lemmas, we will assume that (H1)-(H2) holds associated to some constants $\kappa_i \in \mathbb{R}$, $\kappa_0 < \kappa_1$, and we always make the following splitting structure hypothesis

(HS1) there exists a splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ such that $\mathcal{B} - \alpha$ is invertible for any $\alpha \ge \kappa_0$ and

$$ (2.21) \qquad \mathcal{V}(\alpha) := \sum_{i=0}^{N-1} (\mathcal{R}_{\mathcal{B}}(\alpha)\mathcal{A})^i \mathcal{R}_{\mathcal{B}}(\alpha), \quad \mathcal{W}(\alpha) := (\mathcal{R}_{\mathcal{B}}(\alpha)\mathcal{A})^N, $$

are bounded in $\mathcal{B}(X)$ uniformly with respect to $\alpha \ge \kappa_0$ and for some $N \ge 1$, where we recall that $\mathcal{R}_{\mathcal{B}}(\alpha) := (\alpha - \mathcal{B})^{-1}$ is the resolvent of $\mathcal{B}$.

We first present a result also based on a strong compactness property which is assumed to hold however at the level of the resolvent operator. It turns out that we will be able to use that result in most of the applications.

**Lemma 2.8.** (1) We assume (H1)-(H2)-(HS1) and there exists $N \ge 1$ such that

$$ (2.22) \qquad \mathcal{W}(\alpha) \text{ is strongly compact locally uniformly on } \alpha \ge \kappa_0, $$

in the sense that if $\alpha_n \to \alpha$, $\alpha_n \ge \kappa_0$, and $(g_n)$ is a bounded sequence in $X$, then there exist $f \in X$ and a subsequence $(g_{n_k})$ such that $\mathcal{W}(\alpha_{n_k})g_{n_k} \to f$ strongly in $X$. Then condition (H3) holds.

(2) We assume (H1)-(H2) and (HS1) where $\mathcal{R}_{\mathcal{B}}(\alpha)$ is bounded uniformly in $\alpha \ge \kappa_0$, $\mathcal{A} \in \mathcal{B}(X)$ and $\mathcal{W}(\alpha) \in \mathcal{K}(X)$ for any fixed $\alpha \ge \kappa_0$ and some $N \ge 1$. Then condition (H3) holds both for the primal and the dual problems.

**Remark 2.9.** (1) The property (2.22) holds if we assume $\mathcal{W}(\alpha) : X \to \mathcal{X}_1$ is bounded uniformly in $\alpha \ge \kappa_0$ and $\mathcal{X}_1 \subset X$ with strong compact embedding.

(2) The property (2.22) holds if we assume (H1)-(H2)-(HS1) together with the facts that $\mathcal{R}_{\mathcal{B}}(\alpha)$ and $\mathcal{R}_{\mathcal{B}}(\alpha)\mathcal{A}$ are bounded uniformly in $\alpha \ge \kappa_0$ and $\mathcal{W}(\alpha) \in \mathcal{K}(X)$ for any fixed $\alpha \ge \kappa_0$. Consider indeed $\alpha_n \to \alpha$, $\alpha_n \ge \kappa_0$, and $(g_n)$ a bounded sequence in $X$. On the one hand, there exist $f \in X$ and a subsequence $(g_{n_k})$ such that $\mathcal{W}(\alpha)g_{n_k} \to f$ strongly in $X$, because $\mathcal{W}(\alpha) \in \mathcal{K}(X)$. On the other hand, using the resolvent identity $\mathcal{R}_{\mathcal{B}}(\lambda) - \mathcal{R}_{\mathcal{B}}(\mu) = (\mu - \lambda)\mathcal{R}_{\mathcal{B}}(\lambda)\mathcal{R}_{\mathcal{B}}(\mu)$, we have

$$ \mathcal{W}(\alpha) - \mathcal{W}(\alpha_n) = (\alpha_n - \alpha) \sum_{j=1}^{N} (\mathcal{R}_{\mathcal{B}}(\alpha)\mathcal{A})^{N-j} \mathcal{R}_{\mathcal{B}}(\alpha) (\mathcal{R}_{\mathcal{B}}(\alpha_n)\mathcal{A})^j \to 0, $$

so that $\mathcal{W}(\alpha_{n_k})g_{n_k} \to f$ strongly in $X$, and (2.22) holds true.

(3) In the applications part, we will mainly use Lemma 2.8 for establishing (H3). It is very flexible and allows for generalizations such as using two or more different splittings, see Section 9.3 for an example.

(4) We observe that the same conclusion holds when $\mathcal{R}_{\mathcal{L}} = \mathcal{V} + \mathcal{W}\mathcal{R}_{\mathcal{L}}$ for two family of operators $\mathcal{V}(\alpha)$ and $\mathcal{W}(\alpha)$ such that $\mathcal{V}(\alpha)$ is bounded in $\mathcal{B}(X)$ uniformly on $\alpha \ge \kappa_0$ and $\mathcal{W}$ satisfies (2.22). That follows from the fact that (2.20) writes equivalently

$$ \hat{f}_n = \mathcal{V}(\lambda_n)\varepsilon_n + \mathcal{W}(\lambda_n)\hat{f}_n, \quad \lambda_n \searrow \lambda_1, \varepsilon_n \to 0, \|\hat{f}_n\| = 1, $$

and we may then proceed exactly in the same way as below.

