigma_P^1(\mathcal{L})$ differs from the boundary point spectrum $\Sigma_P^+(\mathcal{L})$, which is defined in Section 1.1 as

$$ \Sigma_P^+(\mathcal{L}) := \Sigma_P(\mathcal{L}) \cap \Sigma_+(\mathcal{L}). $$

Because $\lambda_1 \in \Sigma(\mathcal{L})$ and thus $\lambda_1 \le s(\mathcal{L})$, we always have $\Sigma_P^+(\mathcal{L}) \subset \Sigma_P^1(\mathcal{L})$. However, these two sets coincide in the case when $\lambda_1 = s(\mathcal{L})$, which is the conclusion (1.3) in (S1). For further reference, we introduce the translated operator $\tilde{\mathcal{L}} := \mathcal{L} - \lambda_1$ and the associated principal point spectrum

$$ \Sigma_P^1(\tilde{\mathcal{L}}) := \Sigma_P(\tilde{\mathcal{L}}) \cap \bar{\Delta}_0 = \Sigma_P^1(\mathcal{L}) - \lambda_1. $$

Analyzing these dominant parts of the spectrum will be made possible by first introducing a suitable and usual complexification framework and next by assuming a stronger positivity property on $\mathcal{L}$ or on the associated semigroup.

### 5.1. Complexification and the sign operator.

We present some materials, most of them being very classical, about the sign operator in a complex Banach lattice and we refer to [336, 15] for more details.

**Complexification.** The complexification space $X_C$ associated to a real Banach lattice $X$ is defined by $X_C := X + iX$ so that $f \in X_C$ if $f = g + ih, g, h \in X$. In general, we just write $X$ without mentioning the field, although when we need to specify it, we write $X_C$ or $X_\mathbb{R}$. We extend on $X_C$ the order defined on $X_\mathbb{R}$ by writing

$$ f = g + ih \ge 0 \quad \text{if} \quad g \ge 0 \text{ and } h = 0. $$

The conjugate $\bar{f}$ of a complex vector $f = g + ih$ is defined by $\bar{f} = g - ih$. We then define the modulus

$$ (5.1) \qquad |f| := \sup_{\theta \in [0, 2\pi]} (g \cos \theta + h \sin \theta), $$

which indeed exists for such a family of vectors. One checks the usual modulus properties:

$$ |f| \ge 0, \quad |f| = 0 \quad \text{iff} \quad f = 0, \quad |\lambda f| = |\lambda| |f|, \quad |f+g| \le |f| + |g|, $$

for any $f, g \in X$ and $\lambda \in \mathbb{C}$. We finally define the norm on $X_C$ by

$$ \|f\| := \|g + ih\|_{X_\mathbb{R}}, $$

and we observe that $X_C$ has a complex Banach lattice structure. We extend the definition of $A \in \mathcal{B}(X_\mathbb{R})$ to $X_C$ by setting

$$ A(g + ih) = Ag + iAh, \quad \forall g + ih \in X_C. $$
Step 1. We consider $f$ associated to an eigenvalue $i\alpha \in \Sigma_P(\tilde{\mathcal{L}})\setminus\{0\}$, and we define

$$T(t) := (\operatorname{sign} f)e^{-i\alpha t}\tilde{S}(t)(\operatorname{sign} \bar{f}).$$

Observing that $\tilde{S}(t)f = e^{i\alpha t}f$, we have

$$T(t)|f| = (\operatorname{sign} f)e^{-i\alpha t}\tilde{S}(t)f = (\operatorname{sign} f)f = |f| = \tilde{S}(t)|f|.$$

On the other hand, we have

$$|T(t)g| \le |\tilde{S}(t)(\operatorname{sign} \bar{f})g| \le \tilde{S}(t)|g|, \quad \forall g \in X,$$

which, by positivity of $\tilde{S}(t)$, yields

$$|T(t)g| \le A_f(g)\tilde{S}(t)|f| = A_f(g)|f|, \quad \forall g \in X_f.$$

Because $|f| = f_1 \gg 0$ from Lemma 5.4, we can apply Lemma 5.3 to $|f|$ and $Q := T(t)$. We deduce that $T(t) \ge 0$ on $X_{|f|} = X_f$, and then on $X = \bar{X}_f$. As a consequence, $0 \le T(t)g = |T(t)g| \le \tilde{S}(t)g$ for any $g \ge 0$. In other words, we have $0 \le \tilde{S}(t) - T(t)$ and then $0 \le \tilde{S}(t)^* - T(t)^*$. We must have $\tilde{S}(t)^* - T(t)^* = 0$. Otherwise, there would exist $\psi \in Y_+\setminus\{0\}$ such that $(\tilde{S}(t)^* - T(t)^*)\psi \in Y_+\setminus\{0\}$, and we find a contradiction by computing

$$0 < \langle (\tilde{S}(t)^* - T(t)^*)\psi, f_1 \rangle = \langle \psi, (\tilde{S}(t) - T(t))f_1 \rangle = 0.$$

We have thus established that $\tilde{S}(t) = T(t)$.

Step 2. Consider $\alpha, \beta \in \mathbb{R}$ and $f, g \in X\setminus\{0\}$ such that $\tilde{\mathcal{L}}f = i\alpha f$ and $\tilde{\mathcal{L}}g = i\beta g$, and suppose first that $(\operatorname{sign} \bar{f}) : D(\mathcal{L}) \to D(\mathcal{L})$. From Step 1 and the fact that $(\operatorname{sign} \bar{f}) \circ \operatorname{sign} f = I$, for any $h \in D(\mathcal{L})$, we may compute

$$\begin{align*}
\tilde{\mathcal{L}}h &= \lim_{t \to 0} \frac{1}{t}(\tilde{S}(t)h - h) \\
&= (\operatorname{sign} f) \lim_{t \to 0} \frac{1}{t}(e^{-i\alpha t}\tilde{S}(t)(\operatorname{sign} \bar{f})h - (\operatorname{sign} \bar{f})h) \\
&= (\operatorname{sign} f)(\tilde{\mathcal{L}} - i\alpha)(\operatorname{sign} \bar{f})h,
\end{align*}$$

or in other words $\tilde{\mathcal{L}} - i\alpha = (\operatorname{sign} \bar{f})\tilde{\mathcal{L}}(\operatorname{sign} f)$. We have similarly $\tilde{\mathcal{L}} - i\beta = (\operatorname{sign} \bar{g})\tilde{\mathcal{L}}(\operatorname{sign} g)$. Both equations together imply

$$\tilde{\mathcal{L}} - i(\alpha + \beta) = (\operatorname{sign} \bar{f})(\operatorname{sign} \bar{g})\tilde{\mathcal{L}}(\operatorname{sign} g)(\operatorname{sign} f).$$

Defining $h := (\operatorname{sign} \bar{f})(\operatorname{sign} \bar{g})f_1$, so that $(\operatorname{sign} g)(\operatorname{sign} f)h = f_1$, we get $\tilde{\mathcal{L}}h = i(\alpha + \beta)h$, and finally $i(\alpha + \beta) \in \mathcal{S}$, so that the additive subgroup structure is established.

When the condition $(\operatorname{sign} \bar{f}) : D(\mathcal{L}) \to D(\mathcal{L})$ is not granted, we modify the above argument by using a resolvent approach. For some $\lambda > 0$, we compute thanks to (2.13)

$$\begin{align*}
(\lambda - \tilde{\mathcal{L}})^{-1} &= \int_{0}^{\infty} e^{-\lambda t} \tilde{S}(t) dt \\
&= (\operatorname{sign} f) \int_{0}^{\infty} e^{-(\lambda + i\alpha)t} \tilde{S}(t) dt (\operatorname{sign} \bar{f}) \\
&= (\operatorname{sign} f)(\lambda + i\alpha - \tilde{\mathcal{L}})^{-1}(\operatorname{sign} \bar{f}).
\end{align*}$$

Repeating the argument, we obtain

$$(\lambda + i(\alpha + \beta) - \tilde{\mathcal{L}})^{-1} = (\operatorname{sign} \bar{f})(\operatorname{sign} \bar{g})(\lambda - \tilde{\mathcal{L}})^{-1}(\operatorname{sign} g)(\operatorname{sign} f).$$

Applying that last identity to the vector $h = (\operatorname{sign} \bar{f})(\operatorname{sign} \bar{g})f_1$ and using that $(\lambda - \tilde{\mathcal{L}})^{-1}f_1 = \lambda^{-1}f_1$, we deduce $(\lambda + i(\alpha + \beta) - \tilde{\mathcal{L}})^{-1}h = \lambda^{-1}h$. In other words, we have again $\tilde{\mathcal{L}}h = i(\alpha + \beta)h$, and we conclude as above.

Step 3. From the fact that $(\operatorname{sign} f)$ is an invertible operator and the equation

$$(\tilde{\mathcal{L}} - i\alpha)^k = (\operatorname{sign} f)^{-1}(\tilde{\mathcal{L}})^k(\operatorname{sign} f),$$

we see from Theorem 4.13-(ii) that $N(\tilde{\mathcal{L}} - i\alpha)^k = (\operatorname{sign} f)^{-1}N(\tilde{\mathcal{L}})^k = (\operatorname{sign} f)^{-1}\operatorname{Span} f_1$ for any $k \ge 1$, so that its dimension is one. $\square$
**The operator sign.** We extend the sign operator defined in Section 4.1 to the present complex Banach lattice framework. Instead of dealing with the most general case, we will use some regularity assumption on the Banach lattice $X$ which is suitable for our purpose and that we present below. Similarly as in Remark 4.5, for $f \in X$, we define

$$X_f := \bigcup_n \{g \in X; |g| \le n|f|\},$$

and next, similarly as in Theorem 2.24, we define

$$A_f[g] := \inf\{C > 0; |g| \le C|f|\}, \quad \forall g \in X_f.$$

We summarize the regularity conditions we need on the Banach lattice $X$ by assuming:

(X2) For any $f \in X$ such that $|f| \in X_{++}$, there exists a sign operator $\text{sign } f \in \mathcal{B}(X)$, with the following properties

$$(5.2) \qquad \text{sign } f \circ \text{sign } \bar{f} = I, \quad (\text{sign } f)f = |f|,$$

$$(5.3) \qquad (\text{sign } f)g = (\text{sign } (uf))(ug), \quad |(\text{sign } f)g| \le |g|, \quad \forall g \in X, \forall u \in \mathbb{S}^1,$$

and furthermore

(X3) for any $f \in X$ such that $|f| \in X_{++}$, the inclusion $X_f \subset X$ is dense for the strong, the weak, or the weak-* topology, and for all $f \in X$ and $g \in X_f$

$$(5.4) \qquad (g \in X_{\mathbb{R}} \text{ and } |g| \le C|f|) \Leftrightarrow A_f[g - ir|f|] \le \sqrt{C^2 + r^2}, \quad \forall r \in \mathbb{R}.$$

For a space of functions, the sign operator is defined as the multiplication by (abusing notations)

$$(5.5) \qquad \text{sign } f := \bar{f}/|f|, \quad \forall f \in X, \quad |f| \in X_{++}.$$

**Lemma 5.1.** With (5.5), the properties (X2) and (X3) hold when $X = L^p(E, \mathcal{E}, \mu)$ or $X = C_0(E)$.

*Proof of Lemma 5.1.* For $f \in X$, $|f| \in X_{++}$, we just indicate the proof of $\bar{X}_f = X$, the other algebraic properties being clear from the definition (5.5). When $f \in L^p$ such that $|f| > 0$ $\mu$-a.e. and $0 \le g \in L^p$, we set $g_n := g \wedge (n|f|)$. We have $0 \le g_n \le g$ and $g_n \to g$ strongly $L^p$ if $p < \infty$ and weakly-* $L^\infty$ if $p = \infty$. The general case $g \in L^p$ is dealt in the usual way by introducing positive and negative parts and next real and imaginary part. That concludes the proof of $\bar{X}_f = L^p$. The proof of $\bar{X}_f = C_0(E)$ is similar. $\square$

A sign operator satisfying (X2) and (X3) can actually be built by using Kakutani's theorem in general Banach lattices whenever $|f|$ is a quasi-interior point, see for instance [41, Chapter 14.3]. In $X = L^\infty(E, \mathcal{E}, \mu)$, being a quasi-interior point is more demanding than belonging to $X_{++}$, and our framework is thus more general in that case. In $X = M^1(E)$, the situation is even worst since there is no quasi-interior point, so the approach via Kakutani's theorem does not provide any sign operator. However, we can associate to $f \in M^1(E)$ such that $|f| \gg 0$ a sign operator by means of the Radon-Nikodym theorem. Denoting $\alpha : E \to \mathbb{S}^1$ the measurable function such that $f = \alpha|f|$, the multiplication by $\bar{\alpha}/|\alpha|$ defines a sign operator $\text{sign } f \in \mathcal{B}(X)$, or in other words (abusing notations)

$$(5.6) \qquad \text{sign } f := \bar{\alpha}/|\alpha|, \quad \forall f = \alpha|f| \in M^1, \quad |f| \in M_{++}^1.$$

**Lemma 5.2.** With the definition (5.6), $X = M^1(E)$ enjoys the properties (X2) and (X3).

*Proof of Lemma 5.2.* As for Lemma 5.1, we only sketch the proof of the density property $\bar{X}_f = X$, which holds here for the weak-* topology, the other algebraic properties being clear from the definition (5.6). Without loss of generality, we may take $f \in X_{++}$, meaning that $f(\mathcal{O}) > 0$ for any open set $\mathcal{O} \subset E$. For $\varepsilon > 0$ and $\varphi \in C_0(E)$, we can find a partition $E_1, \dots, E_n$ of $E$ and some elements $x_1, \dots, x_n$ of $E$ such that for any $i \in \{1, \dots, n\}$:

$$f(E_i) > 0, \quad x_i \in E_i \quad \text{and} \quad \sup_{x \in E_i} |\varphi(x) - \varphi(x_i)| < \varepsilon.$$

For $g \in X$ and $\varepsilon > 0$, defining $g_\varepsilon$ by

$$g_\varepsilon := \sum_{i=1}^n \frac{g(E_i)}{f(E_i)} f_{|E_i} \in X_f,$$
Making an additional splitting structure hypothesis as yet introduced in Section 2.2, we may significantly improve the conclusion. We first recall a classical result on the spectrum of an operator which holds under some power compactness assumption on the resolvent.

**Theorem 5.6.** We assume that $\mathcal{L}$ satisfies the splitting structure (HS1) introduced in Section 2.2 with $W(z) \in \mathcal{H}(X)$ for some $N \ge 1$ and any $z \in \Delta_{\kappa_0}$. Then $\Sigma(\mathcal{L}) \cap \Delta_{\kappa_0} \subset \Sigma_d(\mathcal{L})$.

Theorem 5.6 is a consequence of [355, Cor. 1.1]. We also refer to [278, proof of Thm. 3.1] for a possible elementary proof.

A sketch of the proof of Theorem 5.6. Iterating the formula $\mathcal{R}_\mathcal{L} = \mathcal{R}_B + \mathcal{R}_B A \mathcal{R}_\mathcal{L}$, we deduce

$$ \mathcal{J}(z)R_{\mathcal{L}}(z) = \mathcal{V}(z) $$

with $\mathcal{J} := I - (AR_B)^N$ and $\mathcal{V} := R_B + \dots + R_B(AR_B)^{N-1}$. Because $\mathcal{J}$ is holomorphic on $\Delta_{\kappa_0}$, it is a compact perturbation of the identity and $\mathcal{J}(z) \to I$ when $\Re z \to \infty$, one may use the theory of degenerate-meromorphic functions of Ribarič and Vidav [331] (also established independently by Steinberg, see in particular [341, Cor. 1]), and conclude that $\mathcal{J}(z)$ is invertible outside of a discrete set $\mathcal{D}$ of $\Delta_{\kappa_0}$. That implies that $\Sigma(\mathcal{L}) \cap \Delta_{\kappa_0} = \mathcal{D}$ is a discrete set of $\Delta_{\kappa_0}$. On the other hand, thanks to the Fredholm alternative [164], one deduces that the eigenspace associated to each spectral value $\lambda \in \mathcal{D}$ is non zero and finite dimensional, so that $\lambda \in \Sigma_d(\mathcal{L})$. See also [342, 357] for pioneering works in the subject. $\square$

We end this section by a result which gives a more accurate description of the geometry of the boundary spectrum, and is a variant of the classical results [15, C-III, Thm. 3.12], [152, Thm. VI.1.12], [41, Thm. 14.17].

**Theorem 5.7.** Assume (C2), (X2), (X3), that the complex Kato's inequality (5.7) holds true and additionally that the splitting structure (HS1) holds with $W(z) \in \mathcal{H}(X)$ for some $N \ge 1$ and any $z \in \Delta_{\kappa_0}$. Then the set $\Sigma_P(\tilde{\mathcal{L}}) \cap i\mathbb{R}$ is a discrete additive subgroup of $i\mathbb{R}$ and any of its elements is an algebraically simple eigenvalue. More precisely,

- either $\Sigma_P(\tilde{\mathcal{L}}) \cap i\mathbb{R} = \{0\}$ and the projection on the first eigenspace (associated to $\lambda_1$) writes

$$ \Pi f := \langle f, \phi_1 \rangle f_1; $$

- or $\Sigma_P(\tilde{\mathcal{L}}) \cap i\mathbb{R} = i\alpha\mathbb{Z}$ for some $\alpha > 0$ and there exists a sequence $(g_k, \psi_k)_{k \in \mathbb{Z}}$ such that $\mathcal{L}g_k = (\lambda_1 + ik\alpha)g_k$, $\mathcal{L}^*\psi_k = (\lambda_1 + ik\alpha)\psi_k$, and $\langle g_k, \psi_\ell \rangle = \delta_{k\ell}$.

*Proof of Theorem 5.7.* Combining Theorem 5.5 and Theorem 5.6, we immediately get that the subgroup $S := \Sigma_P(\tilde{\mathcal{L}}) \cap i\mathbb{R}$ satisfies $S \subset \Sigma_d(\mathcal{L})$, it is thus discrete and made of algebraically simple eigenvalues. The first case $\Sigma_P(\tilde{\mathcal{L}}) \cap i\mathbb{R} = \{0\}$ falls yet in the conclusions of Theorem 4.13. $\square$

In the second case, where the boundary spectrum is not trivial, the existence of a projection on the boundary eigenspace $\overline{\text{Span}}(g_k)_{k \in \mathbb{Z}}$ is ensured by the Jacobs-de Leeuw-Glicksberg theorem provided that $\mathcal{L}$ is the generator of a relatively compact semigroup, see for instance [41, Thm. A.39 and Prop. A.40] and the references therein. We also refer to [229, paragraphs III.6.4 and III.6.5] for very classical results on the projector on the direct sum of eigenspaces associated to eigenvalues belonging to a subset of the spectrum. We can even give an explicit expression of this projection in terms of $(g_k)$ and $(\psi_k)$ under the form of a Fejér type sum, see Theorem 5.25.

### 5.3. Stronger positivity.

In order to go one step further and establish the triviality of the boundary point spectrum, we need to reinforce the positivity of the semigroup or its generator. One possible condition is based on the following notion.

**The reverse strong positivity condition**

For $A \ge 0$, we recall that from (2.6), we have

$$ (5.8) \qquad |Af| \le A|f|, \quad \forall f \in X, $$

and we observe that the above inequality is an equality when $Af = uA|f|$ for some $u \in \mathbb{S}^1$. We focus now on the case of equality in (5.8).
we have

$$ \left| \langle g_{\varepsilon}, \varphi \rangle - \sum_{i=1}^{n} \varphi(x_i) g(E_i) \right| \le \sum_{i=1}^{n} |g(E_i)| \int_{E_i} |\varphi(x) - \varphi(x_i)| \frac{f(dx)}{f(E_i)} \le \varepsilon \|g\|_X, $$

as well as

$$ \left| \langle g, \varphi \rangle - \sum_{i=1}^{n} \varphi(x_i) g(E_i) \right| \le \sum_{i=1}^{n} \int_{E_i} |\varphi(x) - \varphi(x_i)| |g|(dx) \le \varepsilon \|g\|_X. $$

We have established that $|\langle g_{\varepsilon} - g, \varphi \rangle| \le 2\varepsilon\|g\|$ for any $\varepsilon > 0$, from what we deduce that $g$ belongs to the weak-* closure of $X_f$. $\square$

We establish a technical result which will be useful in the next section.

**Lemma 5.3.** Assume (X2)-(X3), and $f \in X_{++}$. Consider a linear operator $Q: X_f \to X_f$ such that $Qf = f$ and $A_f(Qg) \le A_f(g)$ for any $g \in X_f$. Then $Q \ge 0$.

*Proof of Lemma 5.3.* Take $0 \le g \in X_f$ such that $g \le 2Cf$, $C > 0$, and observe that

$$ -Cf \le g - Cf \le Cf. $$

For any $r \in \mathbb{R}$, we compute

$$ \begin{aligned} A_f[(Qg) - Cf - irf] &= A_f[Q(g - Cf - irf)] \\ &\le A_f[g - Cf - irf] \\ &\le \sqrt{C^2 + r^2}, \end{aligned} $$

by using the non expansion property of $Q$ and the claim (5.4). Using again (5.4), we deduce $-Cf \le (Qg) - Cf \le Cf$ and the conclusion. $\square$

We generalize Kato's inequality (4.4) to the present complex framework by saying that an operator $\mathcal{L}$ on $X$ satisfies (the complex) Kato's inequality if

$$ (5.7) \qquad \forall f \in D(\mathcal{L}), \quad \Re(\operatorname{sign}f)\mathcal{L}f \le \mathcal{L}|f|, $$

possibly in a dual sense as in (4.5). As for the real Kato's inequality, when $\mathcal{L}$ is the generator of a semigroup, Kato's inequality (5.7) is a consequence of the positivity of the semigroup, and we refer to Remark 4.1 for references about this claim.

## 5.2. On the subgroup and discrete structure of the principal point spectrum.

In this section, we establish that the principal point spectrum enjoys a subgroup structure under the same kind of hypotheses as considered in the previous sections. Such a deep result is classical and it is usually a central argument in the analysis of positive semigroups, in particular because it is used during the proof of Theorem 5.28. In the next sections, we will develop alternative and more constructive approaches, but we will still make use of the material of this section or its consequences a couple of times in the applications (in particular in Section 9.2 and in the proof of Theorem 11.11). Here, we develop a method that is less abstract and a bit more general than the usual approach, although it remains somewhat technical.

**Lemma 5.4.** Under assumptions (C2), (X2) and the complex Kato's inequality (5.7), we have $\Sigma_P^1(\mathcal{L}) \cap \Delta_{\lambda_1} = \emptyset$, or in other words $\Sigma_P^1(\tilde{\mathcal{L}}) = \Sigma_P(\tilde{\mathcal{L}}) \cap i\mathbb{R}$. Furthermore, for any $\lambda \in \Sigma_P^1(\mathcal{L})$ the associated normalized eigenfunction $f$ satisfies $|f| = f_1$.

*Proof of Lemma 5.4.* Let $\lambda \in \mathbb{C}$ and $f \in D(\mathcal{L}) \setminus \{0\}$ such that $\mathcal{L}f = \lambda f$. By linearity of the operator sign and using (5.2) and Kato's inequality (5.7), we have

$$ (\Re e \lambda) |f| = \Re e[\lambda(\operatorname{sign} f) f] = \Re e(\operatorname{sign} f)(\lambda f) = \Re e(\operatorname{sign} f) \mathcal{L}f \le \mathcal{L}|f|. $$

Testing this inequality against $\phi_1 \gg 0$, we get

$$ (\Re e \lambda) \langle \phi_1, |f| \rangle \le \lambda_1 \langle \phi_1, |f| \rangle $$

and so $\Re e \lambda \le \lambda_1$. This ensures that $\Sigma_P^1(\mathcal{L}) \cap \Delta_{\lambda_1} = \emptyset$. Now if $\lambda \in \Sigma_P^1(\mathcal{L})$, which imposes that $\Re e \lambda = \lambda_1$, we deduce by the duality argument introduced during the proof of Lemma 4.17, that the above inequality $\lambda_1|f| \le \mathcal{L}|f|$ must be an equality. We conclude thanks to the simplicity of $\lambda_1$. $\square$
**Definition 5.8.** We say that A satisfies the “reverse strong positivity condition” for a subclass of vectors $C \subset X$ if for any $f \in C$

$$ (5.9) \qquad |Af| = A|f| \quad \text{implies} \quad \exists u \in \mathbb{S}^1, Af = uA|f|. $$

We start observing that $A > 0$ (as defined in Section 4.2) implies the strict positivity for non-signed vectors in $X_{\mathbb{R}}$.

**Lemma 5.9.** Consider an operator $A > 0$ and assume $X$ is reflexive. For $f \in X_{\mathbb{R}}$ such that $\pm f \notin X_+$, there holds

$$ |Af| \ll A|f|. $$

*Proof of Lemma 5.9.* Let us consider $f \in X_{\mathbb{R}}$ such that $f_{\pm} \neq 0$. We claim that $|Af| \ll A|f|$. Observing that

$$ Af_+ = Af + Af_- \geq Af, $$

we deduce $Af_+ \geq (Af)_+$, and similarly $Af_- \geq (Af)_-$. We first consider the case $(Af)_+ > 0$. For $\phi \gg 0$, we have

$$ 0 < \langle (Af)_+, \phi \rangle = \sup_{0 \leq \psi \leq \phi} \langle Af, \psi \rangle = \langle Af, \psi^* \rangle = \langle f, A^* \psi^* \rangle, $$

for some $0 \leq \psi^* \leq \phi$, where we have used the very definition of $X_{++}$, the definition of $(Af)_+$ as an element of $X''$ and that $B_{X'}$ is compact for the weakly * topology $\sigma(X', X)$. We deduce in particular that $\psi^* \neq 0$, so that $\psi^* > 0$ and finally $A^*\psi^* \gg 0$ because $A^* > 0$ (as an elementary consequence of the fact that $A > 0$ listed in Section 4.2). We deduce

$$ \langle (Af)_+, \phi \rangle = \langle f, A^* \psi^* \rangle < \langle f_+, A^* \psi^* \rangle = \langle Af_+, \psi^* \rangle \leq \langle Af_+, \phi \rangle $$

where for the strict inequality we have first used the assumption $f_- \neq 0$ and next elementary arguments. We thus have $(Af)_+ < Af_+$. Similarly, we establish $(Af)_- < Af_-$ when $(Af)_- > 0$. As a conclusion, in the three cases $Af = 0$, $(Af)_+ \neq 0$ and $(Af)_- \neq 0$, we have

$$ |Af| = (Af)_+ + (Af)_- \ll Af_+ + Af_- = A|f|, $$

which is the desired strict inequality.

□

We believe that a similar result also holds true for complex vectors in a general Banach lattice framework. We do not try to prove such a statement but we rather establish the corresponding complex version for our examples of concrete Banach spaces in which the definition of the absolute value $|f|$ of a vector $f \in X$ is more tractable.

**Lemma 5.10.** Consider an operator $A > 0$ on $X \subset L_{\text{loc}}^1(E)$ for some locally and $\sigma$-compact metric space $E$. For $f \in X$ such that $|f| \gg 0$, we have

$$ |Af| = A|f| \quad \text{implies} \quad \exists u \in \mathbb{S}^1, f = u|f|, $$

and thus (5.9) holds.

*Proof of Lemma 5.10.* We assume by contradiction that $\forall v \in \mathbb{S}^1$, $|f| > \Re(vf)$, in particular writing $f = g + ih$, $g, h \in X_{\mathbb{R}}$, we have $g, h \in X \setminus \{0\}$. On the one hand, because of $A > 0$ and $A$ is linear, for any $v = e^{i\alpha} \in \mathbb{S}^1$, we have

$$ A|f| \gg A\Re(e^{i\alpha}f)) = \cos\alpha(Ag) - \sin\alpha(Ah). $$

On the other hand, in the Banach lattice we consider here, there exists $\beta : E \to \mathbb{R}$ measurable such that $|Af| = e^{i\beta} Af$ and thus

$$ A|f| = |Af| = \Re|Af| = \cos\beta(Ag) - \sin\beta(Ah), $$

and a contradiction. We have established that there exists $v \in \mathbb{S}^1$ such that $|f| \equiv \Re(vf)$. Now, we have

$$ \sqrt{(\Re(fv))^2 + (\Im(fv))^2} = |fv| = |f| = \Re(fv), $$

which in turn implies $\Im(m(fv) = 0$, since $\Re(fv) \gg 0$. That is here that we use the assumption $|f| \gg 0$ and not only $f \in X_+ \setminus \{0\}$. We conclude that $|f| = fv$ and thus that $f = u|f|$, with $u := v^{-1} \in \mathbb{S}^1$. □
**Theorem 5.5.** Assume (C2), (X2), (X3) and that complex Kato's inequality (5.7) holds true. Denoting $\tilde{\mathcal{L}} = \mathcal{L} - \lambda_1$, the set $\Sigma_P^1(\tilde{\mathcal{L}}) = \Sigma_P(\tilde{\mathcal{L}}) \cap i\mathbb{R}$ is an additive subgroup of $i\mathbb{R}$ and $\dim N(\tilde{\mathcal{L}} - i\alpha)^k = 1$ for any $i\alpha \in \Sigma_P^1(\tilde{\mathcal{L}})$ and $k \ge 1$.

Theorem 5.5 is similar but more general than [15, C-III, Cor. 2.12] and [41, Prop. 14.15]. Our proof is also very similar to the proof of [41, Prop. 14.15]. However, it is more direct and avoid the use of the $C(K)$ algebra and Kakutani's Theorem [226] (see also [263, Thm. 2.1.3]).

*Proof of Theorem 5.5.* We split the proof into three steps.

**Step 1.** We consider $f$ associated to an eigenvalue $i\alpha \in \Sigma_P(\tilde{\mathcal{L}})\setminus\{0\}$, and we define

$$T(t) := (\operatorname{sign} f)e^{-i\alpha t}\tilde{S}(t)(\operatorname{sign} \bar{f}).$$

Observing that $\tilde{S}(t)f = e^{i\alpha t}f$, we have

$$T(t)|f| = (\operatorname{sign} f)e^{-i\alpha t}\tilde{S}(t)f = (\operatorname{sign} f)f = |f| = \tilde{S}(t)|f|.$$

On the other hand, we have

$$|T(t)g| \le |\tilde{S}(t)(\operatorname{sign} \bar{f})g| \le \tilde{S}(t)|g|, \quad \forall g \in X,$$

which, by positivity of $\tilde{S}(t)$, yields

$$|T(t)g| \le A_f(g)\tilde{S}(t)|f| = A_f(g)|f|, \quad \forall g \in X_f.$$

Because $|f| = f_1 \gg 0$ from Lemma 5.4, we can apply Lemma 5.3 to $|f|$ and $Q := T(t)$. We deduce that $T(t) \ge 0$ on $X_{|f|} = X_f$, and then on $X = \bar{X}_f$. As a consequence, $0 \le T(t)g = |T(t)g| \le \tilde{S}(t)g$ for any $g \ge 0$. In other words, we have $0 \le \tilde{S}(t) - T(t)$ and then $0 \le \tilde{S}(t)^* - T(t)^*$. We must have $\tilde{S}(t)^* - T(t)^* = 0$. Otherwise, there would exist $\psi \in Y_+\setminus\{0\}$ such that $(\tilde{S}(t)^* - T(t)^*)\psi \in Y_+\setminus\{0\}$, and we find a contradiction by computing

$$0 < \langle (\tilde{S}(t)^* - T(t)^*)\psi, f_1 \rangle = \langle \psi, (\tilde{S}(t) - T(t))f_1 \rangle = 0.$$

We have thus established that $\tilde{S}(t) = T(t)$.

**Step 2.** Consider $\alpha, \beta \in \mathbb{R}$ and $f, g \in X\setminus\{0\}$ such that $\tilde{\mathcal{L}}f = i\alpha f$ and $\tilde{\mathcal{L}}g = i\beta g$, and suppose first that $(\operatorname{sign} \bar{f}) : D(\tilde{\mathcal{L}}) \to D(\tilde{\mathcal{L}})$. From Step 1 and the fact that $(\operatorname{sign} \bar{f}) \circ \operatorname{sign} f = I$, for any $h \in D(\tilde{\mathcal{L}})$, we may compute

$$\begin{aligned} \tilde{\mathcal{L}}h &= \lim_{t \to 0} \frac{1}{t}(\tilde{S}(t)h - h) \\ &= (\operatorname{sign} f) \lim_{t \to 0} \frac{1}{t}(e^{-i\alpha t}\tilde{S}(t)(\operatorname{sign} \bar{f})h - (\operatorname{sign} \bar{f})h) \\ &= (\operatorname{sign} f)(\tilde{\mathcal{L}} - i\alpha)(\operatorname{sign} \bar{f})h, \end{aligned}$$

or in other words $\tilde{\mathcal{L}} - i\alpha = (\operatorname{sign} \bar{f})\tilde{\mathcal{L}}(\operatorname{sign} f)$. We have similarly $\tilde{\mathcal{L}} - i\beta = (\operatorname{sign} \bar{g})\tilde{\mathcal{L}}(\operatorname{sign} g)$. Both equations together imply

$$\tilde{\mathcal{L}} - i(\alpha + \beta) = (\operatorname{sign} \bar{f})(\operatorname{sign} \bar{g})\tilde{\mathcal{L}}(\operatorname{sign} g)(\operatorname{sign} f).$$

Defining $h := (\operatorname{sign} \bar{f})(\operatorname{sign} \bar{g})f_1$, so that $(\operatorname{sign} g)(\operatorname{sign} f)h = f_1$, we get $\tilde{\mathcal{L}}h = i(\alpha + \beta)h$, and finally $i(\alpha + \beta) \in \Sigma_P^1(\tilde{\mathcal{L}})$, so that the additive subgroup structure is established.

When the condition $(\operatorname{sign} \bar{f}) : D(\tilde{\mathcal{L}}) \to D(\tilde{\mathcal{L}})$ is not granted, we modify the above argument by using a resolvent approach. For some $\lambda > 0$ and thanks to (2.13), we compute

$$\begin{aligned} (\lambda - \tilde{\mathcal{L}})^{-1} &= \int_{0}^{\infty} e^{-\lambda t} \tilde{S}(t) dt \\ &= (\operatorname{sign} f) \int_{0}^{\infty} e^{-(\lambda+i\alpha)t} \tilde{S}(t) dt (\operatorname{sign} \bar{f}) \\ &= (\operatorname{sign} f)(\lambda + i\alpha - \tilde{\mathcal{L}})^{-1}(\operatorname{sign} \bar{f}). \end{aligned}$$

Repeating the argument, we obtain

$$(\lambda + i(\alpha + \beta) - \tilde{\mathcal{L}})^{-1} = (\operatorname{sign} \bar{f})(\operatorname{sign} \bar{g})(\lambda - \tilde{\mathcal{L}})^{-1}(\operatorname{sign} g)(\operatorname{sign} f).$$
A similar result also holds in the Radon space of measures. For a measurable space $(E, \mathcal{E})$, we call transition kernel, a mapping $Q : E \times \mathcal{E} \to [0, \infty]$ such that

(i) $\forall B \in \mathcal{E}, x \mapsto Q(x, B)$ is measurable;

(ii) $\forall x \in E, B \mapsto Q(x, B)$ is a measure.

We recall the classical Markov-Riesz representation theorem which claims that for any linear and positive operator $B : C_0(E) \to C_0(E)$ there holds

$$ (B\phi)(x) = \int_E \phi(x)Q(x, dy), \quad \forall \phi \in C_0(E), $$

for a transition kernel $Q$ such that in the condition (i) above the mapping is furthermore continuous.

**Lemma 5.11.** Consider an operator $A > 0$ in $X = M^1 = M^1(E, \mathcal{E})$, for some Borel space $(E, \mathcal{E})$ where $E$ is a locally and $\sigma$-compact metric set. For $f \in X$ such that $|f| \gg 0$, we have (5.9).

*Proof of Lemma 5.11.* By definition, the operator $A$ is the dual of a positive operator on $C_0(E)$. Using the representation formula recalled above for that adjoint operator, we get

$$ (Af)(dy) = \int_E Q(x, dy)f(dx), \quad \forall f \in M^1, $$

for a transition kernel $Q$. We deduce that

$$ \langle Af, \phi \rangle = \int_{E \times E} \phi(y)Q(x, dy)f(dx), $$

for any bounded Borel function $\phi : E \to \mathbb{C}$. In particular, the strict positivity $A > 0$ translates as $Q(x, \cdot) \gg 0$ in $M^1$ for any $x \in E$. We fix now $\phi \in C_0(E)$ such that $\phi \gg 0$ and $f \in M^1$ such that $|f| \gg 0$, and we observe that from the Radon-Nikodym theorem, there exist two measurable functions $\alpha, \beta : E \to [0, 2\pi)$ such that $f = e^{i\alpha}|f|$ and $Af = e^{i\beta}|Af|$. We next compute

$$ \begin{aligned} \langle A|f| - |Af|, \phi \rangle &= \Re e \{ \langle A|f|, \phi \rangle - \langle Af, e^{-i\beta}\phi \rangle \} \\ &= \int_{E \times E} \Re e \{1 - e^{i(\alpha(x) - \beta(y))}\} \phi(y)Q(x, dy)|f|(dx) \\ &= \int_{E \times E} \{1 - \cos(\alpha(x) - \beta(y))\} \phi(y)Q(x, dy)|f|(dx). \end{aligned} $$

In the case of equality $A|f| = |Af|$, we must have $1 - \cos(\alpha(y) - \beta(x)) = 0$ for $\mu$-a.e. $x \in E$ and $|f|$-a.e. $y \in \text{supp } f = E$. We deduce that $\beta$ is a constant function, so that $Af = e^{i\beta}|Af| = uA|f|$, for the constant $u = e^{i\beta} \in \mathbb{S}^1$. $\square$

## The reverse Kato's inequality condition

We recall that it has been stated in section 4.1 that the generator $\mathcal{L}$ of a positive semigroup $S(t)$ satisfies Kato's inequality (4.5) which in a complex framework writes

$$ (5.10) \qquad \forall f \in X, \quad \Re(\operatorname{sign} f) \mathcal{L} f \le \mathcal{L} |f|. $$

We also observe that if $f = u|f|$ for some $u \in \mathbb{S}^1$, we have

$$ \Re(\operatorname{sign} f) \mathcal{L} f = \operatorname{sign}(u^{-1} f) \mathcal{L}(u^{-1} f) = \mathcal{L}|f|, $$

which is the case of equality in Kato's inequality.