**Proof of Lemma 2.8.** We first assume (1). Taking advantage of the splitting structure (HS1), we write equation (2.20) as

$$ (2.23) \qquad (\lambda_n - \mathcal{B})\hat{f}_n = \mathcal{A}\hat{f}_n + \varepsilon_n, $$

or equivalently

$$ \hat{f}_n = \mathcal{R}_{\mathcal{B}}(\lambda_n)\mathcal{A}\hat{f}_n + \mathcal{R}_{\mathcal{B}}(\lambda_n)\varepsilon_n. $$

Iterating that last identity and using the notations (2.21), we get

$$ (2.24) \qquad \hat{f}_n = w_n + v_n, \quad w_n := \mathcal{W}(\lambda_n)\hat{f}_n, \quad v_n := \mathcal{V}(\lambda_n)\varepsilon_n. $$

We observe that $(w_n)$ is strongly relatively compact from (2.22) and $\|\hat{f}_n\|_X = 1$, so that there exist a subsequence $(w_{n_k})$ and $f_1 \in X$ such that $w_{n_k} \to f_1$ strongly in $X$. Since $v_n \to 0$ strongly in $X$, we deduce that $\hat{f}_{n_k} \to f_1$ strongly in $X$, in particular $f_1 \ge 0$ and $\|f_1\|_X = 1$. We conclude that condition (H3) holds as in the proof of Lemma 2.7.
**Proof of Lemma 2.11.** We go back to the proof of Lemma 2.8 and we start with (2.25). We recall that $\|\hat{f}_n\|_{X_0} = 1$ and $\|v_n\|_{X_0} \to 0$ from (2.20) and that $w_n \ge 0$ because $W(\lambda_n)$ is a positive operator. We also observe that

$$ \|w_n\|_{X_1} \le C_W \|\hat{f}_n\|_{X_0} = C_W $$

and

$$ \|w_n\|_{X_0} \ge 1 - \|v_n\|_{X_0} \ge 1/2 $$

for any $n \ge n_*$, with $n_* \ge 1$ large enough, so that $w_n \in C := C_{1/2, C_W}$ for any $n \ge n_*$. By the compactness properties of $C$, there exist a subsequence $(w_{n_k})$ and $f_1 \in X_+ \setminus \{0\}$ such that $w_{n_k} \to f_1$ weakly $\sigma(X, Y)$. Since $v_n \to 0$ strongly in $X$, we deduce that $\hat{f}_{n_k} \to f_1$ weakly $\sigma(X, Y)$ and that ends the proof of (H3). $\square$

We present a typical concrete application of the preceding result.

**Lemma 2.13.** We assume $X = L^p(E, \mathcal{E}, \mu)$, $p \in [1, \infty)$, (H1)-(H2)-(HS1) with $A \ge 0$, $R_B(\alpha) \ge 0$ for $\alpha \ge \kappa_0$, and there exists $N \ge 1$ such that

$$ (2.27) \qquad W(\alpha) : X \to \mathcal{X}_1 \text{ is uniformly bounded in } \alpha \ge \kappa_0, $$

for a subspace $\mathcal{X}_1 \subset X$ such that $\{g^p; g \ge 0, \|g\|_{X_1} \le R_1\}$ is a weakly compact subset of $L^1(E)$ for any $R_1 > 0$. Then condition (H3) holds.

**Remark 2.14.** (1) A typical example in the above statement is $\mathcal{X}_1 := L^q \cap L_m^p$ for some exponent $q > p$ and some weight function $m : E \to [1, \infty)$ such that $m(x) \to \infty$ as $x \to \infty$.

(2) The same result holds under the condition that if $(u_n)$ is a nonnegative and bounded sequence in $L^p$ then the nonnegative sequence $w_n := W(\lambda_n)u_n$ is such that $w_n^p$ is weakly compact in $L^1$.

**Proof of Lemma 2.13.** For $0 < R_0 < R_1$, we define $C$ by (2.26) with $\mathcal{X}_0 := L^p$. From the weak compactness property made on $\mathcal{X}_1$, we observe that

$$ \alpha(R) := \sup_{g \in C} \|g \mathbf{1}_{E_R^c}\|_{L^p} \to 0, \text{ as } R \to \infty, $$

and

$$ \beta(M) := \sup_{g \in C} \|g \mathbf{1}_{g \ge M}\|_{L^p} \to 0, \text{ as } M \to \infty. $$

For $g \in C$, we may then write

$$ R_0 \le \|g\|_{L^p} \le \|g \wedge M \mathbf{1}_{E_R}\|_{L^p} + \|g \mathbf{1}_{E_R^c}\|_{L^p} + \|g \mathbf{1}_{g \ge M}\|_{L^p} $$

and thus

$$ M^{1-1/p} \|g \mathbf{1}_{E_R}\|_{L^1}^{1/p} \ge \|g \wedge M \mathbf{1}_{E_R}\|_{L^p} \ge R_0 - \alpha(R) - \beta(M) \ge R_0/2, $$

for some $R, M > 0$ large enough. On the one hand, from the reflexivity of $L^p$ or the Dunford-Pettis theorem, the set $C$ is relatively sequentially compact for the weak topology $\sigma(L^p, L^{p'})$. On the other hand, because $\mathbf{1}_{E_R} \in L^{p'}$ the last estimate implies that any element $g^* \in \bar{C}$, where the closure is taken in the sense of the weak topology $\sigma(L^p, L^{p'})$, satisfies

$$ \langle g^*, \mathbf{1}_{E_R} \rangle = \|g^* \mathbf{1}_{E_R}\|_{L^1} \ge M^{1-p} (R_0/2)^p > 0, $$

and in particular $0 \notin \bar{C}$. We deduce that (H3) holds as a consequence of Lemma 2.11. $\square$

We present a second kind of result where some weak compactness is involved.

**Lemma 2.15.** We assume (H1)-(H2)-(HS1) and there exists $N \ge 1$ such that

$$ (2.28) \qquad W(\alpha) : \mathcal{X}_0 \to X \subset \mathcal{X}_0 \text{ is uniformly bounded in } \alpha \ge \kappa_0 $$

and, denoting $\mathcal{X}_1 := X$, the set $C$ defined by (2.26) satisfies the same properties as the ones stated in Lemma 2.11. Then condition (H3) holds.

**Remark 2.16.** If we replace the norm $\|\cdot\|_{X_0}$ by a seminorm $\|f\|_{X_0} := \langle|f|, \varphi_0\rangle$, $\varphi_0 \in Y_+$, and we define $C$ accordingly by (2.26), and if we assume that $X = Y'$ with $Y$ separable, then $C$ satisfies the same compactness properties as required in the statement of Lemma 2.11. If we further assume that (2.28) holds where $\mathcal{X}_0$ is endowed with the above seminorm, we may repeat the proof below in order to obtain that (H3) holds in that situation (see also Lemma 2.19 and its proof for a slightly generalized situation).
We next assume (2). As observed in Remark 2.9-(2), the property (2.22) holds and thus also the condition (H3) for the primal problem. We claim that the same locally uniform strong compactness property (2.22) holds for the dual problem at order $N + 1$ and thus condition (H3) holds for the dual problem. We may indeed use Remark 2.9-(2) since then $\mathcal{R}_{\mathcal{B}^*}(\alpha)$ and $\mathcal{A}^*\mathcal{R}_{\mathcal{B}^*}(\alpha)$ are bounded uniformly in $\alpha \ge \kappa_0$ and

$$ (\mathcal{A}^*\mathcal{R}_{\mathcal{B}^*}(\alpha))^{N+1} = \mathcal{A}^*\mathcal{W}(\alpha)^*\mathcal{R}_{\mathcal{B}^*}(\alpha) \in \mathcal{K}(Y), \quad \forall \alpha \ge \kappa_0, $$

as a product of two bounded operator with a compact operator.

□

**Remark 2.10.** Instead of (HS1) in Lemma 2.8, one can assume that there exists a splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ and $N \ge 1$ such that $\mathcal{B} - \alpha$ is invertible for any $\alpha \ge \kappa_0$ and

$$ \mathcal{R}_{\mathcal{B}}(\alpha) := (\alpha - \mathcal{B})^{-1}, \quad \check{\mathcal{V}}(\alpha) := \sum_{i=0}^{N-1} (\mathcal{A}\mathcal{R}_{\mathcal{B}}(\alpha))^i, \quad \check{\mathcal{W}}(\alpha) := (\mathcal{A}\mathcal{R}_{\mathcal{B}}(\alpha))^N $$

are respectively bounded in $\mathcal{B}(X)$ uniformly with respect to $\alpha \ge \kappa_0$ and strongly compact locally uniformly on $\alpha \ge \kappa_0$. Starting indeed again from (2.23) and defining $h_n := (\lambda_n - \mathcal{B})\hat{f}_n$, we may write

$$ h_n = \mathcal{A}\mathcal{R}_{\mathcal{B}}(\lambda_n)h_n + \varepsilon_n. $$

Observing that $\|h_n\|_X \ge \|\mathcal{R}_{\mathcal{B}}(\lambda_n)\|_{\mathcal{B}(X)}^{-1} \ge c > 0$ by assumption, we deduce that $\hat{h}_n := h_n/\|h_n\|_X$ satisfies

$$ \hat{h}_n = \tilde{w}_n + \tilde{v}_n, \quad \tilde{w}_n := \check{\mathcal{W}}(\lambda_n)\hat{h}_n, \quad \tilde{v}_n := \check{\mathcal{V}}(\lambda_n)\hat{\varepsilon}_n, $$

with $\|\hat{h}_n\| = 1$ and $\hat{\varepsilon}_n := \varepsilon_n/\|h_n\|_X \to 0$. Similarly as in the proof of Lemma 2.8, we conclude to the existence of subsequence $(\hat{h}_{n_k})$ and $h_1 \in X_+ \setminus \{0\}$ such that $\hat{h}_{n_k} \to h_1$ strongly in $X$. Defining $f_1 := \mathcal{R}_{\mathcal{B}}(\lambda_1)h_1/\|\mathcal{R}_{\mathcal{B}}(\lambda_1)h_1\|$, we have again $\hat{f}_{n_k} \to f_1$ strongly in $X$ and next that condition (H3) holds.

As we see now, strong compactness is not really necessary.

**Lemma 2.11.** We assume (H1)-(H2)-(HS1) and there exists $N \ge 1$ such that

$$ \mathcal{W}(\alpha) : X \to \mathcal{X}_1 \subset X \text{ is uniformly bounded in } \alpha \ge \kappa_0 $$

and, denoting $\mathcal{X}_0 := X$, we assume that for any $R_1 \ge R_0 > 0$ the set

$$ (2.25) \qquad C = C_{R_0, R_1} := \{g \in X_+; \|g\|_{\mathcal{X}_0} \ge R_0, \|g\|_{\mathcal{X}_1} \le R_1\} $$

is relatively sequentially compact for the weak topology $\sigma(X, Y)$ and $0 \notin \bar{C}$, where the closure is taken in the sense of the weak topology $\sigma(X, Y)$. We further assume that either (1) $\mathcal{W}(\alpha)$ is positive or (2) there exists a constant $C_{\mathcal{X}_1} > 0$ such that $\|g_+\|_{\mathcal{X}_1} \le C_{\mathcal{X}_1}\|g\|_{\mathcal{X}_1}$ for any $g \in \mathcal{X}_1$. Then condition (H3) holds.

**Remark 2.12.** When $\mathcal{X}_1$ is a Banach lattice, the condition (2) holds because of (2.1). When $\mathcal{X}_1 \subset \mathcal{X}_0$ with strongly compact embedding the above set $C$ clearly satisfies the required conditions. In particular, Lemma 2.11 somehow generalizes the result stated in Remark 2.9-(1).

*Proof of Lemma 2.11.* We go back to the proof of Lemma 2.8 and we start with (2.24). We recall that $\|\hat{f}_n\|_{\mathcal{X}_0} = 1$ and $\|v_n\|_{\mathcal{X}_0} \to 0$ from (2.20). We first assume (1) so that $w_n \ge 0$ because $\mathcal{W}(\lambda_n)$ is a positive operator. We also observe that

$$ \|w_n\|_{\mathcal{X}_1} \le C_W \|\hat{f}_n\|_{\mathcal{X}_0} = C_W $$

and

$$ \|w_n\|_{\mathcal{X}_0} \ge 1 - \|v_n\|_{\mathcal{X}_0} \ge 1/2 $$

for any $n \ge n_*$, with $n_* \ge 1$ large enough, so that $w_n \in C := C_{1/2, C_W}$ for any $n \ge n_*$. By the compactness properties of $C$, there exist a subsequence $(w_{n_k})$ and $f_1 \in X_+ \setminus \{0\}$ such that $w_{n_k} \to f_1$ weakly $\sigma(X, Y)$. Since $v_n \to 0$ strongly in $X$, we deduce that $\hat{f}_{n_k} \to f_1$ weakly $\sigma(X, Y)$ and that ends the proof of (H3).