**Definition 5.12.** We say that $\mathcal{L}$ satisfies a “reverse Kato's inequality condition” for a class of vectors $C \subset D(\mathcal{L})$ if for any $f \in C$ the case of equality in Kato's inequality

$$ \mathcal{L}|f| = \Re(\operatorname{sign} f) \mathcal{L} f $$

implies

$$ \exists u \in \mathbb{C}, f = u|f|. $$

In some situation, we may prove directly that the “reverse Kato's inequality condition” holds by reasoning at the level of the operator $\mathcal{L}$, see for instance [231, Proof of Theorem 5.1]. That is also a consequence of the strong positivity of the semigroup as we see below.
Applying that last identity to the vector $h = (\operatorname{sign} \bar{f})(\operatorname{sign} \bar{g})f_1$ and using that $(\lambda - \tilde{\mathcal{L}})^{-1}f_1 = \lambda^{-1}f_1$, we deduce $(\lambda + i(\alpha + \beta) - \tilde{\mathcal{L}})^{-1}h = \lambda^{-1}h$. In other words, we have again $\tilde{\mathcal{L}}h = i(\alpha + \beta)h$, and we conclude as before.

**Step 3.** From the fact that $(\operatorname{sign} f)$ is an invertible operator and the equation

$$ (\tilde{\mathcal{L}} - i\alpha)^k = (\operatorname{sign} f)^{-1}(\tilde{\mathcal{L}})^k(\operatorname{sign} f), $$

we see from Theorem 4.13-(ii) that $N(\tilde{\mathcal{L}} - i\alpha)^k = (\operatorname{sign} f)^{-1}N(\tilde{\mathcal{L}})^k = (\operatorname{sign} f)^{-1}\operatorname{Span}f_1$ for any $k \ge 1$, so that its dimension is one. $\square$

Making an additional splitting structure hypothesis as yet introduced in Section 2.2, we may significantly improve the conclusion. We first recall a classical result on the spectrum of an operator which holds under some power compactness assumption on the resolvent.

**Theorem 5.6.** We assume that $\mathcal{L}$ satisfies (H1)-(H2) and the splitting structure (HS1) introduced in Section 2.2 with $W(z) \in K(X)$ for some $N \ge 1$ and any $z \in \Delta_{\kappa_0}$. Then $\Sigma(\mathcal{L}) \cap \Delta_{\kappa_0} \subset \Sigma_d(\mathcal{L})$.

Theorem 5.6 is a consequence of [354, Cor. 1.1]. We also refer to [277, proof of Thm. 3.1] for a possible elementary proof.

A sketch of the proof of Theorem 5.6. Iterating the formula $\mathcal{R}_{\mathcal{L}} = \mathcal{R}_B + \mathcal{R}_B\mathcal{A}\mathcal{R}_{\mathcal{L}}$, we deduce

$$ \mathcal{J}(z)\mathcal{R}_{\mathcal{L}}(z) = \mathcal{V}(z) $$

with $\mathcal{J} := I - (\mathcal{A}\mathcal{R}_B)^N$ and $\mathcal{V} := \mathcal{R}_B + \dots + \mathcal{R}_B(\mathcal{A}\mathcal{R}_B)^{N-1}$. Because $\mathcal{J}$ is holomorphic on $\Delta_{\kappa_0}$, it is a compact perturbation of the identity and $\mathcal{J}(z) \to I$ when $\Re z \to \infty$, one may use the theory of degenerate-meromorphic functions of Ribarič and Vidav [330] (also established independently by Steinberg, see in particular [340, Cor. 1]), and conclude that $\mathcal{J}(z)$ is invertible outside of a discrete set $\mathcal{D}$ of $\Delta_{\kappa_0}$. That implies that $\Sigma(\mathcal{L}) \cap \Delta_{\kappa_0} = \mathcal{D}$ is a discrete set of $\Delta_{\kappa_0}$. On the other hand, thanks to the Fredholm alternative [163], one deduces that the eigenspace associated to each spectral value $\lambda \in \mathcal{D}$ is non zero and finite dimensional, so that $\lambda \in \Sigma_d(\mathcal{L})$. See also [341, 356] for pioneering works in the subject. $\square$

We end this section by a result which gives a more accurate description of the geometry of the boundary spectrum, and is a variant of the classical results [15, C-III, Thm. 3.12], [151, Thm. VI.1.12], [41, Thm. 14.17].

**Theorem 5.7.** Under the assumptions of Theorem 5.5 and Theorem 5.6, we have $\Sigma_P^+(\tilde{\mathcal{L}}) = \Sigma_P^+(\tilde{\mathcal{L}})$, this set is a discrete additive subgroup of $i\mathbb{R}$ and any of its elements is an algebraically simple eigenvalue. More precisely,

- either $\Sigma_P^+(\tilde{\mathcal{L}}) = \{0\}$ and the projection on the first eigenspace (associated to $\lambda_1$) writes

$$ \Pi f := \langle f, \phi_1 \rangle f_1; $$

- or $\Sigma_P^+(\tilde{\mathcal{L}}) = i\alpha\mathbb{Z}$ for some $\alpha > 0$ and there exists a sequence $(g_k, \psi_k)_{k \in \mathbb{Z}}$ such that $\mathcal{L}g_k = (\lambda_1 + ik\alpha)g_k$, $\mathcal{L}^*\psi_k = (\lambda_1 + ik\alpha)\psi_k$, and $\langle g_k, \psi_\ell \rangle = \delta_{k\ell}$.

*Proof of Theorem 5.7.* Because of Lemma 2.8 and (C2), we have $\lambda_1 = s(\mathcal{L})$ so that the principal point spectrum is the boundary point spectrum. Combining Theorem 5.5 and Theorem 5.6, we immediately get that the subgroup $\mathcal{S} := \Sigma_P(\tilde{\mathcal{L}}) \cap i\mathbb{R}$ satisfies $\mathcal{S} \subset \Sigma_d(\mathcal{L})$, it is thus discrete and made of algebraically simple eigenvalues. The first case $\Sigma_P(\tilde{\mathcal{L}}) \cap i\mathbb{R} = \{0\}$ falls yet in the conclusions of Theorem 4.13. $\square$

In the second case, where the boundary spectrum is not trivial, the existence of a projection on the boundary eigenspace $\operatorname{Span}(g_k)_{k \in \mathbb{Z}}$ is ensured by the Jacobs-de Leeuw-Glicksberg theorem provided that $\mathcal{L}$ is the generator of a relatively compact semigroup, see for instance [41, Thm. A.39 and Prop. A.40] and the references therein. We also refer to [228, paragraphs III.6.4 and III.6.5] for very classical results on the projector on the direct sum of eigenspaces associated to eigenvalues belonging to a subset of the spectrum. We can even give an explicit expression of this projection in terms of $(g_k)$ and $(\psi_k)$ under the form of a Fejér type sum, see Theorem 5.25.
**Lemma 5.13.** Consider a semigroup $S$ and its generator $\mathcal{L}$. On the set $C$ of vectors $f \in X\setminus\{0\}$ such that

$$ (5.11) \qquad \exists \lambda \in \mathbb{C}, \quad \mathcal{L}f = \lambda f, \quad \mathcal{L}|f| = (\Re e \lambda)|f|, $$

there is equivalence between:

(i) $S(t)$ satisfies the “reverse strong positivity condition” for some (and thus any) $t > 0$;

(ii) $\mathcal{L}$ satisfies the “reverse Kato’s inequality condition”.

**Remark 5.14.** When $X \subset L_{\text{loc}}^1$, the “reverse Kato’s inequality condition” (ii) implies the “reverse strong positivity condition” (i) on the class $C$ of vectors such that $f \in D(\mathcal{L})$, $0 \ll |f| \in D(\mathcal{L})$. Assume indeed that $\mathcal{L}$ satisfies (ii) and consider $f \in C$ such that $|S_t f| = S_t |f|$ for any $t \ge 0$. By differentiating, we get

$$ (5.12) \qquad (\text{sign } f)\mathcal{L}f = \mathcal{L}|f|. $$

From the “reverse Kato’s inequality condition”, we deduce that $f = u|f|$ for some $u \in \mathbb{S}^1$, so that (i) holds.

**Proof of Lemma 5.13.** In what follows, we fix $f \in X\setminus\{0\}$ such that (5.11) holds, and we compute

$$ (5.13) \qquad \Re(\text{sign } f)\mathcal{L}f = \Re(\text{sign } f)(\lambda f) = (\Re e \lambda)|f| = \mathcal{L}|f|. $$

For any $t > 0$, we also have $S_t f = e^{\lambda t} f$, $S_t |f| = e^{\Re e \lambda t} |f|$, and thus

$$ (5.14) \qquad |S_t f| = S_t |f|. $$

Assuming the “reverse Kato’s inequality condition”, we deduce from (5.13) that $f = u|f|$ for some $u \in \mathbb{S}^1$, thus $S_t f = uS_t |f|$ for some $u \in \mathbb{S}^1$, which is the conclusion of the “reverse strong positivity condition” when (5.14) holds.

On the other way round, assuming the “reverse strong positivity condition” for some $T > 0$, we deduce from (5.14) for $T > 0$ that there exists $v \in \mathbb{S}^1$ such that

$$ e^{\lambda T} f = S_T f = v S_T |f| = v e^{\Re e \lambda T} |f|. $$

That implies that $f = u|f|$ with $u = v e^{-i(\Re m \lambda)T}$, which is nothing but the conclusion of the “reverse Kato’s inequality condition” when (5.13) holds. $\square$

We summarize the material developed above in the following main result of the section.

**Theorem 5.15.** Assume that $S$ is a positive semigroup on $X$ with $X \subset L_{\text{loc}}^1(E)$ or $X = M^1(E)$ for some locally and $\sigma$-compact metric space $E$ and denote by $(E_k)$ a sequence of increasing compact sets such that $E = \lim E_k$. We furthermore assume that for any $k \ge 1$ there exists $T > 0$ such that $S_T$ is strictly positive on $E_k$, in the sense that

$$ (5.15) \qquad \forall f \in X_+\setminus\{0\}, \quad f|_{E_k} \neq 0, \quad \forall \phi \in X'_+\setminus\{0\}, \quad \text{supp } \phi \subset E_k, \quad \langle S_T f, \phi \rangle > 0. $$

Then $\mathcal{L}$ satisfies the “reverse Kato’s inequality condition” on the set $C$ of eigenvectors introduced in Lemma 5.13.

**Proof of Theorem 5.15.** Let us consider $f \in X\setminus\{0\}$ such that (5.11) holds, so that $S_t|f| = e^{(\Re e \lambda)t}|f|$ for any $t \ge 0$. On the one hand, we may fix $k \ge 1$ such that $|f| \neq 0$ on $E_k$. Then for any $\ell \ge k$, there exists $T_\ell > 0$ such that (5.15) holds, so that

$$ e^{(\Re e \lambda)T_\ell} \langle |f|, \phi \rangle = \langle S_{T_\ell} |f|, \phi \rangle > 0, $$

for any $\phi \in Y_+\setminus\{0\}$, $\text{supp } \phi \subset E_\ell$. That implies $\langle |f|, \phi \rangle > 0$ on for any $\phi \in Y_+\setminus\{0\}$, and thus $|f| \gg 0$. Next, as in the proof of Lemma 5.13, we observe that

$$ |S_{T_\ell} f| = S_{T_\ell} |f|, \quad \forall \ell \ge k. $$

Repeating the proof of Lemma 5.10 and Lemma 5.11, we deduce that there exists $u_\ell \in \mathbb{S}^1$ such that $S_{T_\ell} f = u_\ell S_{T_\ell} |f|$ on $E_\ell$, or equivalently there exists $v_\ell \in \mathbb{S}^1$ such that $f = v_\ell|f|$ on $E_\ell$, with $v_\ell := u_\ell e^{-i(\Re m \lambda)T_\ell}$. Because $E_\ell \supset E_1$, we have established that $f = v_1|f|$ on $E$ which is the conclusion of the “reverse Kato’s inequality condition” when (5.11) holds. $\square$
### 5.3. Stronger positivity.

In order to go one step further and establish the triviality of the principal point spectrum, we need to reinforce the positivity of the semigroup or its generator. One possible condition is based on the following notion.

**The reverse strong positivity condition**

For $A \in \mathcal{B}(X) \cap \mathcal{B}(X_+)$, we recall that from (2.6), we have

$$ (5.8) \qquad |Af| \le A|f|, \quad \forall f \in X, $$

and we observe that the above inequality is an equality when $Af = uA|f|$ for some $u \in \mathbb{S}^1$. We focus now on the case of equality in (5.8).

**Definition 5.8.** We say that A satisfies the “reverse strong positivity condition” for a subclass of vectors $C \subset X$ if for any $f \in C$

$$ (5.9) \qquad |Af| = A|f| \quad \text{implies} \quad \exists u \in \mathbb{S}^1, Af = uA|f|. $$

We start observing that $A > 0$ (as defined in Section 4.2) implies the strict positivity for non-signed vectors in $X_\mathbb{R}$.

**Lemma 5.9.** Consider an operator $A > 0$ and assume $X$ is reflexive. For $f \in X_\mathbb{R}$ such that $\pm f \notin X_+$, there holds

$$ |Af| \ll A|f|. $$

*Proof of Lemma 5.9.* Let us consider $f \in X_\mathbb{R}$ such that $f_\pm \neq 0$. We claim that $|Af| \ll A|f|$. Observing that

$$ Af_+ = Af + Af_- \ge Af, $$

we deduce $Af_+ \ge (Af)_+$, and similarly $Af_- \ge (Af)_-$. We first consider the case $(Af)_+ > 0$. For $\phi \gg 0$, we have

$$ 0 < \langle (Af)_+, \phi \rangle = \sup_{0 \le \psi \le \phi} \langle Af, \psi \rangle = \langle Af, \psi^* \rangle = \langle f, A^* \psi^* \rangle, $$

for some $0 \le \psi^* \le \phi$, where we have used the very definition of $X_+$, the definition of $(Af)_+$ as an element of $X''$ and that $B_{X'}$ is compact for the weakly * topology $\sigma(X', X)$. We deduce in particular that $\psi^* \ne 0$, so that $\psi^* > 0$ and finally $A^*\psi^* \gg 0$ because $A^* > 0$ (as an elementary consequence of the fact that $A > 0$ listed in Section 4.2). We deduce

$$ \langle (Af)_+, \phi \rangle = \langle f, A^* \psi^* \rangle < \langle f_+, A^* \psi^* \rangle = \langle Af_+, \psi^* \rangle \le \langle Af_+, \phi \rangle $$

where for the strict inequality we have first used the assumption $f_- \ne 0$ and next elementary arguments. We thus have $(Af)_+ < Af_+$. Similarly, we establish $(Af)_- < Af_-$ when $(Af)_- > 0$. As a conclusion, in the three cases $Af = 0$, $(Af)_+ \ne 0$ and $(Af)_- \ne 0$, we have

$$ |Af| = (Af)_+ + (Af)_- \ll Af_+ + Af_- = A|f|, $$

which is the desired strict inequality. $\square$

<figure><img src="image_5.png" /></figure>

We believe that a similar result also holds true for complex vectors in a general Banach lattice framework. We do not try to prove such a statement but we rather establish the corresponding complex version for our examples of concrete Banach spaces in which the definition of the absolute value $|f|$ of a vector $f \in X$ is more tractable.

**Lemma 5.10.** Consider an operator $A > 0$ on $X \subset L_{\text{loc}}^1(E)$ for some locally and $\sigma$-compact metric space $E$. For $f \in X$ such that $|f| \gg 0$, we have

$$ |Af| = A|f| \quad \text{implies} \quad \exists u \in \mathbb{S}^1, f = u|f|, $$

and thus (5.9) holds.

*Proof of Lemma 5.10.* We assume by contradiction that $\forall v \in \mathbb{S}^1$, $|v| > \Re(vf)$, in particular writing $f = g + ih$, $g, h \in X_\mathbb{R}$, we have $g, h \in X \setminus \{0\}$. On the one hand, because of $A > 0$ and $A$ is linear, for any $v = e^{i\alpha} \in \mathbb{S}^1$, we have

$$ A|f| \gg A\Re(e^{i\alpha f})) = \cos \alpha (Ag) - \sin \alpha (Ah). $$
**5.4. On the triviality of the boundary spectrum.** As in section 4.3, we still assume the existence (C1) of a solution $(\lambda_1, f_1, \phi_1) \in \mathbb{R} \times X_+ \times Y_+$ to the first eigenvalue problem (4.1) and that $\mathcal{L}$ enjoys the weak maximum principle (4.16) and Kato's inequalities (4.17) as formulated in condition (H1') as well as the strong maximum principle (H4). Because we deal with complex eigenvalue, we also assume that the **complex** Kato's inequality variant (5.10) holds.

We introduce a first additional assumption:

(H5) the“reverse Kato’s inequality condition” (as defined in Definition 5.12) holds true for the class $\mathcal{C}$ defined in Lemma 5.13: for $f \in X \setminus \{0\}$ such that

$$ (5.16) \qquad \exists \lambda \in \mathbb{C}, \quad \mathcal{L}f = \lambda f, \quad \mathcal{L}|f| = (\Re e \lambda)|f| = \Re e(\operatorname{sign} f)\mathcal{L}f, $$

we have

$$ \exists u \in \mathbb{C}, f = u|f|. $$

On the other hand, we do not need the structure assumption (X3).

We are then able to make a more accurate analyse of the geometry of the spectrum.

**Theorem 5.16.** Consider an unbounded operator $\mathcal{L}$ on a Banach lattice $X$ which satisfy (C2), (H4), (4.17) and (H5). Then the conclusion (S32) about the uniqueness of $\lambda_1$ as the eigenvalue with largest real part holds: $\Sigma_P^+(\mathcal{L}) = \{\lambda_1\}$.

**Remark 5.17.** (1) It is worth emphasizing again that (4.17) is true when $\mathcal{L}$ is the generator of a positive semigroup and that (H5) is true when $S_\mathcal{L}(T)$ satisfies the “reverse strong positivity condition” for some $T > 0$ as a consequence of Lemma 5.13, see also Theorem 5.15.

(2) During the proof we use similar arguments as in [231, Thm. 5.1].

(3) Condition (H5) is reminiscent of PDE arguments as we may find for instance in [231, Proof of Thm. 5.1] or in the discussion in [252, 4th course] about an uniqueness argument due to L. Tartar.

**Proof of Theorem 5.16.** Consider an eigenvalue $\lambda \in \mathbb{C}$ with normalized eigenvector $f \in X \setminus \{0\}$, and more precisely $\langle |f|, \phi_1 \rangle = 1$ and $\mathcal{L}f = \lambda f$. Thanks to the complex Kato's inequality (4.17), we have

$$ (\Re e \lambda)|f| = \Re e \operatorname{sign}(f)(\lambda f) = \Re e \operatorname{sign}(f)(\mathcal{L}f) \le \mathcal{L}|f|. $$

We consider two cases:

When the above inequality is not an equality, we have

$$ (\Re e \lambda)\langle |f|, \phi_1 \rangle < \langle \mathcal{L}|f|, \phi_1 \rangle = \langle |f|, \mathcal{L}^* \phi_1 \rangle = \lambda_1 \langle |f|, \phi_1 \rangle, $$

and thus $\Re e \lambda < \lambda_1$.