We now assume that (2) holds. We define $w'_n := (w_n)_+$ and $v'_n := v_n - (w_n)_-$. We observe that $(w_n)_- = (\hat{f}_n - v_n)_- \le (v_n)_+$ and thus $\|(w_n)_-\|_{\mathcal{X}_0} \le \|v_n\|_{\mathcal{X}_0}$ because of the compatibility of norm and order structures in the Banach space $\mathcal{X}_0 = X$. We immediately deduce that

$$ \hat{f}_n = w'_n + v'_n, \quad w'_n \ge 0, \quad \|v'_n\|_{\mathcal{X}_0} \to 0. $$
Proof of Lemma 2.15. We start here again with (2.25). We have

$$1 = \|\hat{f}_n\|_{X_1} \le C_W \|\hat{f}_n\|_{X_0} + \|v_n\|_{X_1},$$

and thus

$$\|\hat{f}_n\|_{X_0} \ge C_W^{-1}(1 - \|v_n\|_{X_1}) \ge (2C_W)^{-1}$$

for any $n \ge n_*$, with $n_* \ge 1$ large enough, so that $\hat{f}_n \in C := C_{(2C_W)^{-1},1}$, for $n \ge n_*$. By the compactness properties of $C$, there exist a subsequence $(\hat{f}_{n_k})$ and $f_1 \in X_+ \setminus \{0\}$ such that $\hat{f}_{n_k} \to f_1$ weakly $\sigma(X, Y)$. $\square$

We present a variant of Lemma 2.13 which is also a concrete consequence of Lemma 2.11 and Lemma 2.15.

**Corollary 2.17.** We assume (H1)-(H2)-(HS1) in $X = L_{m_0}^{p_0}$, $1 \le p_0 < \infty$, together with the facts that $\mathcal{R}_B(\alpha)$ is positive and bounded in $\mathcal{B}(L_{m_0}^{p_0})$ and $\mathcal{B}(L_{m_1}^{p_1})$ uniformly in $\alpha \ge \kappa_0$, $0 \le \mathcal{A} \in \mathcal{B}(L_{m_0}^{p_0})$ and $(\mathcal{R}_B(\alpha)\mathcal{A})^N$ is bounded in $\mathcal{B}(L_{m_0}^{p_0}, L_{m_1}^{p_1})$ uniformly in $\alpha \ge \kappa_0$ for some $N \ge 1$, with $p_1 > p_0$ and $m_1$ such that $m_0/m_1 \in L^\vartheta$, $1/\vartheta := 1/p_0 - 1/p_1$. Then condition (H3) holds for both the primal and the dual problems.

Proof of Corollary 2.17. On the one hand, we have

$$\mathcal{R}_B(\alpha) + \dots + (\mathcal{R}_B(\alpha)\mathcal{A})^{N-1}\mathcal{R}_B(\alpha) \text{ is bounded in } \mathcal{B}(X) \text{ uniformly in } \alpha \ge \kappa_0,$$

$$\mathcal{W}(\alpha) := (\mathcal{R}_B(\alpha)\mathcal{A})^N \text{ is bounded in } \mathcal{B}(X, \mathcal{X}_1) \text{ uniformly in } \alpha \ge \kappa_0,$$

with $\mathcal{X}_1 := L_{m_1}^{p_1} \subset X$ and thus $\{(gm_0)^{p_0}; g \ge 0, \|g\|_{X_1} \le R_1\}$ is a weakly compact subset of $L^1(E)$ for any $R_1 > 0$. Condition (H3) holds for the direct problem thanks to Lemma 2.13.

On the other hand, we set $Y := X' = L_{\nu_0}^{q_0}$, $q_0 := p'_0$, $\nu_0 := m_0^{-1}$, and we first observe that

$$\mathcal{R}_{B^*}(\alpha) + \dots + (\mathcal{R}_{B^*}(\alpha)\mathcal{A}^*)^{N-1}\mathcal{R}_{B^*}(\alpha) \text{ is bounded in } \mathcal{B}(Y) \text{ uniformly in } \alpha \ge \kappa_0.$$

We next observe that

$$(\mathcal{A}^*\mathcal{R}_{B^*}(\alpha))^{N+1} = \mathcal{A}^*\mathcal{W}(\alpha)^*\mathcal{R}_{B^*}(\alpha) \text{ is bounded in } \mathcal{B}(\mathcal{Y}_0, Y) \text{ uniformly in } \alpha \ge \kappa_0,$$

with $\mathcal{Y}_0 := L_{\nu_1}^{q_1}$, $q_1 := p'_1$, $\nu_1 := m_1^{-1}$. Because $\{(g\nu_1)^{q_1}; g \ge 0, \|g\|_Y \le R_1\}$ is a weakly compact subset of $L^1(E)$ for any $R_1 > 0$, we have from the proof of Lemma 2.13 that the set $\mathcal{C}$ defined by (2.26) for the norms of $\mathcal{Y}_0$ and $\mathcal{Y}_1 := Y$ satisfies the weak compactness property required in the statement of Lemma 2.11. We may thus apply Lemma 2.15 and we deduce that condition (H3) holds for the dual problem. $\square$

Another concrete consequence of Lemma 2.11 and Lemma 2.15 is the following.

**Lemma 2.18.** We assume $X = M_{m_i}^1(E)$ for a continuous weight function $m_i$ on $E$, $i = 0$ or $i = 1$, (H1)-(H2)-(HS1) and there exists $N \ge 1$ such that $(\mathcal{R}_B(\alpha)\mathcal{A})^N : M_{m_0}^1(E) \to M_{m_1}^1(E)$ uniformly in $\alpha \ge \kappa_0$ for another continuous weight function $m_{1-i}$ on $E$ such that $m_1(x)/m_0(x) \to \infty$ as $x \to \infty$. We additionally assume that $\mathcal{A} \ge 0$ and $\mathcal{R}_B(\alpha) \ge 0$ for $\alpha \ge \kappa_0$ when $i = 0$. Then condition (H3) holds.

Proof of Lemma 2.18. We define $\mathcal{X}_i := M_{m_i}^1(E)$ and we consider the set $\mathcal{C}$ defined by (2.26) which is clearly compact for the weak $\sigma(M_{m_1}^1, C_{m_1,0})$ topology. When $X = M_{m_0}^1$, the result follows from Lemma 2.11 while when $X = M_{m_1}^1$, the result is a consequence of Lemma 2.15. $\square$

We may slightly improve the preceding results by considering a more abstract framework and a somehow more general boundedness condition.

**Lemma 2.19.** We assume $X = Y'$, $Y$ separable, (H1)-(H2)-(HS1) and there exist $N \ge 1$, $\gamma \in [0, 1)$ and $\varphi \in Y_+ \setminus \{0\}$ such that for any $\alpha \ge \kappa_0$, there holds

$$(2.29) \qquad \|\mathcal{W}(\alpha)f\|_X \le \gamma \|f\|_X + \langle f, \varphi \rangle_{X,Y},$$

for all $f \in X_+$, or there holds

$$(2.30) \qquad \|\mathcal{W}(\alpha)f\|_X \le \gamma \|f\|_X + \langle \mathcal{W}(\alpha)f, \varphi \rangle_{X,Y},$$

for all $f \in X_+$. Then condition (H3) holds true, and the limit $f_1$ satisfies $\langle f_1, \varphi \rangle_{X,Y} \ge 1 - \gamma > 0$.
Arguing as above and using the additional property of $\mathcal{X}_1$, we have

$$ \|w'_{n}\|_{\mathcal{X}_1} \le C_{\mathcal{X}_1} \|w_n\|_{\mathcal{X}_1} \le C_{\mathcal{X}_1} C_W \quad \text{and} \quad \|w'_{n}\|_{\mathcal{X}_0} \ge 1/2 $$

for any $n \ge n_*$, with $n_* \ge 1$ large enough. We conclude the proof in the same way as previously. $\square$

We present a typical concrete application of the preceding result.

**Lemma 2.13.** We assume $X = L^p(E, \mathcal{E}, \mu)$, $p \in [1, \infty)$, $(\mathbf{H}1)-(\mathbf{H}2)-(\mathbf{HS1})$ and

$$ (2.26) \qquad \mathcal{W}(\alpha) : X \to \mathcal{X}_1 \text{ is uniformly bounded in } \alpha \ge \kappa_0, $$

for some $N \ge 1$ and a subspace $\mathcal{X}_1 \subset X$ such that $\{g^p; g \ge 0, \|g\|_{\mathcal{X}_1} \le R_1\}$ is a weakly compact subset of $L^1(E)$ for any $R_1 > 0$. Then condition (H3) holds.

**Remark 2.14.** (1) A typical example in the above statement is $\mathcal{X}_1 := L^q \cap L_m^p$ for some exponent $q > p$ and some weight function $m : E \to [1, \infty)$ such that $m(x) \to \infty$ as $x \to \infty$.

(2) The same result holds under the condition that if $(u_n)$ is a nonnegative and bounded sequence in $L^p$ then the nonnegative sequence $w_n := \mathcal{W}(\lambda_n)u_n$ is such that $w_n^p$ is weakly compact in $L^1$.

(3) We will use Lemma 2.13 only for the transport equation with an abstract kernel term in Section 8.5.

**Proof of Lemma 2.13.** For $0 < R_0 < R_1$, we define $\mathcal{C}$ by (2.25) with $\mathcal{X}_0 := L^p$. From the weak compactness property made on $\mathcal{X}_1$, we observe that

$$ \alpha(R) := \sup_{g \in \mathcal{C}} \|g \mathbf{1}_{E_R^c}\|_{L^p} \to 0, \text{ as } R \to \infty, $$

and

$$ \beta(M) := \sup_{g \in \mathcal{C}} \|g \mathbf{1}_{g \ge M}\|_{L^p} \to 0, \text{ as } M \to \infty. $$

For $g \in \mathcal{C}$, we may then write

$$ R_0 \le \|g\|_{L^p} \le \|g \wedge M \mathbf{1}_{E_R}\|_{L^p} + \|g \mathbf{1}_{E_R^c}\|_{L^p} + \|g \mathbf{1}_{g \ge M}\|_{L^p} $$

and thus

$$ M^{1-1/p} \|g \mathbf{1}_{E_R}\|_{L^1}^{1/p} \ge \|g \wedge M \mathbf{1}_{E_R}\|_{L^p} \ge R_0 - \alpha(R) - \beta(M) \ge R_0/2, $$

for some $R, M > 0$ large enough. On the one hand, from the reflexivity of $L^p$ or the Dunford-Pettis theorem, the set $\mathcal{C}$ is relatively sequentially compact for the weak topology $\sigma(L^p, L^{p'})$. On the other hand, because $\mathbf{1}_{E_R} \in L^{p'}$ the last estimate implies that any element $g^* \in \bar{\mathcal{C}}$, where the closure is taken in the sense of the weak topology $\sigma(L^p, L^{p'})$, satisfies

$$ \langle g^*, \mathbf{1}_{E_R} \rangle = \|g^* \mathbf{1}_{E_R}\|_{L^1} \ge M^{1-p} (R_0/2)^p > 0, $$

and in particular $0 \notin \bar{\mathcal{C}}$. We deduce that (H3) holds as a consequence of Lemma 2.11. $\square$

We present a second kind of result where some weak compactness is involved.

**Lemma 2.15.** We assume (H1)-(H2)-(HS1) and there exists $N \ge 1$ such that

$$ (2.27) \qquad \mathcal{W}(\alpha) : \mathcal{X}_0 \to X \subset \mathcal{X}_0 \text{ is uniformly bounded in } \alpha \ge \kappa_0 $$

and, denoting $\mathcal{X}_1 := X$, the set $\mathcal{C}$ defined by (2.25) satisfies the same properties as the ones stated in Lemma 2.11. Then condition (H3) holds.

**Remark 2.16.** If we replace the norm $\|\cdot\|_{\mathcal{X}_0}$ by a seminorm $\|f\|_{\mathcal{X}_0} := \langle |f|, \varphi_0 \rangle$, $\varphi_0 \in Y_+$, we define $\mathcal{C}$ accordingly by (2.25) and we assume that $X = Y'$ with $Y$ separable, then $\mathcal{C}$ satisfies the same compactness properties as required in the statement of Lemma 2.11. If we further assume that (2.27) holds, where $\mathcal{X}_0$ is endowed with the above seminorm, we may repeat the proof below in order to obtain that (H3) holds in that situation (see also Lemma 2.19 and its proof for a slightly generalized situation).
The case $X = M_{m_1}^1(E)$ in Lemma 2.18 corresponds here to the first situation where (2.29) holds with $X := M_{m_1}^1(E)$, $\gamma := 0$, $Y := C_{m_0,0}(E)$ and $\varphi := m_0/m_1$.