When on the contrary the above inequality is an equality, then $|f|$ is a positive eigenvector associated to the eigenvalue $\Re e \lambda$. Because of (H4), we have $|f| \in X_{++}$ and repeating the proof of Lemma 4.17, we get $\Re e \lambda = \lambda_1$. The condition (C2) implies $|f| = f_1$. On the other hand, $f$ satisfies (5.16) and thus $f \in \operatorname{Span}(f_1)$ from assumption (H5), in particular $\lambda = \lambda_1$. $\square$

When $\mathcal{L}$ is the generator of a positive and irreducible semigroup $S$, we may introduce the alternative assumption:

(H5') the semigroup $S$ is aperiodic as defined in (4.13), namely

$$ \forall f \in X_+ \setminus \{0\}, \forall \phi \in Y_+ \setminus \{0\}, \exists T > 0, \forall \tau \ge T \quad \langle S_\tau f, \phi \rangle > 0. $$

**Theorem 5.18.** Let $X$ be Banach lattice in which the property (4.10) holds true. Consider a positive and irreducible semigroup $S$ on $X$ which satisfies the aperiodicity condition (H5') and such that its generator $\mathcal{L}$ satisfies (C2). Then the conclusion (S32) holds: $\Sigma_P^+(\mathcal{L}) = \{\lambda_1\}$.

**Remark 5.19.** It is worth pointing out that since (H5') is stronger than (H4), see the points (2) and (3) in Lemma 4.8, we can use Theorem 4.13 and replace in Theorem 5.18 the assumption that (C2) is satisfied by the assumption that (C1) and (H1') for both $\mathcal{L}$ and $\mathcal{L}^*$ are satisfied, together with the structure assumption (X1) on $X$ and $Y$.
On the other hand, in the Banach lattice we consider here, there exists $\beta : E \to \mathbb{R}$ measurable such that $|Af| = e^{i\beta}Af$ and thus

$$A|f| = |Af| = \Re e|Af| = \cos \beta (Ag) - \sin \beta (Ah),$$

and a contradiction. We have established that there exists $v \in \mathbb{S}^1$ such that $|v| = \Re e(v)$. Now, we have

$$\sqrt{(\Re e(fv))^2 + (\Im m(fv))^2} = |fv| = |f| = \Re e(fv),$$

which in turn implies $\Im m(fv) = 0$, since $\Re e(fv) \gg 0$. That is here that we use the assumption $|f| \gg 0$ and not only $f \in X_+ \setminus \{0\}$. We conclude that $|f| = fv$ and thus that $f = u|f|$, with $u := v^{-1} \in \mathbb{S}^1$. $\square$

A similar result also holds in the Radon space of measures. For a measurable space $(E, \mathcal{E})$, we call transition kernel, a mapping $Q : E \times \mathcal{E} \to [0, \infty]$ such that

$$(i) \quad \forall B \in \mathcal{E}, x \mapsto Q(x, B) \text{ is measurable;}$$

$$(ii) \quad \forall x \in E, B \mapsto Q(x, B) \text{ is a measure.}$$

We recall the classical Markov-Riesz representation theorem which claims that for any linear and positive operator $B : C_0(E) \to C_0(E)$ there holds

$$(B\phi)(x) = \int_E \phi(x)Q(x, dy), \quad \forall \phi \in C_0(E),$$

for a transition kernel $Q$ such that in the condition (i) above the mapping is furthermore continuous.

**Lemma 5.11.** Consider an operator $A > 0$ in $X = M^1 = M^1(E, \mathcal{E})$, for some Borel space $(E, \mathcal{E})$ where $E$ is a locally and $\sigma$-compact metric set. For $f \in X$ such that $|f| \gg 0$, we have (5.9).

*Proof of Lemma 5.11.* By definition, the operator $A$ is the dual of a positive operator on $C_0(E)$. Using the representation formula recalled above for that adjoint operator, we get

$$(Af)(dy) = \int_E Q(x, dy)f(dx), \quad \forall f \in M^1,$$

for a transition kernel $Q$. We deduce that

$$\langle Af, \phi \rangle = \int_{E \times E} \phi(y)Q(x, dy)f(dx),$$

for any bounded Borel function $\phi : E \to \mathbb{C}$. In particular, the strict positivity $A > 0$ translates as $Q(x, \cdot) \gg 0$ in $M^1$ for any $x \in E$. We fix now $\phi \in C_0(E)$ such that $\phi \gg 0$ and $f \in M^1$ such that $|f| \gg 0$, and we observe that from the Radon-Nikodym theorem, there exist two measurable functions $\alpha, \beta : E \to [0, 2\pi)$ such that $f = e^{i\alpha}|f|$ and $Af = e^{i\beta}|Af|$. We next compute

$$\begin{aligned} \langle A|f| - |Af|, \phi \rangle &= \Re e\{\langle A|f|, \phi \rangle - \langle Af, e^{-i\beta}\phi \rangle\} \\ &= \int_{E \times E} \Re e\{1 - e^{i(\alpha(x) - \beta(y))}\}\phi(y)Q(x, dy)|f|(dx) \\ &= \int_{E \times E} \{1 - \cos(\alpha(x) - \beta(y))\}\phi(y)Q(x, dy)|f|(dx). \end{aligned}$$

In the case of equality $A|f| = |Af|$, we must have $1 - \cos(\alpha(y) - \beta(x)) = 0$ for $\mu$-a.e. $x \in E$ and $|f|$-a.e. $y \in \text{supp } f = E$. We deduce that $\beta$ is a constant function, so that $Af = e^{i\beta}|Af| = uA|f|$, for the constant $u = e^{i\beta} \in \mathbb{S}^1$. $\square$

## The reverse Kato's inequality condition

We recall that it has been stated in section 4.1 that the generator $\mathcal{L}$ of a positive semigroup $S(t)$ satisfies Kato's inequality (4.4) which in a complex framework writes

$$\forall f \in X, \quad \Re e(\text{sign} f) \mathcal{L} f \le \mathcal{L} |f|.$$

We also observe that if $f = u|f|$ for some $u \in \mathbb{S}^1$, we have

$$\Re e(\text{sign } f) \mathcal{L} f = \text{sign}(u^{-1}f) \mathcal{L}(u^{-1}f) = \mathcal{L}|f|,$$

which is the case of equality in Kato's inequality.
Proof of Theorem 5.18. We introduce the notations $\tilde{S}_t := S_t e^{-\lambda_1 t}$ and $\tilde{\mathcal{L}} := \mathcal{L} - \lambda_1$. Assume that $f = g + ih \in X$, $g, h \in X_{\mathbb{R}}$, is an eigenfunction associated to the eigenvalue $\lambda = \lambda_1 + i\alpha \in \mathbb{C}$, $\alpha > 0$, so that

$$ \tilde{\mathcal{L}}(g + ih) = i\alpha(g + ih) = \frac{2\pi i}{t_0}(g + ih), $$

for some $t_0 > 0$. On the one hand, because $\alpha \neq 0$, we must have $g \neq 0$ and $h \neq 0$, and because of

$$ \alpha\langle g, \phi_1 \rangle = \langle \tilde{\mathcal{L}}h, \phi_1 \rangle = \langle h, \tilde{\mathcal{L}}^*\phi_1 \rangle = 0, $$

and $\phi_1 \gg 0$, we have $g_+ \neq 0$ and $g_- \neq 0$. As a consequence, and because of (4.10), there exists $\psi \in Y_+ \setminus \{0\}$ such that $\langle g_+, \psi \rangle = 0$. On the other hand, we compute

$$ \tilde{S}_{t_0}(g + ih) = e^{i\alpha t_0}(g + ih) = g + ih, $$

from what we deduce $\tilde{S}_{t_0}g = g$, because $S_t$ is real. On the other hand, because $S_t$ is positive, we have

$$ g_+ = (\tilde{S}_{t_0}g)_+ \le \tilde{S}_{t_0}g_+, $$

and next

$$ \langle \phi_1, g_+ \rangle \le \langle \phi_1, \tilde{S}_{t_0}g_+ \rangle = \langle \tilde{S}_{t_0}^*\phi_1, g_+ \rangle = \langle \phi_1, g_+ \rangle, $$

so that the inequalities are equalities (remind again that $\phi_1 \gg 0$), and thus

$$ \tilde{S}_{t_0}g_+ = g_+. $$

We conclude that

$$ \langle \tilde{S}_{kt_0}g_+, \psi \rangle = \langle g_+, \psi \rangle = 0, \quad \forall k \ge 0, $$

what is in contradiction with (H5'). We have established that $\Sigma_P^+(\mathcal{L}) = \{\lambda_1\}$. $\square$

<figure><img src="image_3.png" /></figure>

We end this section with a third situation where the triviality of the boundary spectrum is an immediate consequence of Theorem 5.5 and Theorem 5.6.

**Theorem 5.20.** (1) We make the same assumptions as in Theorem 5.5 and also that there exists $M > 0$ large enough such that $\lambda - \mathcal{L}$ is invertible in $\mathcal{B}(X)$ for any $\lambda \in \mathbb{C}$, $\Re\lambda = \lambda_1$, $|\lambda| \ge M$. Then $\lambda_1$ is the unique eigenvalue with largest real part as formulated in (S3$_2$).

(2) We furthermore assume that the hypothesis of Theorem 5.6 are met and that $\lambda - \mathcal{L}$ is invertible in $\mathcal{B}(X)$ for any $\lambda \in \mathbb{C}$, $\Re\lambda \ge \lambda_1 - \varepsilon$, $|\lambda| \ge M$. Then a (non constructive) spectral gap (S3$_3$) holds.

We summarize the main results established in this section as follows.

(C3) the first eigentriplet problem (4.1) has a solution $(\lambda_1, f_1, \phi_1)$, furthermore this one is unique, $f_1 \gg 0$, $\phi_1 \gg 0$, $\lambda_1$ is algebraically simples (for both $\mathcal{L}$ and $\mathcal{L}^*$) and $\Sigma_P^+(\mathcal{L}) = \{\lambda_1\}$.

**5.5. Ergodicity.** Thanks to the above analyze, we are able to formulate some convergence results on the trajectories associated to a semigroup. More precisely, assuming the existence and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$ for the generator $\mathcal{L}$ of a semigroup $S$ and still denoting the rescaled semigroup $\tilde{S}_t := e^{-\lambda_1 t} S(t)$, we wish in particular to establish the following ergodic property

(E2) for any $f \in X$, there holds

$$ (5.17) \qquad \tilde{S}_t f \to \langle f, \phi_1 \rangle f_1, \quad \text{as } t \to \infty, $$

in a sense to be specified.

We start with a simple result which take advantage of some dissipativity property of the semigroup formulated by a "reverse positivity condition". We next present some more involved results which use directly the spectral information. It is worth emphasizing that our results in this section do not use any spectral gap property what contrasts with the results we will present in the next section.
**Definition 5.12.** We say that $\mathcal{L}$ satisfies a “reverse Kato’s inequality condition” for a class of vectors $C \subset D(\mathcal{L})$ if for any $f \in C$ the case of equality in Kato’s inequality

$$ \mathcal{L}|f| = \Re(\mathrm{sign} f)\mathcal{L}f $$

implies

$$ \exists u \in \mathbb{C}, f = u|f|. $$

In some situation, we may prove directly that the “reverse Kato’s inequality condition” holds by reasoning at the level of the operator $\mathcal{L}$, see for instance [230, Proof of Thm. 5.1]. That is also a consequence of the strong positivity of the semigroup as we see below.

**Lemma 5.13.** Consider a semigroup $S$ and its generator $\mathcal{L}$. On the set $C$ of vectors $f \in X\setminus\{0\}$ such that

$$ (5.10) \qquad \exists \lambda \in \mathbb{C}, \quad \mathcal{L}f = \lambda f, \quad \mathcal{L}|f| = (\Re e \lambda)|f|, $$

there is equivalence between:

(i) $S(t)$ satisfies the “reverse strong positivity condition” for some (and thus any) $t > 0$ as stated in Definition 5.8;

(ii) $\mathcal{L}$ satisfies the above “reverse Kato’s inequality condition”.

**Remark 5.14.** When $X \subset L_{\mathrm{loc}}^1$, the “reverse Kato’s inequality condition” (ii) implies the “reverse strong positivity condition” (i) on the class $C$ of vectors such that $f \in D(\mathcal{L})$, $0 \ll |f| \in D(\mathcal{L})$. Assume indeed that $\mathcal{L}$ satisfies (ii) and consider $f \in C$ such that $|S_t f| = S_t|f|$ for any $t \ge 0$. By differentiating, we get

$$ (5.11) \qquad (\mathrm{sign} f)\mathcal{L}f = \mathcal{L}|f|. $$

From the “reverse Kato’s inequality condition”, we deduce that $f = u|f|$ for some $u \in \mathbb{S}^1$, so that (i) holds.

**Proof of Lemma 5.13.** In what follows, we fix $f \in X\setminus\{0\}$ such that (5.10) holds, and we compute

$$ (5.12) \qquad \Re(\mathrm{sign} f)\mathcal{L}f = \Re(\mathrm{sign} f)(\lambda f) = (\Re e \lambda)|f| = \mathcal{L}|f|. $$

For any $t > 0$, we also have $S_t f = e^{\lambda t} f$, $S_t|f| = e^{\Re e \lambda t}|f|$, and thus

$$ (5.13) \qquad |S_t f| = S_t|f|. $$

Assuming the “reverse Kato’s inequality condition”, we deduce from (5.12) that $f = u|f|$ for some $u \in \mathbb{S}^1$, thus $S_t f = uS_t|f|$ for some $u \in \mathbb{S}^1$, which is the conclusion of the “reverse strong positivity condition” when (5.13) holds.

On the other way round, assuming the “reverse strong positivity condition” for some $T > 0$, we deduce from (5.13) for $T > 0$ that there exists $v \in \mathbb{S}^1$ such that

$$ e^{\lambda T} f = S_T f = v S_T |f| = v e^{\Re e \lambda T} |f|. $$

That implies that $f = u|f|$ with $u = v e^{-i(\Re e \lambda)T}$, which is nothing but the conclusion of the “reverse Kato’s inequality condition” when (5.12) holds. $\square$

We summarize the material developed above in the following main result of the section.

**Theorem 5.15.** Assume that $S$ is a positive semigroup on $X$ with $X \subset L_{\mathrm{loc}}^1(E)$ or $X = M^1(E)$ for some locally and $\sigma$-compact metric space $E$ and denote by $(E_k)$ a sequence of increasing compact sets such that $E = \lim E_k$. We furthermore assume that for any $k \ge 1$ there exists $T > 0$ such that $S_T$ is strictly positive on $E_k$, in the sense that

$$ (5.14) \qquad \forall f \in X_+\setminus\{0\}, f|_{E_k} \ne 0, \forall \phi \in X'_+\setminus\{0\}, \mathrm{supp} \phi \subset E_k, \langle S_T f, \phi \rangle > 0. $$

Then $\mathcal{L}$ satisfies the “reverse Kato’s inequality condition” on the set $C$ of eigenvectors introduced in Lemma 5.13.

**Proof of Theorem 5.15.** Let us consider $f \in X\setminus\{0\}$ such that (5.10) holds, so that $S_t|f| = e^{(\Re e \lambda)t}|f|$ for any $t \ge 0$. On the one hand, we may fix $k \ge 1$ such that $|f| \ne 0$ on $E_k$. Then for any $\ell \ge k$, there exists $T_\ell > 0$ such that (5.14) holds, so that

$$ e^{(\Re e \lambda)T_\ell} \langle |f|, \phi \rangle = \langle S_{T_\ell} |f|, \phi \rangle > 0, $$
**Theorem 5.21.** Consider a positive semigroup S on a Banach lattice X such that its generator $\mathcal{L}$ enjoys the conclusions (C2) of existence, uniqueness and strict positivity of the first eigentriplet $(\lambda_1, f_1, \phi_1)$ and let us set $\tilde{S}_t := e^{-\lambda_1 t} S_t$. We denote $\mathcal{X}$ the space X endowed with the norm $[\cdot]$, with $[f] := \langle |f|, \phi_1 \rangle$. Assume furthermore that

(1) for any $f \in X$, the trajectory $(\tilde{S}_t f)_{t \ge 0}$ is continuous in $\mathcal{X}$ and belongs to a compact set of a normed space $\mathcal{X}_1$, with $\mathcal{X}_1 \subset \mathcal{X}$;

(2) $(S_t)$ satisfies the reverse positivity condition for semigroups

$$ (5.18) \qquad |S_t f| = S_t |f|, \quad \forall t > 0, \quad \text{implies} \quad \exists T > 0, \exists u_T \in \mathbb{S}^1, S_T f = u_T S_T |f|. $$

Then, the ergodicity property (E2) holds in the sense of the norm of $\mathcal{X}_1$.

Let us comment on hypotheses made in the statement of Theorem 5.21. Hypothesis (1) can be obtained as a consequence of a Lyapunov (or growth) condition reminiscent of the structure condition (HS3) introduced in Section 3.3 and an irreducibility condition. Typically, we assume first

$$ \|\tilde{S}(t)f\| \le M\|f\| + K \sup_{0 \le \tau \le t} [\tilde{S}(\tau)f]_0, $$

with $[g]_0 := \langle |g|, \psi_0 \rangle$, $\psi_0 \in Y_+ \setminus \{0\}$, what can be established under the very general condition (ii) of Theorem 3.4. Next we need to be able to prove that $\psi_0 \le r\phi_1$ for some $r > 0$. In concrete situations, we may take $\psi_0$ with compact support and then the above inequality is a consequence of the standard strong maximum principle. We deduce

$$ \begin{aligned} \|\tilde{S}(t)f\| & \le M\|f\| + Kr \sup_{0 \le \tau \le t} \langle |\tilde{S}(\tau)f|, \phi_1 \rangle \\ & \le M\|f\| + Kr \sup_{0 \le \tau \le t} \langle \tilde{S}(\tau)|f|, \phi_1 \rangle \\ & = M\|f\| + Kr \langle |f|, \phi_1 \rangle, \end{aligned} $$

so that $(\tilde{S}_t)$ is bounded. The hypothesis (1) is in fact a bit more demanding, but also quite natural. Assume that $S_{\mathcal{L}}$ enjoys the splitting structure introduced in section 3.1 and section 3.3, so that

$$ (5.19) \qquad \tilde{S} = V + K, $$

with

$$ V := \tilde{S}_{\mathcal{B}} + \dots + (\tilde{S}_{\mathcal{B}}\mathcal{A})^{*(N-1)} * \tilde{S}_{\mathcal{B}}, \quad K := (\tilde{S}_{\mathcal{B}}\mathcal{A})^{*(N)} * \tilde{S}, \quad \tilde{S}_{\mathcal{B}}(t) := e^{-\lambda_1 t} S_{\mathcal{B}}(t). $$

In some applications, we typically have

$$ \|V(t)f_0\| \le \Theta(t)\|f_0\|, \quad \|(\tilde{S}_{\mathcal{B}}\mathcal{A})^{*(N)}\|_{\mathcal{B}(\mathcal{X}, \mathcal{X}_1)} \le \Theta $$

with $\Theta \in L^1(\mathbb{R}_+) \cap C_0(\mathbb{R}_+)$, $\mathcal{X}_1 \subset \mathcal{X}$ compact. In that situation, we deduce (1).

*Proof of Theorem 5.21.* We fix $f \in X$ and without loss of generality, we may assume that $\langle f, \phi_1 \rangle = 0$. We observe that

$$ (5.20) \qquad \langle |\tilde{S}_t f|, \phi_1 \rangle = \langle |\tilde{S}_{t-s} \tilde{S}_s f|, \phi_1 \rangle \le \langle \tilde{S}_{t-s} |\tilde{S}_s f|, \phi_1 \rangle = \langle |\tilde{S}_s f|, \phi_1 \rangle, $$

for any $t \ge s$. We deduce that $(\tilde{S}_t)$ is a dynamical system with compact trajectoires in $\mathcal{X}_1$ and $\mathcal{H}(f) := \langle |f|, \phi_1 \rangle$ is a Lyapunov functional. As a consequence, from the La Salle invariance principle, we have

$$ (5.21) \qquad \inf_{g \in \omega_{\mathcal{H}}} \langle |\tilde{S}_t f - g|, \phi_1 \rangle \to 0 \quad \text{as} \quad t \to \infty, $$

with

$$ (5.22) \qquad \omega_{\mathcal{H}} := \{g \in X; \langle g, \phi_1 \rangle = 0, \forall t \in \mathbb{R}, \mathcal{H}(\tilde{S}_t g) = \inf_{s>0} \mathcal{H}(\tilde{S}_s f)\}. $$

We next characterize $\omega_{\mathcal{H}}$. Picking up $g \in \omega_{\mathcal{H}}$, we observe that

$$ \langle |\tilde{S}_t g|, \phi_1 \rangle = \langle |g|, \phi_1 \rangle = \langle |g|, \tilde{S}_t^* \phi_1 \rangle = \langle \tilde{S}_t |g|, \phi_1 \rangle, \quad \forall t \ge 0, $$

so that

$$ \langle \tilde{S}_t |g| - |\tilde{S}_t g|, \phi_1 \rangle = 0, \quad \forall t \ge 0. $$

In particular, using that $|\tilde{S}_t g| \le \tilde{S}_t |g|$, we have

$$ (5.23) \qquad \tilde{S}_t |g| = |\tilde{S}_t g|, \quad \forall t \ge 0. $$
for any $\phi \in Y_+\setminus\{0\}$, $\text{supp}\phi \subset E_\ell$. That implies $\langle |f|, \phi \rangle > 0$ on for any $\phi \in Y_+\setminus\{0\}$, and thus $|f| \gg 0$. Next, as in the proof of Lemma 5.13, we observe that

$$|S_{T_\ell} f| = S_{T_\ell} |f|, \quad \forall \ell \ge k.$$

Repeating the proof of Lemma 5.10 and Lemma 5.11, we deduce that there exists $u_\ell \in \mathbb{S}^1$ such that $S_{T_\ell} f = u_\ell S_{T_\ell} |f|$ on $E_\ell$, or equivalently there exists $v_\ell \in \mathbb{S}^1$ such that $f = v_\ell |f|$ on $E_\ell$, with $v_\ell := u_\ell e^{-i(\Im\lambda)T_\ell}$. Because $E_\ell \supset E_1$, we have established that $f = v_1 |f|$ on $E$ which is the conclusion of the “reverse Kato’s inequality condition” when (5.10) holds. $\square$

**5.4. On the triviality of the principal point spectrum.** We assume here the existence (C2) of a unique solution $(\lambda_1, f_1, \phi_1) \in \mathbb{R} \times X_{++} \times Y_{++}$ to the first eigenvalue problem (1.1)-(1.2) and that $\mathcal{L}$ enjoys the weak maximum principle (4.15) and Kato's inequalities (4.16) as formulated in condition (H1') as well as the strong maximum principle (H4). Because we deal with complex eigenvalue, we also assume that the **complex** Kato's inequality variant (5.7) holds.

We introduce a first additional assumption:

(H5) the “reverse Kato’s inequality condition” (as defined in Definition 5.12) holds true for the class $\mathcal{C}$ defined in Lemma 5.13: for $f \in X\setminus\{0\}$ such that

$$(5.15) \qquad \exists \lambda \in \mathbb{C}, \quad \mathcal{L}f = \lambda f, \quad \mathcal{L}|f| = (\Re\lambda)|f| = \Re\epsilon(\operatorname{sign}f)\mathcal{L}f,$$

we have

$$\exists u \in \mathbb{C}, f = u|f|.$$

On the other hand, we do not need the structure assumption (X3).

We are then able to make a more accurate analysis of the geometry of the spectrum.

**Theorem 5.16.** Consider an unbounded operator $\mathcal{L}$ on a Banach lattice $X$ which satisfy (X1), (X2), (C2), (H4), (5.7) and (H5). Then $\lambda_1$ is the unique eigenvalue of $\mathcal{L}$ with largest real part: $\Sigma_+^1(\mathcal{L}) = \{\lambda_1\}$.

**Remark 5.17.** (1) It is worth emphasizing again that (5.7) is true when $\mathcal{L}$ is the generator of a positive semigroup and that (H5) is true when $S_{\mathcal{L}}(T)$ satisfies the “reverse strong positivity condition” for some $T > 0$ as a consequence of Lemma 5.13, see also Theorem 5.15.

(2) During the proof we use similar arguments as in [230, Thm. 5.1].

(3) Condition (H5) is reminiscent of PDE arguments as we may find for instance in [230, Proof of Thm. 5.1] or in the discussion in [251, 4th course] about an uniqueness argument due to L. Tartar.

**Proof of Theorem 5.16.** Consider an eigenvalue $\lambda \in \mathbb{C}$ with normalized eigenvector $f \in X\setminus\{0\}$, and more precisely $\langle |f|, \phi_1 \rangle = 1$ and $\mathcal{L}f = \lambda f$. Thanks to the complex Kato's inequality (5.7), we have

$$(\Re\lambda)|f| = \Re\epsilon(\operatorname{sign}(f))(\lambda f) = \Re\epsilon(\operatorname{sign}(f))(\mathcal{L}f) \le \mathcal{L}|f|.$$

We consider two cases:

When the above inequality is not an equality, we have

$$(\Re\lambda)\langle |f|, \phi_1 \rangle < \langle \mathcal{L}|f|, \phi_1 \rangle = \langle |f|, \mathcal{L}^*\phi_1 \rangle = \lambda_1 \langle |f|, \phi_1 \rangle,$$

and thus $\Re\lambda < \lambda_1$.

When on the contrary the above inequality is an equality, then $|f|$ is a positive eigenvector associated to the eigenvalue $\Re\lambda$. Because of (H4), we have $|f| \in X_{++}$ and we get $\Re\lambda = \lambda_1$ thanks to Lemma 4.16. The condition (C2) implies $|f| \in \operatorname{Span}(f_1)$. On the other hand, $f$ satisfies (5.15) and thus $f \in \operatorname{Span}(f_1)$ from assumption (H5), in particular $\lambda = \lambda_1$. $\square$

When $\mathcal{L}$ is the generator of a positive and irreducible semigroup $S$, we may introduce the alternative assumption:

(H5') the semigroup $S$ is aperiodic as defined in (4.12), namely

$$\forall f \in X_+\setminus\{0\}, \forall \phi \in Y_+\setminus\{0\}, \exists T > 0, \forall \tau \ge T \quad \langle S_{\tau}f, \phi \rangle > 0.$$
Because of the reverse positivity condition for semigroups (5.18), there exist $T > 0$ and $u_T \in \mathbb{S}^1$ such that

$$ \tilde{S}_{Tg} = u_T \tilde{S}_T |g|. $$

As a consequence, by definition of the set $\omega_H$, we have

$$ 0 = \langle g, \phi_1 \rangle = \langle \tilde{S}_T g, \phi_1 \rangle = u_T \langle \tilde{S}_T |g|, \phi_1 \rangle = u_T \langle |g|, \phi_1 \rangle. $$

Because $u_T \neq 0$, we conclude that $g = 0$. In other words, we have established that $\omega_H = \{0\}$ and together with (5.21), we obtain (5.17). $\square$

We present a more concrete situation where the previous result can be invoked. Although the hypotheses are somehow restrictive, it is yet useful in many applications and its proof is very simple.

**Corollary 5.22.** Consider a strongly continuous and positive semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ enjoys the conclusions (C2) of existence, uniqueness and strict positivity of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. Assume further that the reverse Kato's inequality condition (as defined in Definition 5.12) holds true for the (large) class

$$ \mathcal{C} := \{f \in D(\mathcal{L}); \mathcal{L}|f| = \Re(\operatorname{sign} f)\mathcal{L}f\}, $$

that $X \subset L_{\text{loc}}^1(E, \mathcal{E}, \mu)$ and that the space $\mathcal{X}^k$ defined in (4.26) satisfies $\mathcal{X}^k \subset L_{\text{loc}}^1$ with strongly compact embedding for some $k \ge 1$. Then the ergodicity property (E2) holds in the sense of strong topology of $L_{\phi_1}^1$.

**Proof of Corollary 5.22.** Because of Step 3 in the proof Theorem 4.23, we see that condition (1) in Theorem 5.21 holds with $\mathcal{X}_1 := \mathcal{X}^k$. On the other hand, because of Remark 5.14 and the reverse Kato's inequality condition in $\mathcal{C}$, we see that condition (2) also holds, so that we may apply Theorem 5.21 and conclude. $\square$

We present now a variant of the previous result which provides a convergence for various topologies, and relies on the (very general) assumption that the boundary spectrum is trivial rather than on the reverse positivity condition.

**Theorem 5.23.** Consider a positive semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ enjoys the conclusions (C3) on the existence, uniqueness and strict positivity of the first eigentriplet problem $(\lambda_1, f_1, \phi_1)$ and triviality of the boundary point spectrum. Setting $\tilde{S}_t := e^{-\lambda_1 t} S_t$, we assume that we are in one of the following situations:

(1) $S$ is strongly continuous and the trajectories $(\tilde{S}_t f)_{t \ge 0}$ are relatively compact for all $f \in X$, and we denote by $\mathcal{T}$ the strong topology of $X$;

(2) $X = Y'$, $Y$ separable, and the trajectories $(\tilde{S}_t f)_{t \ge 0}$ are bounded for all $f \in X$, and we denote by $\mathcal{T}$ the weak $\sigma(Y', Y)$ topology;

(3) $X \subset L_{\text{loc}}^1(E, \mathcal{E}, \mu)$, and we denote by $\mathcal{T}$ the weak topology of $L_{\phi_1}^1$;

(4) $X \subset L_{\text{loc}}^1$, $S$ is strongly continuous, and for some $k \ge 1$ the space $\mathcal{X}^k$ defined in (4.26) satisfies $\mathcal{X}^k \subset L_{\text{loc}}^1$ with strongly compact embedding, and we denote by $\mathcal{T}$ the strong topology of $L_{\phi_1}^1$.

Then the ergodicity property (E2) holds in the sense of the topology $\mathcal{T}$.

**Remark 5.24.** The case (4) of Theorem 5.23 enjoys some strong similarities with the main consequences of the General Relative Entropy technique developed in [269], see in particular [269, Thm. 3.2], [269, Thm. 4.3] and [269, Thm. 5.2]. In particular, the aperiodicity condition that the boundary point spectrum is trivial may be compared with the fact that the first eigenvector $f_1$ is the unique (normalized and nonnegative) vector $f \in X$ with vanishing dissipation of entropy $\mathcal{D}(f) = 0$ as defined in [269] or more generally that $\text{Span}(f_1)$ is the unique invariant space on which the functional $\mathcal{D}$ vanishes. The present formulation is more abstract and probably more general. The drawback is the condition $\mathcal{X}^k \subset L_{\text{loc}}^1$ with strongly compact embedding which can be avoided in [269], by using some weak compactness argument and the lower semicontinuity property of $\mathcal{D}$. That is explained by the fact that our proof uses rather the La Salle invariance principle (similarly as in the proof of [153, Thm. 3.2]) instead of an entropy dissipation argument.

In the case when the boundary point spectrum is not trivial but a discrete set, the same method of proof as for Theorem 5.22 allows us to accurately describe the periodic long time behaviour of the semigroup.
**Theorem 5.18.** Let $X$ be a Banach lattice satisfying conditions (X1), (X2) and in which the property (4.9) holds true with $\mathcal{E} = X$. Consider a positive and irreducible semigroup $S$ on $X$ which satisfies the aperiodicity condition (H5') and such that its generator $\mathcal{L}$ satisfies (C2). Then $\lambda_1$ is the unique eigenvalue of $\mathcal{L}$ with largest real part: $\Sigma_P^1(\mathcal{L}) = \{\lambda_1\}$.

**Remark 5.19.** (1) Because of the restriction made on the Banach lattice structure, this framework excludes the case when $X$ is a (weighted) signed measures space but includes the cases when $X$ is a (weighted) Lebesgue space.

(2) It is worth pointing out that since (H5') is stronger than (H4)-(i), see the points (2) and (3) in Lemma 4.8, we can use Theorem 4.13 and replace in Theorem 5.18 the assumption that (C2) is satisfied by the assumption that (C1) and (H1') for both $\mathcal{L}$ and $\mathcal{L}^*$ are satisfied.

*Proof of Theorem 5.18.* We introduce the notations $\tilde{S}_t := S_t e^{-\lambda_1 t}$ and $\tilde{\mathcal{L}} := \mathcal{L} - \lambda_1$. The complex Kato's inequality (5.7) automatically holds because $\mathcal{L}$ is the generator of a positive semigroup. We may thus use Lemma 5.4 and we deduce that $\Sigma_P^1(\tilde{\mathcal{L}}) = \Sigma_P(\tilde{\mathcal{L}}) \cap i\mathbb{R}$. Assume that $f = g + ih \in X$, $g, h \in X_{\mathbb{R}}$, is an eigenfunction associated to the eigenvalue $\lambda = \lambda_1 + i\alpha \in \mathbb{C}$, $\alpha > 0$, so that

$$ \tilde{\mathcal{L}}(g + ih) = i\alpha(g + ih) = \frac{2\pi i}{t_0}(g + ih), $$

for some $t_0 > 0$. On the one hand, because $\alpha \neq 0$, we must have $g \neq 0$ and $h \neq 0$, and because of

$$ \alpha\langle g, \phi_1 \rangle = \langle \tilde{\mathcal{L}}h, \phi_1 \rangle = \langle h, \tilde{\mathcal{L}}^*\phi_1 \rangle = 0, $$

and $\phi_1 \gg 0$, we have $g_+ \neq 0$ and $g_- \neq 0$. As a consequence, and because of (4.9), there exists $\psi \in Y_+ \setminus \{0\}$ such that $\langle g_+, \psi \rangle = 0$. On the other hand, we compute

$$ \tilde{S}_{t_0}(g + ih) = e^{i\alpha t_0}(g + ih) = g + ih, $$

from what we deduce $\tilde{S}_{t_0}g = g$, because $S_t$ is real. On the other hand, because $S_t$ is positive, we have

$$ g_+ = (\tilde{S}_{t_0}g)_+ \le \tilde{S}_{t_0}g_+, $$

and next

$$ \langle \phi_1, g_+ \rangle \le \langle \phi_1, \tilde{S}_{t_0}g_+ \rangle = \langle \tilde{S}_{t_0}^* \phi_1, g_+ \rangle = \langle \phi_1, g_+ \rangle, $$

so that both inequalities are equalities (remind again that $\phi_1 \gg 0$), and thus

$$ \tilde{S}_{t_0}g_+ = g_+. $$

We conclude that

$$ \langle \tilde{S}_{kt_0}g_+, \psi \rangle = \langle g_+, \psi \rangle = 0, \quad \forall k \ge 0, $$

what is in contradiction with (H5'). We have established that $\Sigma_P^1(\mathcal{L}) = \{\lambda_1\}$. $\square$

We end this section with a third situation where the triviality of the principal point spectrum is an immediate consequence of Theorem 5.5 and Theorem 5.6.

**Theorem 5.20.** (1) We make the same assumptions as in Theorem 5.5 and also that there exists $M > 0$ large enough such that $\lambda - \mathcal{L}$ is invertible in $\mathcal{B}(X)$ for any $\lambda \in \mathbb{C}$, $\Re\lambda = \lambda_1$, $|\lambda| \ge M$. Then $\lambda_1$ is the unique eigenvalue with largest real part: $\Sigma_P^1(\mathcal{L}) = \{\lambda_1\}$.

(2) We furthermore assume that the hypothesis of Theorem 5.6 are met and that $\lambda - \mathcal{L}$ is invertible in $\mathcal{B}(X)$ for any $\lambda \in \mathbb{C}$, $\Re\lambda \ge \lambda_1 - \varepsilon$, $|\lambda| \ge M$. Then a (non constructive) spectral gap (S3) holds.

*Proof of Theorem 5.20.* (1) Combining the property of $\Sigma_P^1(\tilde{\mathcal{L}})$ established in Theorem 5.5 and the above invertibility condition, we see that $\Sigma_P^1(\tilde{\mathcal{L}})$ is a bounded additive subgroup of $i\mathbb{R}$, so that $\Sigma_P^1(\tilde{\mathcal{L}}) = \{0\}$, what is precisely the first claim.

(2) From Theorem 5.6, we know that $\Sigma(\mathcal{L}) \cap \Delta_{\lambda_1 - \varepsilon} \subset \Sigma_d(\mathcal{L})$ and thus together with (1), we get $\Sigma(\mathcal{L}) \cap \Delta_{\kappa} = \{0\}$ for $\lambda_1 - \kappa > 0$ small enough, what is precisely the second claim. $\square$

We summarize the main results established in this section as follows.

(C3) the first eigentriplet problem (1.1)-(1.2) has a solution $(\lambda_1, f_1, \phi_1)$, furthermore this one is unique, $f_1 \gg 0$, $\phi_1 \gg 0$, $\lambda_1$ is algebraically simple (for both $\mathcal{L}$ and $\mathcal{L}^*$) and $\Sigma_P^1(\mathcal{L}) = \{\lambda_1\}$.
**Theorem 5.25.** Consider a positive semigroup S on a Banach lattice X such that its generator $\mathcal{L}$ enjoys the conclusions (C2) on the existence and uniqueness of the first eigentriplet problem $(\lambda_1, f_1, \phi_1)$, and satisfies the complex Kato's inequality (5.7). Suppose furthermore that X and Y both enjoy the structure conditions (X2) and (X3), that $\lambda_1$ is an isolated eigenvalue and that the boundary spectrum is not trivial, i.e. $\Sigma_P^+ \neq \{\lambda_1\}$. Setting $\tilde{S}_t := e^{-\lambda_1 t} S_t$, we assume that we are in one of the situations (1), (2), (3) or (4) listed in statement of Theorem 5.23. Then $\Sigma_P^+ = \{\lambda_1+ik\alpha, k \in \mathbb{Z}\}$ for some $\alpha > 0$, there exists a sequence $(g_k, \psi_k)_{k \in \mathbb{Z}}$ such that $\mathcal{L}g_k = (\lambda_1+ik\alpha)g_k$, $\mathcal{L}^*\psi_k = (\lambda_1+ik\alpha)\psi_k$ and $\langle g_k, \psi_k \rangle = 1$, and for all $f \in X$, in the sense of the topology $\mathcal{T}$, the projection

$$ \Pi f = \lim_{n \to \infty} \frac{1}{n} \sum_{\ell=0}^{n} \sum_{k=-\ell}^{\ell} \langle f, \psi_k \rangle g_k $$

is well defined and

$$ \tilde{S}_t f - \tilde{S}_t \Pi f \to 0 \quad \text{as } t \to +\infty. $$

**Remark 5.26.** In Theorem 5.25, the assumptions that $\lambda_1$ is isolated and $\Sigma_P^+ \neq \{\lambda_1\}$ might seem difficult to check in practice. We indicate here some ways to verify them.