**Proof of Lemma 2.19.** Starting with (2.25) and using (2.29), we have

$$ \begin{aligned} \|\hat{f}_n\|_X &\le \|\mathcal{W}(\lambda_n)\hat{f}_n\|_X + \|\mathcal{V}(\lambda_n)\varepsilon_n\|_X \\ &\le \gamma \|\hat{f}_n\|_X + \langle \hat{f}_n, \varphi \rangle_{X,Y} + \|v_n\|_X, \end{aligned} $$

so that

$$ \langle \hat{f}_n, \varphi \rangle_{X,Y} \ge 1 - \gamma - \|v_n\|_X. $$

By compactness, there are $f_1 \ge 0$ and a subsequence $(\hat{f}_{n'})$ such that $\hat{f}_{n'} \to f_1$ weak * $\sigma(X, Y)$.
Passing to the limit as $n' \to \infty$ in the above estimate, we find

$$ (2.31) \qquad \langle f_1, \varphi \rangle_{X,Y} = \lim_{n' \to \infty} \langle \hat{f}_{n'}, \varphi \rangle_{X,Y} \ge 1 - \gamma, $$

and in particular $f_1 \ne 0$.

Under the assumption (2.30), modifying slightly the previous argument, we have

$$ \|\hat{f}_n\|_X \le \gamma \|\hat{f}_n\|_X + \langle w_n, \varphi \rangle_{X,Y} + \|v_n\|_X, $$

which, together with

$$ \langle \hat{f}_n, \varphi \rangle_{X,Y} = \langle w_n, \varphi \rangle_{X,Y} + \langle v_n, \varphi \rangle_{X,Y}, $$

implies

$$ \langle \hat{f}_n, \varphi \rangle_{X,Y} \ge 1 - \gamma - \|v_n\|_X + \langle v_n, \varphi \rangle_{X,Y}. $$

By compactness again, there are $f_1 \ge 0$ and a subsequence $(\hat{f}_{n'})$ such that $\hat{f}_{n'} \to f_1$ weak * $\sigma(X, Y)$, and passing to the limit as $n' \to \infty$ in the above estimate, we conclude again to (2.31). $\square$

Let us comment on Lemma 2.19 and in particular the condition (2.30).

In the case when $X = L^\infty(E, \mathcal{E}, \mu) = (L^1(E, \mathcal{E}, \mu))'$, we can relate condition (2.30) to the assumption that there exist $f_0 \in X_+$ and $\varphi \in Y_+ \setminus \{0\}$ such that

$$ (2.32) \qquad \|S_{\mathcal{L}}(t)f_0\|_X \le \langle S_{\mathcal{L}}(t)f_0, \varphi \rangle, \quad \forall t \ge 0. $$

This last condition is reminiscent from conditions that appear in probabilistic inspired methods for the ergodicity of semigroups, see the condition (1b) in Theorem 1.7 but also Assumption (A2) in [103], both in the vein of [131, Condition $\mathcal{Z}$]. Assume indeed (2.32), let $\eta > \kappa_1 - \kappa_0 > 0$ and consider the trivial decomposition $\mathcal{L} = \mathcal{A} + \mathcal{B} = \eta + (\mathcal{L} - \eta)$. Then set $\kappa_B := \kappa_1 - \eta < \kappa_0$, so that for any $\alpha > \kappa_B$, $\mathcal{B} - \alpha = \mathcal{L} - (\eta + \alpha)$ is invertible since $\eta + \alpha > \eta + \kappa_B = \kappa_1$. We thus have for any $\alpha > \kappa_B$

$$ \mathcal{W}(\alpha) := \eta(\alpha - \mathcal{B})^{-1} = \eta \int_0^\infty e^{-(\eta+\alpha)t} S_{\mathcal{L}}(t) dt $$

and (2.32) then ensures that

$$ \|\mathcal{W}(\alpha)f_0\|_X \le \langle \mathcal{W}(\alpha)f_0, \varphi \rangle. $$

We recover (2.30) with $\gamma = 0$ and the difference that $f_0$ is fixed here.

As a Corollary of Lemma 2.18 or Lemma 2.19 and anticipating on the material of part 3, we present now a situation very classical in stochastic processes theory.

**Corollary 2.20.** We consider a positive semigroup $S = S_{\mathcal{L}}$ defined on a Radon space $X = M_{\psi}^1(E)$ for some positive weight functions $\psi$ on $E$, in particular (H1) holds. We also assume that (H2) holds for some $\kappa_0 \in \mathbb{R}$. We finally assume the Lyapunov condition

$$ (2.33) \qquad \mathcal{L}^*\psi \le \kappa_B\psi + M\chi, $$

with $\kappa_B < \kappa_0$, $M \ge 0$ and $\chi \in C_{\psi,0}(E)$, $0 \le \chi \le \psi$. Then condition (H3) holds true.

Let us emphasize that we may assume some regularity on $\psi$ by considering $\psi \in D(\mathcal{L}^*)$ so that (2.33) makes sense in $X$ or just understand (2.33) in the weak sense:

$$ \langle \mathcal{L}f, \psi \rangle \le \kappa_B \langle f, \psi \rangle + M \langle f, \chi \rangle, \quad \forall f \in D(\mathcal{L}) \cap X_+. $$

**Proof of Corollary 2.20.** We introduce the splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ where $\mathcal{A}$ is the bounded multiplicator operator $\mathcal{A} := M\chi/\psi$. As a bounded perturbation of $\mathcal{L}$, the operator $\mathcal{B}$ is the generator of a
Proof of Lemma 2.15. We start here again with (2.24). We have

$$1 = \|\hat{f}_n\|_{X_1} \le C_W \|\hat{f}_n\|_{X_0} + \|v_n\|_{X_1},$$

and thus

$$\|\hat{f}_n\|_{X_0} \ge C_W^{-1}(1 - \|v_n\|_{X_1}) \ge (2C_W)^{-1}$$

for any $n \ge n_*$, with $n_* \ge 1$ large enough, so that $\hat{f}_n \in C := C_{(2C_W)^{-1},1}$, for $n \ge n_*$. By the compactness properties of $C$, there exist a subsequence $(\hat{f}_{n_k})$ and $f_1 \in X_+ \setminus \{0\}$ such that $\hat{f}_{n_k} \to f_1$ weakly $\sigma(X, Y)$. □

We present a variant of Lemma 2.13 which is also a concrete consequence of Lemma 2.11 and Lemma 2.15.

**Corollary 2.17.** We assume (H1)-(H2)-(HS1) in $X = L_{m_0}^{p_0}$, $1 \le p_0 < \infty$, together with the facts that $\mathcal{R}_B(\alpha)$ is bounded in $\mathcal{B}(L_{m_0}^{p_0})$ and $\mathcal{B}(L_{m_1}^{p_1})$ uniformly in $\alpha \ge \kappa_0$, $A \in \mathcal{B}(L_{m_0}^{p_0})$ and $(\mathcal{R}_B(\alpha)A)^N$ is bounded in $\mathcal{B}(L_{m_0}^{p_0}, L_{m_1}^{p_1})$ uniformly in $\alpha \ge \kappa_0$ for some $N \ge 1$, with $p_1 > p_0$ and $m_1$ such that $m_0/m_1 \in L^\vartheta$, $1/\vartheta := 1/p_0 - 1/p_1$. Then condition (H3) holds for both the primal and the dual problems.

Proof of Corollary 2.17. On the one hand, we have

$$\mathcal{R}_B(\alpha) + \dots + (\mathcal{R}_B(\alpha)A)^{N-1}\mathcal{R}_B(\alpha) \text{ is bounded in } \mathcal{B}(X) \text{ uniformly in } \alpha \ge \kappa_0, \\ W(\alpha) := (\mathcal{R}_B(\alpha)A)^N \text{ is bounded in } \mathcal{B}(X, \mathcal{X}_1) \text{ uniformly in } \alpha \ge \kappa_0,$$

with $\mathcal{X}_1 := L_{m_1}^{p_1} \subset X$ and thus $\{(gm_0)^{p_0}; g \ge 0, \|g\|_{X_1} \le R_1\}$ is a weakly compact subset of $L^1(E)$ for any $R_1 > 0$. Condition (H3) holds for the direct problem thanks to Lemma 2.13. On the other hand, we set $Y := X' = L_{\nu_0}^{q_0}, q_0 := p'_0, \nu_0 := m_0^{-1}$, and we first observe that

$$\mathcal{R}_{B^*}(\alpha) + \dots + (\mathcal{R}_{B^*}(\alpha)A^*)^{N-1}\mathcal{R}_{B^*}(\alpha) \text{ is bounded in } \mathcal{B}(Y) \text{ uniformly in } \alpha \ge \kappa_0.$$

We next observe that

$$(A^*\mathcal{R}_{B^*}(\alpha))^{N+1} = A^*W(\alpha)^*\mathcal{R}_{B^*}(\alpha) \text{ is bounded in } \mathcal{B}(Y_0, Y) \text{ uniformly in } \alpha \ge \kappa_0,$$

with $Y_0 := L_{\nu_1}^{q_1}, q_1 := p'_1, \nu_1 := m_1^{-1}$. Because $\{(g\nu_1)^{q_1}; g \ge 0, \|g\|_Y \le R_1\}$ is a weakly compact subset of $L^1(E)$ for any $R_1 > 0$, we have from the proof of Lemma 2.13 that the set $C$ defined by (2.25) for the norms of $Y_0$ and $Y_1 := Y$ satisfies the weak compactness property required in the statement of Lemma 2.11. We may thus apply Lemma 2.15 and we deduce that condition (H3) holds for the dual problem. □

Another concrete consequence of Lemma 2.11 and Lemma 2.15 is the following.

**Lemma 2.18.** We assume $E$ not compact, $X = M_{m_i}^1(E)$ for a continuous weight function $m_i$ on $E$, $i = 0$ or $i = 1$, (H1)-(H2)-(HS1) and there exists $N \ge 1$ such that $(\mathcal{R}_B(\alpha)A)^N : M_{m_0}^1(E) \to M_{m_1}^1(E)$ uniformly in $\alpha \ge \kappa_0$ for another continuous weight function $m_{1-i}$ on $E$ such that $m_1(x)/m_0(x) \to \infty$ as $x \to \infty$. Then condition (H3) holds.

Proof of Lemma 2.18. We define $\mathcal{X}_i := M_{m_i}^1(E)$ and we consider the set $C$ defined by (2.25) which is clearly compact for the weak $\sigma(M_{m_1}^1, C_{m_1^{-1},0})$ topology. When $X = M_{m_0}^1$, the result follows from Lemma 2.11 while when $X = M_{m_1}^1$, the result is a consequence of Lemma 2.15. □

We may slightly improve the preceding results by considering a more abstract framework and a somehow more general boundedness condition.

**Lemma 2.19.** We assume $X = Y'$, $Y$ separable, (H1)-(H2)-(HS1) and there exist $N \ge 1$, $\gamma \in [0, 1)$ and $\varphi \in Y_+ \setminus \{0\}$ such that for any $\alpha \ge \kappa_0$, there holds

$$(2.28) \qquad \|W(\alpha)f\|_X \le \gamma \|f\|_X + \langle f, \varphi \rangle_{X,Y},$$

for all $f \in X_+$, or there holds

$$(2.29) \qquad \|W(\alpha)f\|_X \le \gamma \|f\|_X + \langle W(\alpha)f, \varphi \rangle_{X,Y},$$

for all $f \in X_+$. Then condition (H3) holds true, and the limit $f_1$ satisfies $\langle f_1, \varphi \rangle_{X,Y} \ge 1 - \gamma > 0$.
semigroup $S_B$. Defining $\tilde{S}_t := S_L(t)e^{-Mt} \ge 0$ and $A^c := M(1 - \chi/\psi) \ge 0$, we have the Duhamel formula

$$S_B = \tilde{S} + \tilde{S}A^c * S_B$$

and iterating infinitely many times, we deduce the Dyson-Philips formula

$$S_B = \sum_{k=0}^{\infty} (\tilde{S}A^c)^{(k)} * \tilde{S}.$$

That implies that $S_B \ge 0$ as a combination of positive operators. Alternatively, from the very definition of $B$, we have $\kappa - B \le (M + \kappa) - L$ for any $\kappa \in \mathbb{R}$. Choosing $\kappa > \max(\omega(S_L), \omega(S_B))$ and using the direct implication in Lemma 2.1, we have $R_B(\kappa) \ge R_L(M + \kappa) \ge 0$. Using the reciprocal implication in Lemma 2.1, we obtain again that $S_B \ge 0$.