(i) The condition that $\lambda_1$ is an isolated eigenvalue is for instance guaranteed under the assumptions of Theorem 5.6 or Theorem 6.5.

(ii) The condition that $\Sigma_P^+$ is not restricted to $\{\lambda_1\}$ can be guaranteed by verifying that (E2) does not hold. Indeed, if $\Sigma_P^+ = \{\lambda_1\}$, then Theorem 5.23 imposes (E2) to hold.

The result in Theorem 5.25 can be compared for instance to [41, Thm. 14.19], although our hypotheses are slightly more general. Our proof is also more direct than in [41] and it additionally provides an explicit expression of the projection on the boundary eigenspace $\overline{\text{Span}}(g_k)_{k \in \mathbb{Z}}$. The proof of Theorems 5.23 and 5.25 relies on the theory of almost periodic functions which dates back to H. Bohr. There is a large literature on the subject and we refer for instance to the book of Corduneanu [113] for a comprehensive introduction. There are several equivalent definitions of almost periodic functions and we will use the following one. A function $f \in C_b(\mathbb{R}, X)$, i.e. a bounded continuous function from $\mathbb{R}$ to $X$, is said to be almost periodic if the set $\{f(\cdot + \tau), \tau \in \mathbb{R}\}$ is relatively compact in $C_b(\mathbb{R}, X)$. The set of almost periodic functions is a sub-algebra of $C_b(\mathbb{R}, X)$, and also the closure of the space of periodic functions in $C_b(\mathbb{R}, X)$. We start with the proof of Theorem 5.23 and Theorem 5.25 in the case when S satisfies the condition (1). Then we deduce the cases (2), (3) and (4) from the case (1).

*Proof of Theorems 5.23 and 5.25 in the case (1).* Step 1. Let $f \in X$. Since the trajectory $(\tilde{S}_t f)_{t \ge 0}$ is relatively compact, we infer from [201, Thm. 8] (with $U(\tau, t) = S_t$ and thus no periodicity condition on $U$) the existence of an almost periodic eternal solution $g$ of the rescaled semigroup $\tilde{S}$, i.e. a function $g: \mathbb{R} \to X$ such that $g(t+\tau) = \tilde{S}_\tau g(t)$ for all $t \in \mathbb{R}$ and $\tau \ge 0$, such that

$$ \lim_{t \to +\infty} \| \tilde{S}_t f - g(t) \| = 0. $$

The end of the proof consists in characterizing the function $g$ in the situations of Theorems 5.23 and 5.25. For $\lambda \in \mathbb{R}$, we define the Bohr transformation of the almost-periodic function $g$ by

$$ c_{\lambda}(g) = \lim_{T \to +\infty} \frac{1}{T} \int_{0}^{T} e^{-i\lambda t} g(t) dt, $$

which is known to exists, see [113, Thm. 3.4], since $e^{-i\lambda t} g(t)$ is also almost periodic. Since $e^{-i\lambda t} g(t)$ is besides an eternal solution of the semigroup $e^{-i\lambda t} \tilde{S}_t$ with infinitesimal generator $\mathcal{L}_\lambda = \mathcal{L} - \lambda_1 - i\lambda$, we have that

$$ \mathcal{L}_\lambda \int_0^T e^{-i\lambda t} g(t) dt = g(T) - g(0). $$

Dividing by T the above expression, passing to the limit $T \to +\infty$ and using that $\mathcal{L}_\lambda$ is a closed operator, we get

$$ \mathcal{L}_\lambda c_\lambda(g) = 0. $$

In other words, we have established

$$ \mathcal{L}c_{\lambda}(g) = (\lambda_{1} + i\lambda)c_{\lambda}(g) $$
We notice that (C3) is actually the combination of (C2), (S3₂) and the algebraic simplicity.

5.5. **Ergodicity.** Thanks to the above analysis, we are able to formulate some convergence results on the trajectories associated to a semigroup. More precisely, assuming the existence and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$ for the generator $\mathcal{L}$ of a semigroup $S$ and still denoting the rescaled semigroup $\tilde{S}(t) := e^{-\lambda_1 t} S(t)$, we wish in particular to establish the following ergodic property

(E2) for any $f \in X$, there holds

$$ (5.16) \qquad \tilde{S}_t f \to \langle f, \phi_1 \rangle f_1, \quad \text{as } t \to \infty, $$

in a sense to be specified.

We start with a simple result which takes advantage of some dissipativity property of the semigroup formulated by a "reverse positivity condition". We next present some more involved results which use directly the spectral information. It is worth emphasizing that our results in this section do not use any spectral gap property what contrasts with the results we will present in the next section.

**Theorem 5.21.** Consider a positive semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ enjoys the conclusions (C2) of existence, uniqueness and strict positivity of the first eigentriplet $(\lambda_1, f_1, \phi_1)$ and let us set $\tilde{S}_t := e^{-\lambda_1 t} S_t$. We denote $\mathcal{X}$ the space $X$ endowed with the norm $[\cdot]$, with $[f] := \langle |f|, \phi_1 \rangle$. Assume furthermore that

(1) for any $f \in X$, the trajectory $(\tilde{S}_t f)_{t \ge 0}$ is continuous in $\mathcal{X}$ and belongs to a compact set of a normed space $\mathcal{X}_1$, with $\mathcal{X}_1 \subset \mathcal{X}$;

(2) $(S_t)$ satisfies the reverse positivity condition for semigroups

$$ (5.17) \qquad |S_t f| = S_t |f|, \quad \forall t > 0, \quad \text{implies } \exists T > 0, \exists u_T \in \mathbb{S}^1, S_T f = u_T S_T |f|. $$

Then, the ergodicity property (E2) holds in the sense of the norm of $\mathcal{X}_1$.

Let us comment on hypotheses made in the statement of Theorem 5.21. Hypothesis (1) can be obtained as a consequence of a Lyapunov (or growth) condition reminiscent of the structure condition (HS3) introduced in Section 3.3 and an irreducibility condition. Typically, we assume first

$$ \|\tilde{S}(t)f\| \le M\|f\| + K \sup_{0 \le \tau \le t} [\tilde{S}(\tau)f]_0, $$

with $[g]_0 := \langle |g|, \psi_0 \rangle$, $\psi_0 \in Y_+ \setminus \{0\}$, what can be established under the very general condition (ii) of Theorem 3.4. Next we need to be able to prove that $\psi_0 \le r\phi_1$ for some $r > 0$. In concrete situations, we may take $\psi_0$ with compact support and then the above inequality is a consequence of the standard strong maximum principle. We deduce

$$ \begin{align*} \|\tilde{S}(t)f\| &\le M\|f\| + Kr \sup_{0 \le \tau \le t} \langle |\tilde{S}(\tau)f|, \phi_1 \rangle \\ &\le M\|f\| + Kr \sup_{0 \le \tau \le t} \langle \tilde{S}(\tau)|f|, \phi_1 \rangle \\ &= M\|f\| + Kr\langle |f|, \phi_1 \rangle, \end{align*} $$

so that $(\tilde{S}_t)$ is bounded. The hypothesis (1) is in fact a bit more demanding, but also quite natural. Assume that $S_\mathcal{L}$ enjoys the splitting structure introduced in section 3.1 and section 3.3, so that

$$ (5.18) \qquad \tilde{S} = V + K, $$

with

$$ V := \tilde{S}_B + \dots + (\tilde{S}_B \mathcal{A})^{(N-1)} * \tilde{S}_B, \quad K := (\tilde{S}_B \mathcal{A})^{(N)} * \tilde{S}, \quad \tilde{S}_B(t) := e^{-\lambda_1 t} S_B(t). $$

In some applications, we typically have

$$ \|V(t)f_0\| \le \Theta(t)\|f_0\|, \quad \|(\tilde{S}_B \mathcal{A})^{(N)}\|_{\mathcal{B}(X, \mathcal{X}_1)} \le \Theta $$

with $\Theta \in L^1(\mathbb{R}_+) \cap C_0(\mathbb{R}_+)$, $\mathcal{X}_1 \subset X$ compact. In that situation, we deduce (1).
and $\lambda_1 + i\lambda$ is an eigenvalue of $\mathcal{L}$ if $c_\lambda(g) \neq 0$.

**Step 2.** We deduce that if the boundary spectrum is trivial, as in Theorem 5.23, then necessarily $c_\lambda(g) = 0$ for all $\lambda \neq 0$. By the uniqueness theorem, see for instance [113, Thm. 4.7], we get that $g$ is constant. Due to the conservation law $\langle \tilde{S}_t f, \phi_1 \rangle = \langle f, \phi_1 \rangle$ and the simplicity of the eigenvalue 0, we get that $g = \langle f, \phi_1 \rangle f_1$ and the result of the case (1) in Theorem 5.23 is proved.

**Step 3.** In the case of Theorem 5.25, the boundary spectrum is not trivial and we know from Theorem 5.5 that $\Sigma_P^+(\mathcal{L})$ is an additive subgroup of $i\mathbb{R}$, made of algebraically simple eigenvalues. Due to the assumption that $\lambda_1$ is isolated, this subgroup must be discrete and $\Sigma_P^+(\mathcal{L})$ is thus given by $\{\lambda_1 + i\alpha k, k \in \mathbb{Z}\}$ for some $\alpha > 0$. As a consequence, any $\lambda$ such that $c_\lambda(g) \neq 0$ is necessarily of the form $\lambda = \alpha k$ for some $k \in \mathbb{Z}$. By the uniqueness theorem, $g$ is then a $\alpha$-periodic function which is given, due to Fejér's theorem, by

$$g(t) = \lim_{n \to \infty} \frac{1}{n} \sum_{\ell=0}^{n} \sum_{k=-\ell}^{\ell} c_{\alpha k}(g) e^{i \alpha k t}.$$

Consider $(g_k, \psi_k)$ two positive direct and dual eigenvectors of $\mathcal{L}$ associated to the eigenvalue $i\alpha k$ such that $\langle g_k, \psi_k \rangle = 1$. From the conservation laws $\langle \tilde{S}_t f, \psi_k \rangle = \langle f, \psi_k \rangle e^{i\alpha k t}$ and the algebraic simplicity of the eigenvalues $i\alpha k$, we get that $c_{\alpha k}(g) = \langle f, \phi_k \rangle g_k$, and the result is proved. $\square$

**Proof of Theorems 5.23 and 5.25 in the case (2).** Since $Y$ is separable, we can find a sequence $(\varphi_n)_{n \ge 1} \subset Y$ which satisfies $\|\varphi_n\| = 1$ and $\mathrm{span}(\varphi_n)$ is dense in $Y$. We can then define on $X$ the norm $\|\cdot\|_*$ by setting

$$(5.24) \qquad \|f\|_* = \sum_{n=1}^{\infty} 2^{-n} |\langle f, \varphi_n \rangle|.$$

On bounded subsets of $X$, the topology of this norm is the same as the weak-$*$ topology, or more explicitly it is worth emphasizing

$$f_n \to f * \sigma(Y', Y) \Leftrightarrow (\sup \|f_n\| < \infty \quad \text{and} \quad \|f_n - f\|_* \to 0).$$

Since by assumption the trajectory $(\tilde{S}_t f)$ is bounded, it is weakly-$*$ relatively compact, and so relatively compact in $(X, \|\cdot\|_*)$. It is also clear that the semigroup $S$ is continuous for the weak norm $\|\cdot\|_*$. The normed space $(X, \|\cdot\|_*)$ is not a Banach space, but the proof of Theorem 5.25 actually only requires, for applying [201, Thm. 8], that the closed balls of $X$ are complete metric spaces, which is the case for the distance induced by $\|\cdot\|_*$. Applying the case (1) of Theorems 5.23 and 5.25 then yields the result. $\square$

**Proof of Theorems 5.23 and 5.25 in the case (3).** We consider $f \in X$ and, repeating the proof of Step 2 in Theorem 4.23, we get that $(S_t f)_{t \ge 0}$ belongs to a weak compact set $\mathcal{G}$ of $L_{\phi_1}^1$. We define the norm $\|\cdot\|_*$ by (5.24) for a sequence $(\varphi_n)_{n \ge 1} \subset C_c(E)$ which satisfies $\|\varphi_n\|_{L^\infty} = 1$ and $\mathrm{span}(\varphi_n)$ is dense in $C_0(E)$. This norm induces a metric on $\mathcal{G}$ which is topologically equivalent to the weak convergence on $L_{\phi_1}^1$. The trajectory $(\tilde{S}_t f)$ is then relatively compact in $(\mathcal{G}, \|\cdot\|_*)$ and the semigroup $S$ is continuous for the weak norm $\|\cdot\|_*$. We conclude as in the proof of the case (2). $\square$

**Proof of Theorems 5.23 and 5.25 in the case (4).** From the step 3 of the proof of Theorem 4.23, we know that for any $f \in \mathcal{X}^k$ the trajectory $(\tilde{S}_t f)$ is compact for the strong topology of $L_{\phi_1}^1$. We may then conclude similarly as in the case (1), using that $\mathcal{X}^k$ is dense in $X$ for the norm of $L_{\phi_1}^1$. $\square$

**5.6. A word about spectral analysis argument.** The aim of this section is to recall some more or less classical results which makes possible to slightly improve the conclusions of the results presented in the previous section by additionally assume some spectral gap at the level of the operator or the semigroup. More precisely, we are interested by some accurate versions of a partial, but principal spectral mapping theorem which asserts that

$$(5.25) \qquad \Sigma(e^{t\mathcal{L}}) \cap B^c(0, e^{\kappa t}) = e^{t\Sigma(\mathcal{L}) \cap \Delta_\kappa}, \quad \forall t \ge 0,$$

for some $\kappa < \lambda_1$, and even more precisely, we wish to establish the following geometric (or exponential) asymptotic stability
Proof of Theorem 5.21. We fix $f \in X$ and without loss of generality, we may assume that $\langle f, \phi_1 \rangle = 0$. We observe that

$$ (5.19) \qquad \langle |\tilde{S}_t f|, \phi_1 \rangle = \langle |\tilde{S}_{t-s} \tilde{S}_s f|, \phi_1 \rangle \le \langle \tilde{S}_{t-s} |\tilde{S}_s f|, \phi_1 \rangle = \langle |\tilde{S}_s f|, \phi_1 \rangle, $$

for any $t \ge s$. We deduce that $(\tilde{S}_t)$ is a dynamical system with compact trajectories in $\mathcal{X}_1$ and $\mathcal{H}(f) := \langle |f|, \phi_1 \rangle$ is a Lyapunov functional. As a consequence, from the La Salle invariance principle, we have

$$ (5.20) \qquad \inf_{g \in \omega_H} \langle |\tilde{S}_t f - g|, \phi_1 \rangle \to 0 \quad \text{as } t \to \infty, $$

with

$$ (5.21) \qquad \omega_H := \{g \in X; \langle g, \phi_1 \rangle = 0, \forall t \in \mathbb{R}, \mathcal{H}(\tilde{S}_t g) = \inf_{s>0} \mathcal{H}(\tilde{S}_s f)\}. $$

We next characterize $\omega_H$. Picking up $g \in \omega_H$, we observe that

$$ \langle |\tilde{S}_t g|, \phi_1 \rangle = \langle |g|, \phi_1 \rangle = \langle |g|, \tilde{S}_t^* \phi_1 \rangle = \langle \tilde{S}_t |g|, \phi_1 \rangle, \quad \forall t \ge 0, $$

so that

$$ \langle \tilde{S}_t |g| - |\tilde{S}_t g|, \phi_1 \rangle = 0, \quad \forall t \ge 0. $$

In particular, using that $|\tilde{S}_t g| \le \tilde{S}_t |g|$, we have

$$ (5.22) \qquad \tilde{S}_t |g| = |\tilde{S}_t g|, \quad \forall t \ge 0. $$

Because of the reverse positivity condition for semigroups (5.17), there exist $T > 0$ and $u_T \in \mathbb{S}^1$ such that

$$ \tilde{S}_T g = u_T \tilde{S}_T |g|. $$

As a consequence, by definition of the set $\omega_H$, we have

$$ 0 = \langle g, \phi_1 \rangle = \langle \tilde{S}_T g, \phi_1 \rangle = u_T \langle \tilde{S}_T |g|, \phi_1 \rangle = u_T \langle |g|, \phi_1 \rangle. $$

Because $u_T \ne 0$, we conclude that $g = 0$. In other words, we have established that $\omega_H = \{0\}$ and together with (5.20), we obtain (5.16). $\square$

We present a more concrete situation where the previous result can be invoked. Although the hypotheses are somehow restrictive, it is yet useful in many applications and its proof is very simple.

**Corollary 5.22.** Consider a strongly continuous and positive semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ enjoys the conclusions (C2) of existence, uniqueness and strict positivity of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. Assume further that the reverse Kato's inequality condition (as defined in Definition 5.12) holds true for the (large) class

$$ \mathcal{C} := \{f \in D(\mathcal{L}); \mathcal{L}|f| = \Re(\operatorname{sign} f)\mathcal{L}f\}, $$

that $X \subset L_{\text{loc}}^1(E, \mathcal{E}, \mu)$ and that the space $\mathcal{X}^k$ defined in (4.23) satisfies $\mathcal{X}^k \subset L_{\text{loc}}^1$ with strongly compact embedding for some $k \ge 1$. Then the ergodicity property (E2) holds in the sense of strong topology of $L_{\phi_1}^1$.

**Proof of Corollary 5.22.** Because of Step 3 in the proof Theorem 4.22, we see that condition (1) in Theorem 5.21 holds with $\mathcal{X}_1 := \mathcal{X}^k$. On the other hand, because of Remark 5.14 and the reverse Kato's inequality condition in $\mathcal{C}$, we see that condition (2) also holds, so that we may apply Theorem 5.21 and conclude. $\square$

We present now a variant of the previous result which provides a convergence for various topologies, and relies on the (very general) assumption that the principal point spectrum is trivial rather than on the reverse positivity condition.

**Theorem 5.23.** Consider a positive semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ enjoys the conclusions (C3) on the existence, uniqueness and strict positivity of the first eigentriplet problem $(\lambda_1, f_1, \phi_1)$ and triviality of the boundary point spectrum. Setting $\tilde{S}_t := e^{-\lambda_1 t} S_t$, we assume that we are in one of the following situations:

(1) $S$ is strongly continuous and the trajectories $(\tilde{S}_t f)_{t \ge 0}$ are relatively compact for all $f \in X$, and we denote by $\mathcal{T}$ the strong topology of $X$;
(E3₁) there exist some constants $\kappa < \lambda_1$ and $C \ge 1$ such that for any $f \in X$, there holds

$$
(5.26) \quad \|\tilde{S}(t)f - \langle f, \phi_1 \rangle f_1\| \le \Theta(t) \|f - \langle f, \phi_1 \rangle f_1\|, \quad \forall t \ge 0, \forall f \in X,
$$

with the decay rate function $\Theta(t) := C e^{(\kappa-\lambda_1)t}$.

In order to discuss the several results we present, we recall the splitting framework

$$
(5.27) \qquad S = V + W * S, \quad \|V(t)\|_{\mathcal{B}(X)} + \|W(t)\|_{\mathcal{B}(X)} \lesssim e^{\kappa t},
$$

for the same $\kappa \in \mathbb{R}$ as above. We start by recalling the spectral mapping theorem for the point spectrum, and its proof, which is instructive.

**Lemma 5.27** (Spectral mapping theorem for point spectrum). For a semigroup $(S_t)_{t \ge 0}$ with infinitesimal generator $\mathcal{L}$ we have

$$
\Sigma_P(S_t) \setminus \{0\} = e^{t\Sigma_P(\mathcal{L})}, \quad \forall t \ge 0.
$$

Proof of Lemma 5.27. The inclusion $e^{t\Sigma_P(\mathcal{L})} \subset \Sigma_P(S_t) \setminus \{0\}$ is clear. Now let $\xi \in \Sigma_P(S_t) \setminus \{0\}$, that is $\xi \in \mathbb{C} \setminus \{0\}$ such that $S_t f = \xi f$ for some $f \in X_{\mathbb{C}} \setminus \{0\}$, and let $\lambda \in \mathbb{C}$ such that $\xi = e^{\lambda t}$ and $\phi \in X'$ such that $\langle \phi, f \rangle \neq 0$. For any $k \in \mathbb{Z}$ we have $\xi = e^{\lambda t + 2ik\pi}$ and so

$$
0 = e^{-(\lambda + \frac{2ik\pi}{t})t} S_t f - f = \left( \mathcal{L} - \lambda - \frac{2ik\pi}{t} \right) \int_0^t e^{-(\lambda + \frac{2ik\pi}{t})s} S_s f \, ds.
$$

If the last integral is non-zero for some $k \in \mathbb{Z}$, we deduce that $\lambda + \frac{2ik\pi}{t}$ is an eigenvalue of $\mathcal{L}$ and the result is proved. Assume by contradiction that $\int_0^t e^{-(\lambda + \frac{2ik\pi}{t})s} S_s f \, ds = 0$ for all $k \in \mathbb{Z}$. This means that the continuous and periodic complex-valued function $s \mapsto e^{-\lambda s} \langle \phi, S_s f \rangle$ has all its Fourier coefficients equal to zero, which is not possible since this function is not equally zero (its value at $s=0$ is not zero). $\square$

We next present a very classical result about the exponential stability of $f_1$ which is based on the quasi-compact semigroup framework of Voigt [355] (see also [15, B-IV-2] and [152, Sec. V.3]) and which is a more accurate version of Lemma 2.7 and Theorem 5.7.

**Theorem 5.28.** Let $(S_t)_{t \ge 0}$ be a positive irreducible semigroup on a Banach lattice $X$ satisfying the hypotheses of Lemma 2.7 and Theorem 5.7, in particular (H2) holds for a constant $\kappa_0 \in \mathbb{R}$ and there exists $T > 0$ such that the splitting

$$
(5.28) \qquad S_T = V_T + K_T,
$$

holds with $\|V_T\|_{\mathcal{B}(X)} \le e^{\kappa T}$, $\kappa < \kappa_0$, and $K_T \in \mathcal{K}(X)$. Then there exists a unique solution $(\lambda_1, f_1, \phi_1)$ to the eigentriplet and the exponential stability (E3₁) holds (without constructive estimate).

**Remark 5.29.** In the splitting framework (5.27) the critical hypothesis $K_T \in \mathcal{K}(X)$ may be obtained by assuming that

$$
\|W(t)\|_{\mathcal{B}(X, \mathcal{X}_1)} \lesssim e^{\kappa t}, \quad \forall t \ge 0, \quad \mathcal{X}_1 \subset X \text{ compact.}
$$

In fact, in many applications, we are also able to establish $\mathcal{X}_1 \subset D(\mathcal{L}^\beta)$, for some $\beta > 0$, without too much more work.

Theorem 5.28 is in fact nothing but [41, Thm. 14.18] (see also [360, Sec. 2], [152, Thm. V.3.7] or [15, C-IV, Thm. 2.1 & Rk. 2.2]). We give however a short proof of Theorem 5.28 since it is simpler and more direct than the ones we usual find in the literature and in particular does not refer to subtil results about the spectrum and its essential part.

*Proof of Theorem 5.28. First step.* From Lemma 2.7, we already know that (H1), (H2) and (H3) hold. Together with the irreducibility which is nothing but (H4) from Lemma 4.8, we may apply Theorem 4.13 and conclude to the existence, uniqueness and strict positivity result about the eigentriplet solution $(\lambda_1, f_1, \phi_1)$.

*Second step.* We claim that $\Sigma(\mathcal{L}) \cap \{z \in \mathbb{C}, \Re(e(z)) \ge \kappa_0\}$ is also made of a finite number of isolated eigenvalues with finite geometric multiplicity. We indeed set $\beta_0 := e^{\kappa_0 T}$. Since for any $\lambda \in B_{\beta_0}^c := \{z \in \mathbb{C}, |z| \ge \beta_0\}$ the operator $\lambda - V_T$ is invertible, we see that $\lambda \in B_{\beta_0}^c$ is in the
(2) $X = Y'$, $Y$ separable, and the trajectories $(\tilde{S}_t f)_{t \ge 0}$ are bounded for all $f \in X$, and we denote by $\mathcal{T}$ the weak $\ast \sigma(Y', Y)$ topology;

(3) $X \subset L_{\text{loc}}^{1}(E, \mathcal{E}, \mu)$, and we denote by $\mathcal{T}$ the weak topology of $L_{\phi_1}^{1}$;

(4) $X \subset L_{\text{loc}}^{1}$, $S$ is strongly continuous, and for some $k \ge 1$ the space $X^k$ defined in (4.23) satisfies $X^k \subset L_{\text{loc}}^{1}$ with strongly compact embedding, and we denote by $\mathcal{T}$ the strong topology of $L_{\phi_1}^{1}$.

Then the ergodicity property (E2) holds in the sense of the topology $\mathcal{T}$.

**Remark 5.24.** The case (4) of Theorem 5.23 enjoys some strong similarities with the main consequences of the General Relative Entropy technique developed in [268], see in particular [268, Thm. 3.2], [268, Thm. 4.3] and [268, Thm. 5.2]. In particular, the aperiodicity condition that the boundary point spectrum is trivial may be compared with the fact that the first eigenvector $f_1$ is the unique (normalized and nonnegative) vector $f \in X$ with vanishing dissipation of entropy $\mathcal{D}(f) = 0$ as defined in [268] or more generally that $\text{Span}(f_1)$ is the unique invariant space on which the functional $\mathcal{D}$ vanishes. The present formulation is more abstract and probably more general. The drawback is the condition $X^k \subset L_{\text{loc}}^{1}$ with strongly compact embedding which can be avoided in [268], by using some weak compactness argument and the lower semicontinuity property of $\mathcal{D}$. That is explained by the fact that our proof uses rather the La Salle invariance principle (similarly as in the proof of [152, Thm. 3.2]) instead of an entropy dissipation argument.

In the case when the boundary point spectrum is not trivial but a discrete set, the same method of proof as for Theorem 5.23 allows us to accurately describe the periodic long time behaviour of the semigroup.

**Theorem 5.25.** Consider a positive semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ enjoys the conclusions (C2) on the existence and uniqueness of the first eigentriplet problem $(\lambda_1, f_1, \phi_1)$, and satisfies the complex Kato's inequality (5.7). Suppose furthermore that $X$ and $Y$ both enjoy the structure conditions (X2) and (X3), that $\lambda_1$ is an isolated eigenvalue and that the boundary spectrum is not trivial, i.e. $\Sigma_P^+ \ne \{\lambda_1\}$. Setting $\tilde{S}_t := e^{-\lambda_1 t} S_t$, we assume that we are in one of the situations (1), (2), (3) or (4) listed in statement of Theorem 5.23. Then $\Sigma_P^+ = \{\lambda_1 + ik\alpha, k \in \mathbb{Z}\}$ for some $\alpha > 0$, there exists a sequence $(g_k, \psi_k)_{k \in \mathbb{Z}}$ such that $\mathcal{L}g_k = (\lambda_1 + ik\alpha)g_k$, $\mathcal{L}^*\psi_k = (\lambda_1 + ik\alpha)\psi_k$ and $\langle g_k, \psi_k \rangle = 1$, and for all $f \in X$, in the sense of the topology $\mathcal{T}$, the projection

$$ \Pi f = \lim_{n \to \infty} \frac{1}{n} \sum_{\ell=0}^{n} \sum_{k=-\ell}^{\ell} \langle f, \psi_k \rangle g_k $$

is well defined and

$$ \tilde{S}_t f - \tilde{S}_t \Pi f \to 0 \quad \text{as } t \to +\infty. $$

**Remark 5.26.** In Theorem 5.25, the assumptions that $\lambda_1$ is isolated and $\Sigma_P^+ \ne \{\lambda_1\}$ might seem difficult to check in practice. We indicate here some ways to verify them.

(i) The condition that $\lambda_1$ is an isolated eigenvalue is for instance guaranteed under the assumptions of Theorem 5.6 or Theorem 6.6.

(ii) The condition that $\Sigma_P^+$ is not restricted to $\{\lambda_1\}$ can be guaranteed by verifying that (E2) does not hold. Indeed, if $\Sigma_P^+ = \{\lambda_1\}$, then Theorem 5.23 imposes (E2) to hold.

The result in Theorem 5.25 can be compared for instance to [41, Thm. 14.19], although our hypotheses are slightly more general. Our proof is also more direct than in [41] and it additionally provides an explicit expression of the projection on the boundary eigenspace $\text{Span}(g_k)_{k \in \mathbb{Z}}$. The proof of Theorems 5.23 and 5.25 relies on the theory of almost periodic functions which dates back to H. Bohr. There is a large literature on the subject and we refer for instance to the book of Corduneanu [112] for a comprehensive introduction. There are several equivalent definitions of almost periodic functions and we will use the following one. A function $f \in C_b(\mathbb{R}, X)$, i.e. a bounded continuous function from $\mathbb{R}$ to $X$, is said to be almost periodic if the set $\{f(\cdot + \tau), \tau \in \mathbb{R}\}$ is relatively compact in $C_b(\mathbb{R}, X)$. The set of almost periodic functions is a sub-algebra of $C_b(\mathbb{R}, X)$, and also the closure of the space of periodic functions in $C_b(\mathbb{R}, X)$. We start with the proof of Theorem 5.23 and Theorem 5.25 in the case when $S$ satisfies the condition (1). Then we deduce the cases (2), (3) and (4) from the case (1).
spectrum of $S_T$ if and only if 0 is in the spectrum of $I - (\lambda - V_T)^{-1}K_T$, or in the spectrum of $I - K_T(\lambda - V_T)^{-1}$. Indeed, solving $(\lambda - S_T)f = g$ is equivalent to, on the one hand,

$$(I - (\lambda - V_T)^{-1}K_T)f = (\lambda - V_T)^{-1}g,$$

and in the other hand,

$$(I - K_T(\lambda - V_T)^{-1})(\lambda - V_T)f = g.$$

So if $\lambda \in \Sigma(S_T) \cap B_{\beta_0}^c$ then $1 \in \Sigma((\lambda - V_T)^{-1}K_T)$. Since $(\lambda - V_T)^{-1}K_T$ is a compact operator, the classical Fredholm alternative (see for instance [71, Thm. 6.6]) asserts that its spectrum is made of eigenvalues with finite geometric multiplicity, and then so does for $\Sigma(S_T) \cap B_{\beta_0}^c$. We can also prove, by adapting the proof of [71, Lem. 6.2], that these eigenvalues are isolated, and thus $\Sigma(S_T) \cap B_{\beta_0}^c$ is made of a finite number of isolated eigenvalues with finite geometric multiplicity. Since $e^{T\Sigma(\mathcal{L})} \subset \Sigma(S_T)$, we deduce that $\Sigma(\mathcal{L}) \cap \{z \in \mathbb{C}, \Re e(z) \ge \kappa_0\}$ is also made of a finite number of isolated eigenvalues with finite geometric multiplicity.

**Third step.** We prove the existence of a spectral gap and we conclude.

Since $\Sigma(\mathcal{L}) \cap \{z \in \mathbb{C}, \Re e(z) \ge \kappa_0\}$ is finite, $\lambda_1$ is simple, and the boundary spectrum of $\mathcal{L}$ is a group, we deduce the existence of $\varepsilon > 0$ such that $\Sigma(\mathcal{L}) \cap \{z \in \mathbb{C}, \Re e(z) \ge \lambda_1 - \varepsilon\} = \{\lambda_1\}$. The spectral mapping theorem in Lemma 5.27 then ensures that $\Sigma(S_T) \cap \{z \in \mathbb{C}, |z| \ge e^{(\lambda_1 - \varepsilon)T}\} = \{e^{\lambda_1 T}\}$ and that $e^{\lambda_1 T}$ is simple with eigenspace spanned by $f_1$. The restriction $S_T^\perp$ of $S_T$ to the invariant subspace $X_\perp := \{f \in X, \langle \phi_1, f \rangle = 0\}$ thus has a spectral radius smaller than $e^{(\lambda_1 - \varepsilon)T}$. The spectral radius formula (see [334, Thm. 10.13] for instance) then ensures that

$$\lim_{n \to \infty} \|S_{nT}^\perp\|^{1/n} = r(S_T^\perp) \le e^{(\lambda_1 - \varepsilon)T}.$$

This guarantees, for any $\eta \in (0, \varepsilon)$, the existence of a constant $C_\eta > 0$ such that for all $f \in X_\perp$ and all $t \ge 0$

$$\|e^{-\lambda_1 t} S_t f\| \le C_\eta e^{-\eta t} \|f\|,$$

and the proof is complete.

□

Let us now present a variant of another classical result known as the Gearhart-Prüss Theorem in [175, 326], see also the contributions of Herbst [211] and Greiner [15, A-II.7] as well as the more constructive proof [152, Thm. V.1.11] and recently [206] based on techniques developed in or related to [364, 61].

**Theorem 5.30.** Consider a positive semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ satisfies the conclusions (C2) about the existence, positivity and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. We assume furthermore that $X$ is an Hilbert space and that there exist $\kappa < \lambda_1$ and $R > 0$ such that

$$(i) \sup_{z \in \Delta_\kappa \setminus B_R} \|\mathcal{R}(z)\|_{\mathcal{B}(X)} < \infty;$$

$$(ii) \Sigma(\mathcal{L}) \cap \Delta_\kappa \subset \Sigma_d(\mathcal{L}) \cap B_R.$$

Then the exponential stability (E31) holds (without constructive estimate).

**Proof of Theorem 5.30.** The spectral information (C2) and (ii) together imply (C3) (because of Theorem 5.5) and that there exists $\kappa^* \in (\kappa, \lambda_1)$, such that $\Sigma(\mathcal{L}) \cap \Delta_{\kappa^*} = \{\lambda_1\}$. The operator $\mathcal{L}$ on $X_0 := (\text{vect}\{f_1\})^\perp$ thus satisfies $\sup_{z \in \Delta_{\kappa^*}} \|\mathcal{R}(z)\|_{\mathcal{B}(X_0)} < \infty$, and we conclude thanks to [152, Thm. V.1.11]. The lack of constructively here only comes from the fact that our assumptions do not provide any information on the spectral gap $\lambda_1 - \kappa > 0$. □

**Remark 5.31.** Except of the Hilbert space framework, the assumptions made in Theorem 5.30 are slightly weaker than those of Theorem 5.28, and are indeed established during the proof of Theorem 5.28: such an information at the level of the resolvent is a bit easier to establish than a similar estimate at the level of the semigroup. In the splitting framework (5.27) and its resolvent counterpart (2.22), we typically only have to show

$$(5.29) \quad \sup_{\kappa \le \Re e z \le \kappa_1} \|V(z)\|_{\mathcal{B}(X)} < \infty, \quad \lim_{r \to \infty} \sup_{\kappa \le \Re e z \le \kappa_1, |\Im m z| \ge r} \|W(z)\|_{\mathcal{B}(X)} = 0,$$

for some $\kappa < \lambda_1$, and $W(z) \in \mathcal{K}(X)$ for any $z \in \Delta_\kappa$. That last claim is classical (see for instance [190]) and we only briefly sketch the proof. On the one hand, from the first and the last estimates,
Proof of Theorems 5.23 and 5.25 in the case (1). Step 1. Let $f \in X$. Since the trajectory $(\tilde{S}_t f)_{t \ge 0}$ is relatively compact, we infer from [200, Thm. 8] (with $U(\tau, t) = S_t$ and thus no periodicity condition on $U$) the existence of an almost periodic eternal solution $g$ of the rescaled semigroup $\tilde{S}$, i.e. a function $g: \mathbb{R} \to X$ such that $g(t+\tau) = \tilde{S}_\tau g(t)$ for all $t \in \mathbb{R}$ and $\tau \ge 0$, such that

$$ \lim_{t \to +\infty} \|\tilde{S}_t f - g(t)\| = 0. $$

The end of the proof consists in characterizing the function $g$ in the situations of Theorems 5.23 and 5.25. For $\lambda \in \mathbb{R}$, we define the Bohr transformation of the almost-periodic function $g$ by

$$ c_{\lambda}(g) = \lim_{T \to +\infty} \frac{1}{T} \int_{0}^{T} e^{-i\lambda t} g(t) dt, $$

which is known to exists, see [112, Thm. 3.4], since $e^{-i\lambda t}g(t)$ is also almost periodic. Since $e^{-i\lambda t}g(t)$ is besides an eternal solution of the semigroup $e^{-i\lambda t}\tilde{S}_t$ with infinitesimal generator $\mathcal{L}_\lambda = \mathcal{L}-\lambda_1-i\lambda$, we have that

$$ \mathcal{L}_{\lambda} \int_{0}^{T} e^{-i\lambda t} g(t) dt = g(T) - g(0). $$

Dividing by $T$ the above expression, passing to the limit $T \to +\infty$ and using that $\mathcal{L}_\lambda$ is a closed operator, we get

$$ \mathcal{L}_{\lambda} c_{\lambda}(g) = 0. $$

In other words, we have established

$$ \mathcal{L}c_{\lambda}(g) = (\lambda_1 + i\lambda)c_{\lambda}(g) $$

and $\lambda_1 + i\lambda$ is an eigenvalue of $\mathcal{L}$ if $c_\lambda(g) \neq 0$.

Step 2. We deduce that if the boundary spectrum is trivial, as in Theorem 5.23, then necessarily $c_\lambda(g) = 0$ for all $\lambda \neq 0$. By the uniqueness theorem, see for instance [112, Thm. 4.7], we get that $g$ is constant. Due to the conservation law $\langle \tilde{S}_t f, \phi_1 \rangle = \langle f, \phi_1 \rangle$ and the simplicity of the eigenvalue 0, we get that $g = \langle f, \phi_1 \rangle f_1$ and the result of the case (1) in Theorem 5.23 is proved.

Step 3. In the case of Theorem 5.25, the boundary spectrum is not trivial and we know from Theorem 5.5 that $\Sigma_P^+(\tilde{\mathcal{L}})$ is an additive subgroup of $i\mathbb{R}$, made of algebraically simple eigenvalues. Due to the assumption that $\lambda_1$ is isolated, this subgroup must be discrete and $\Sigma_P^+(\mathcal{L})$ is thus given by $\{\lambda_1 + i\alpha k, k \in \mathbb{Z}\}$ for some $\alpha > 0$. As a consequence, any $\lambda$ such that $c_\lambda(g) \neq 0$ is necessarily of the form $\lambda = \alpha k$ for some $k \in \mathbb{Z}$. By the uniqueness theorem, $g$ is then a $\alpha$-periodic function which is given, due to Fejér's theorem, by

$$ g(t) = \lim_{n \to \infty} \frac{1}{n} \sum_{\ell=0}^{n} \sum_{k=-\ell}^{\ell} c_{\alpha k}(g) e^{i\alpha k t}. $$

Consider $(g_k, \psi_k)$ two positive direct and dual eigenvectors of $\mathcal{L}$ associated to the eigenvalue $i\alpha k$ such that $\langle g_k, \psi_k \rangle = 1$. From the conservation laws $\langle \tilde{S}_t f, \psi_k \rangle = \langle f, \psi_k \rangle e^{i\alpha k t}$ and the algebraic simplicity of the eigenvalues $i\alpha k$, we get that $c_{\alpha k}(g) = \langle f, \phi_k \rangle g_k$, and the result is proved. $\square$

Proof of Theorems 5.23 and 5.25 in the case (2). Since $Y$ is separable, we can find a sequence $(\varphi_n)_{n \ge 1} \subset Y$ which satisfies $\|\varphi_n\| = 1$ and span($\varphi_n$) is dense in $Y$. We can then define on $X$ the norm $\| \cdot \|_*$ by setting

$$ (5.23) \qquad \|f\|_* = \sum_{n=1}^{\infty} 2^{-n} |\langle f, \varphi_n \rangle|. $$

On bounded subsets of $X$, the topology of this norm is the same as the weak-* topology, or more explicitly

$$ f_n \rightharpoonup f * \sigma(Y', Y) \Leftrightarrow (\sup \|f_n\| < \infty \text{ and } \|f_n - f\|_* \to 0). $$

Since by assumption the trajectory $(\tilde{S}_t f)$ is bounded, it is weakly-* relatively compact, and so relatively compact in $(X, \| \cdot \|_*)$. It is also clear that the semigroup $S$ is continuous for the weak norm $\| \cdot \|_*$. The normed space $(X, \| \cdot \|_*)$ is not a Banach space, but the proof of Theorem 5.25 actually only requires, for applying [200, Thm. 8], that the closed balls of $X$ are complete metric spaces, which is the case for the distance induced by $\| \cdot \|_*$. Applying the case (1) of Theorems 5.23 and 5.25 then yields the result. $\square$
we deduce that $\Sigma(\mathcal{L}) \cap \Delta_{\kappa} \subset \Sigma_d(\mathcal{L})$ thanks to Theorem 5.6. As in the proof of Theorem 5.6 and with the usual notations, we also have

$$ (I - W(z))\mathcal{R}_{\mathcal{L}}(z) = V(z), \quad \forall z \in \Delta_{\kappa}, $$

where $I - W(z)$ is invertible and $\|(I - W(z))^{-1}\|_{\mathcal{B}(X)} \le 2$ for any $z \in \mathbb{C}$ such that $\kappa \le \Re z \le \kappa_1$, $|\Im z| \ge R$ and $R$ is large enough. We immediately deduce that the condition (i) in Theorem 5.30 holds.

We end this section by a more recent result which is similar to the Gearhart-Prüss Theorem but is not restricted to an Hilbert space.

**Theorem 5.32.** Consider a positive semigroup $S$ on a Banach lattice $X$ such that its generator $\mathcal{L}$ satisfies the conclusions (C2) about the existence, positivity and uniqueness of the first eigentriplet $(\lambda_1, f_1, \phi_1)$. We further assume that $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $0 \le \mathcal{A} \in \mathcal{B}(X)$, $S_{\mathcal{B}} \ge 0$ and the associated operators $V$ and $W$ defined by (3.14) satisfy (5.27) for some $\kappa < \lambda_1$ and that the resolvent counterpart $W$ defined by (2.22) satisfies (5.29) and more precisely

$$ \sup_{\kappa \le \Re z \le \kappa_1} \|\langle z \rangle^{\alpha} W(z)\|_{\mathcal{B}(X)} < \infty, $$

with $\alpha > 1$. Then the exponential stability (E31) holds (without constructive estimate).

The proof of Theorem 5.32 is a mere adaptation of [278, Thm. 3.1] (see also [273]) and it is thus skipped. The needed estimates are a bit stronger than those of Remark 5.31, but in the applications, they are not really more demanding. They also hold at the level of the resolvent instead of what is assumed in the statement of Theorem 5.28.

We conclude by emphasizing again on the fact that all the above results are not constructive. We propose in the next part an alternative approach which is constructive.

## 6. QUANTITATIVE STABILITY

In this section we establish some quantitative stability results in the spirit of the Doblin, Harris, Meyn-Tweedie theory for Markov semigroup.

**6.1. About quantified positivity conditions.** We briefly discuss some positivity conditions related to the strong maximum principle and barriers techniques. The issue is about how quantify the strong maximum principle

$$ f \in X_+ \setminus \{0\}, (\kappa_1 - \mathcal{L})f \ge 0 \text{ imply } f > 0 \text{ or } f \gg 0 $$

or the related strong positivity of the associated semigroup. A possible way can be achieved with the help of a barrier functions family $\mathcal{G} \subset X_+$ and a second weaker (semi)norm $[\cdot]$ used for normalization. Let us then introduce the two conditions

$$ (6.1) \qquad \forall R > 0, \exists g_i \in \mathcal{G}, \forall f \in X_+, [f] = 1, \|f\| \le R, $$

we have

(i) $S_T f \ge g_1$ (for some $T > 0$)

or

(ii) $f \ge g_2$ if $(\kappa_1 - \mathcal{L})f \ge 0$.

Point (ii) is a quantified version of the strong maximum principle when $\mathcal{G} \subset X_+$ and it is always a consequence of the positivity condition (i). Assume indeed that (i) holds (for some $T > 0$) and that $f$ satisfies the requirements (6.1) and $(\kappa_1 - \mathcal{L})f \ge 0$. We then write

$$ \frac{d}{dt}(e^{(\mathcal{L}-\kappa_1)t}f) = e^{(\mathcal{L}-\kappa_1)t}(\mathcal{L}-\kappa_1)f \le 0, $$

so that

$$ f \ge e^{(\mathcal{L}-\kappa_1)T}f = e^{-\kappa_1 T}S_T f \ge e^{-\kappa_1 T}g_2 =: g_1, $$

with $g_2$ given by condition (i). The reciprocal implication is not clear, see however Lemma 4.8-(3).

Let us now make a list of possible quantified positivity conditions of Doblin-Harris type for a linear (and continuous) operator $A: X \to X$:
*Proof of Theorems 5.23 and 5.25 in the case (3).* We consider $f \in X$ and, repeating the proof of Step 2 in Theorem 4.22, we get that $(S_t f)_{t \ge 0}$ belongs to a weak compact set $\mathcal{G}$ of $L_{\phi_1}^1$. We define the norm $\|\cdot\|_*$ by (5.23) for a sequence $(\varphi_n)_{n \ge 1} \subset C_c(E)$ which satisfies $\|\varphi_n\|_{L^\infty} = 1$ and span$(\varphi_n)$ is dense in $C_0(E)$. This norm induces a metric on $\mathcal{G}$ which is topologically equivalent to the weak convergence on $L_{\phi_1}^1$. The trajectory $(\tilde{S}_t f)$ is then relatively compact in $(\mathcal{G}, \|\cdot\|_*)$ and the semigroup $S$ is continuous for the weak norm $\|\cdot\|_*$. We conclude as in the proof of the case (2). $\square$

*Proof of Theorems 5.23 and 5.25 in the case (4).* From the step 3 of the proof of Theorem 4.22, we know that for any $f \in \mathcal{X}^k$ the trajectory $(\tilde{S}_t f)$ is compact for the strong topology of $L_{\phi_1}^1$. We may then conclude similarly as in the case (1), using that $\mathcal{X}^k$ is dense in $X$ for the norm of $L_{\phi_1}^1$. $\square$

**5.6. A word about spectral analysis argument.** The aim of this section is to recall some more or less classical results which make possible to slightly improve the conclusions of the results presented in the previous section by additionally assuming some spectral gap at the level of the operator or the semigroup. More specifically, we are interested in some accurate versions of a partial, but principal spectral mapping theorem which asserts that

$$ (5.24) \qquad \Sigma(e^{t\mathcal{L}}) \cap B^c(0, e^{\kappa t}) = e^{t\Sigma(\mathcal{L}) \cap \overline{\Delta}_\kappa}, \quad \forall t \ge 0, $$

for some $\kappa < \lambda_1$, and even more precisely, we wish to establish the following geometric (or exponential) asymptotic stability

(E31) there exist some constants $\kappa < \lambda_1$ and $C \ge 1$ such that for any $f \in X$, there holds

$$ (5.25) \qquad \|\tilde{S}(t)f - \langle f, \phi_1 \rangle f_1\| \le \Theta(t)\|f - \langle f, \phi_1 \rangle f_1\|, \quad \forall t \ge 0, \forall f \in X, $$

with the decay rate function $\Theta(t) := C e^{(\kappa - \lambda_1)t}$.

In order to discuss the several results we present, we recall the splitting framework

$$ (5.26) \qquad S = V + W * S, \qquad \|V(t)\|_{\mathcal{B}(X)} + \|W(t)\|_{\mathcal{B}(X)} \lesssim e^{\kappa t}, $$

for the same $\kappa \in \mathbb{R}$ as above.

We start by recalling the spectral mapping theorem for the point spectrum, which states that

$$ (5.27) \qquad \Sigma_P(e^{t\mathcal{L}}) \setminus \{0\} = e^{t\Sigma_P(\mathcal{L})}. $$

We actually give a slightly more precise version of this result. For $\xi \in \mathbb{C}$ and $t > 0$, we define $\Lambda_t(\xi) := \{\lambda \in \mathbb{C}, e^{\lambda t} = \xi\}$. Clearly if $\lambda_0 \in \Lambda_t(\xi)$, then $\Lambda_t(\xi) = \{\lambda_0\} + \frac{2i\pi}{t}\mathbb{Z}$.

**Lemma 5.27** (Spectral mapping theorem for the point spectrum). For a semigroup $(S_t)_{t \ge 0}$ with infinitesimal generator $\mathcal{L}$ we have, for any $\kappa \in \mathbb{R}$ and $t > 0$,

$$ (5.28) \qquad \Sigma_P(S_t) \cap \{\xi \in \mathbb{C}; |\xi| \ge e^{\kappa t}\} = e^{t(\Sigma_P(\mathcal{L}) \cap \{\lambda \in \mathbb{C}; \Re\lambda \ge \kappa\})}. $$

More precisely

- If $f \in X$ is an eigenvector of $\mathcal{L}$ with corresponding eigenvalue $\lambda \in \mathbb{C}$, then $f$ is an eigenvector of $S_t$ with corresponding eigenvalue $e^{\lambda t}$,

- If $f \in X$ is an eigenvector of $S_t$ with corresponding eigenvalue $\xi \in \mathbb{C}$, then there exists $\lambda \in \Lambda_t(\xi)$ such that $g := \int_0^t e^{-\lambda s} S_s f \, ds$ is an eigenvector of $\mathcal{L}$ with corresponding eigenvalue $\lambda$.

Clearly, taking the union over all $\kappa \in \mathbb{R}$ of (5.28) yields (5.27).

*Proof of Lemma 5.27.* Fix $\kappa \in \mathbb{R}$, $t > 0$, and define

$$ \Sigma_S := \Sigma_P(S_t) \cap \{\xi \in \mathbb{C}; |\xi| \ge e^{\kappa t}\}, \quad \Sigma_\mathcal{L} := \Sigma_P(\mathcal{L}) \cap \{\lambda \in \mathbb{C}; \Re\lambda \ge \kappa\}. $$

One inclusion is clear: if $\lambda \in \Sigma_\mathcal{L}$ and $\mathcal{L}f = \lambda f$, $f \ne 0$, then $\xi := e^{\lambda t}$ is such that $S_t f = \xi f$ and thus $\xi \in \Sigma_P(S_t)$, $|\xi| = e^{\Re\lambda t} \ge e^{\kappa t}$. We have proved $e^{t\Sigma_\mathcal{L}} \subset \Sigma_S$. In the other way around, let us take $\xi \in \Sigma_S$ and $f \in X_\mathbb{C} \setminus \{0\}$ such that $S_t f = \xi f$. Then let us pick up $\lambda_0 \in \mathbb{C}$ such that $\xi = e^{\lambda_0 t}$ and $\phi \in X'$ such that $\langle \phi, f \rangle \ne 0$. For any $k \in \mathbb{Z}$, we have $\xi = e^{\lambda_0 t + 2ik\pi}$ and so

$$ 0 = e^{-(\lambda_0 + \frac{2ik\pi}{t})t} S_t f - f = \left(\mathcal{L} - \lambda_0 - \frac{2ik\pi}{t}\right) \int_0^t e^{-(\lambda_0 + \frac{2ik\pi}{t})s} S_s f \, ds. $$

If $g_k := \int_0^t e^{-(\lambda_0 + \frac{2ik\pi}{t})s} S_s f \, ds$ is non-zero for some $k \in \mathbb{Z}$, we deduce that $g_k$ is an eigenvector of $\mathcal{L}$ with corresponding eigenvalue $\lambda := \lambda_0 + \frac{2ik\pi}{t}$. Since besides $\Re\lambda = \frac{1}{t} \log|\xi| \ge \kappa$, we get that
(P1') $\exists g_0 \in X_+ \setminus \{0\}$, $\exists \psi_0 \in X_+ \setminus \{0\}$, $\forall f \in X_+$, $Af \ge g_0 \langle f, \psi_0 \rangle$;

(P2') $\exists g_0 \in X_+ \setminus \{0\}$, $\exists \psi_0 \in X'_+$, $\forall f \in X_+$, $Af \ge g_0 \langle f, \psi_0 \rangle$;

(P3') $\exists g_0 \in X_+$, $\exists \psi_0 \in X'_+ \setminus \{0\}$, $\forall f \in X_+$, $Af \ge g_0 \langle f, \psi_0 \rangle$;

(P4') $\exists g_0 \in X_+$, $\exists \psi_0 \in X'_+$, $\forall f \in X_+$, $Af \ge g_0 \langle f, \psi_0 \rangle$.

We summarize some elementary relations between these conditions and those listed in Section 4.2.

**Lemma 6.1.** We have $(P2') \Rightarrow (P2) \Rightarrow (P1)$, $(P3') \Rightarrow (P3) \Rightarrow (P1)$, $(P4') \Rightarrow ((P4), (P3'))$, $(P2')$ as well as $(P4) \Rightarrow ((P3), (P2))$.

We also have: A satisfies $(P2')$ iff $A^*$ satisfies $(P3')$; A satisfies $(P3')$ iff $A^*$ satisfies $(P2')$; A satisfies $(P4')$ iff $A^*$ satisfies $(P4')$.

We finally have: A satisfies $(P2')$ implies $\exists g_0 \in X_+ \setminus \{0\}$, $\exists \kappa > 0$, $Ag_0 \ge \kappa g_0$

*Proof of Lemma 6.1.* We assume $Af \ge g_0 \langle f, \psi_0 \rangle$ for any $f \in X_+$ and some $g_0 \in X_+$, $\psi_0 \in X'_+$. For any $\phi \in X'_+ \setminus \{0\}$ and $f \in X_+$, we have

$$ \langle A^* \phi, f \rangle = \langle \phi, Af \rangle \ge \langle \phi, g_0 \langle f, \psi_0 \rangle \rangle, $$

which implies $A^* \phi \ge \psi_0 \langle \phi, g_0 \rangle$. We thus deduce that A satisfies $(P2')$ (resp. $(P3')$, $(P4))$ implies that $A^*$ satisfies $(P3')$ (resp. $(P2')$, $(P4))$. The other implications can be established in a similar or even simpler way. $\square$

We conclude this introductory section by emphasizing on the fact (as already mentioned above) that $S_L$ satisfies $(Pi')$ implies $R_L(\lambda)$ satisfies $(Pi')$ for any $\lambda \ge \lambda_1$ and $i = 1, \dots, 4$.

**6.2. Asymptotic stability under Doblin condition.** We start with a simple situation. We assume the Doblin condition, namely

$$ (6.2) \qquad \exists T > 0, \exists \psi_0 \gg 0, \exists g_0 > 0, \forall f \ge 0, S_T f \ge g_0 \langle \psi_0, f \rangle, $$

together with the companion positivity condition

$$ (6.3) \qquad \exists r_0 > 0, \quad \langle \phi_1, g_0 \rangle \ge r_0, $$

as well as the strong additional boundedness assumption

$$ (6.4) \qquad \exists R_0 > 0, \quad \phi_1 \le R_0 \psi_0. $$

When $\psi_0 := 1 \in X' \subset L^\infty$, the condition in (6.4) is automatically satisfied with $R_0 := \|\phi_1\| = 1$. Let us first emphasize that (6.3) is a natural condition when $S_L^*$ enjoys a splitting structure similar to (5.19). More precisely, when

$$ \| \tilde{S}^*(t) \phi \| \le \Theta(t) \| \phi \| + \int_0^t \Theta(t-s) [\tilde{S}^*(s) \phi]_{g_0} ds, $$

with $\Theta \in L^1(\mathbb{R}_+) \cap C_0(\mathbb{R}_+)$, we deduce that

$$ 1 = \|\phi_1\| = \|\tilde{S}^*(t)\phi_1\| \le \Theta(t) + \int_0^t \Theta(t-s)[\phi_1]_{g_0} ds, \quad \forall t > 0. $$

Passing to the limit $t \to \infty$, we get (6.3) with $r_0 := \|\Theta\|_{L^1}^{-1}$. Also (6.4) can be deduced from a splitting structure condition on the dual problem. More precisely, we assume that $D(L^\infty) \subset L_{\text{loc}}^1$ and the splitting property $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{A} \in \mathcal{B}(X)$, $R_{\mathcal{B}}(\lambda) \in \mathcal{B}(X) \cap \mathcal{B}(X_+)$ for any $\lambda \ge \kappa$, with $\kappa < \kappa_0 \le \lambda_1$, and the additional regularity condition

$$ (6.5) \qquad (R_{\mathcal{B}^*}(\lambda)A^*)^N : L_{g_0}^1 \to L_{\psi_0^{-1}}^\infty, \quad \forall \lambda > \kappa. $$

Since the dual eigenvector $\phi_1$ satisfies

$$ (\lambda_1 - \mathcal{B}^*) \phi_1 = A^* \phi_1, \quad \lambda_1 > \kappa, $$

and then $\phi_1 = (R_{\mathcal{B}^*}(\lambda_1)A^*)^N \phi_1$, we may use estimate (6.5) and we get that (6.3)-(6.4) holds with the normalization condition $r_0 := 1$ and $R_0 := \|(R_{\mathcal{B}^*}(\lambda)A^*)^N\|_{\mathcal{B}(L_{g_0}^1, L_{\psi_0^{-1}}^\infty)}$.

We are then able to formulate a first quantified stability result.