Now, for $0 \le f_0 \in D(B)$ and setting $f_t := S_B(t)f_0$, we may compute

$$\frac{d}{dt}\langle f_t, \psi \rangle = \langle Bf_t, \psi \rangle \le \kappa_B \langle f_t, \psi \rangle,$$

so that

$$\|S_B(t)f_0\|_{M_\psi^1} \le e^{\kappa_B t} \|f_0\|_{M_\psi^1}.$$

Using (2.13) we immediately and classically deduce

$$\|R_B(\alpha)\|_{B(M_\psi^1)} \le \frac{1}{\alpha - \kappa_B}, \quad \forall \alpha > \kappa_B,$$

so that $R_B(\alpha)$ is bounded in $B(M_\psi^1)$ and $R_B(\alpha)A$ is bounded in $B(M_\chi^1, M_\psi^1)$ uniformly for $\alpha \ge \kappa_0$. We apply Lemma 2.18 or Lemma 2.19 ((2.29) with $N = 1$, $\gamma = 0$ and $\varphi = \frac{M}{\alpha - \kappa_B}\chi$) in order to conclude. $\square$

In the proof of Corollary 2.20, we may alternatively use the trivial splitting $L = \tilde{A} + \tilde{B} = \eta + (L - \eta)$ for some $\eta > \kappa_1 - \kappa_0$, so that $\alpha - \tilde{B}$ is invertible for any $\alpha \ge \kappa_0$, and reformulate the Lyapunov condition

$$(\alpha - \tilde{B}^*)\psi \ge (\alpha + \eta - \kappa_B)\psi - M\chi,$$

for any $\alpha \ge \kappa_0$. Observing that $\tilde{W}(\alpha) := \tilde{A}R_B(\alpha) = \eta(\alpha - \tilde{B})^{-1}$, we deduce

$$\tilde{W}^*(\alpha)\psi \le \frac{\eta}{\eta + \alpha - \kappa_B}\psi + \frac{M}{\eta + \alpha - \kappa_B}\tilde{W}^*(\alpha)\chi.$$

We equivalently have

$$\|\tilde{W}(\alpha)f\|_{M_\psi^1} \le \gamma\|f\|_{M_\psi^1} + \langle \tilde{W}(\alpha)f, \varphi \rangle,$$

uniformly for any $\alpha \ge \kappa_0$, with $\gamma := \frac{\eta}{\eta + \kappa_0 - \kappa_B} < 1$ and $\varphi := \frac{M}{r + \kappa_0 - \kappa_B}\chi$, which is nothing but condition (2.30).

We finally come to the existence of a solution to the first eigenvalue problem and the first eigentriplet problem.

**Theorem 2.21.** Under conditions (H1)-(H2)-(H3), the first eigenvalue problem (2.14) has a solution $(\lambda_1, f_1)$ with $\lambda_1$ satisfying (2.19). When furthermore (H3) holds for the dual problem, then the first eigentriplet problem (1.1)-(1.2) admits a solution $(\lambda_1, f_1, \phi_1) \in \mathbb{R} \times X \times Y$.

Theorem 2.21 generalizes some known versions of the existence part of the Krein-Rutman Theorem where either $L$ is assumed additionally to be the generator of a semigroup or to have strongly power compact resolvent or even where some additional conditions are made on the positive cone $X_+$. As mentioned in the introduction, some possible references for these previous results are Krein-Rutman [238], Greiner in [187, Cor 1.2] and in [15, C-IV Thm. 2.1] and Webb [360, Prop. 2.5], see also [75, Thm. 2], [278, Thm. 5.3], [252], [35, Thm. 2.1], the textbook [41, Thm. 12.15] and the references therein.

**Proof of Theorem 2.21.** We first assume (H1)-(H2)-(H3). Because of Lemma 2.6, there exists a sequence $(\hat{f}_n)$ of $X$ such that (2.20) holds, and in particular

$$(2.34) \qquad \langle \lambda_n \hat{f}_n, \phi \rangle - \langle \hat{f}_n, \mathcal{L}^* \phi \rangle = \langle \varepsilon_n, \phi \rangle,$$

for any $\phi \in D(\mathcal{L}^*)$. Because of condition (H3), we may pass to the limit $n' \to \infty$ in equation (2.34) and we deduce that $(\lambda_1, f_1)$ satisfies (2.14) and (2.19).
The case $X = M_{m_1}^1(E)$ in Lemma 2.18 corresponds here to the first situation where (2.28) holds with $X := M_{m_1}^1(E)$, $\gamma := 0$, $Y := C_{m_0^{-1},0}^1(E)$ and $\varphi := (\sup_{\alpha \ge \kappa_0} \|W(\alpha)\|_{M_{m_0}^1 \to M_{m_1}^1}) m_0$. We will use Lemma 2.19 only for the mutation-selection model with singular mutation kernel in Section 12.2.

*Proof of Lemma 2.19.* Starting with (2.24) and using (2.28), we have

$$ \begin{aligned} \|\hat{f}_n\|_X & \le \|W(\lambda_n)\hat{f}_n\|_X + \|V(\lambda_n)\varepsilon_n\|_X \\ & \le \gamma \|\hat{f}_n\|_X + \langle \hat{f}_n, \varphi \rangle_{X,Y} + \|v_n\|_X, \end{aligned} $$

so that

$$ \langle \hat{f}_n, \varphi \rangle_{X,Y} \ge 1 - \gamma - \|v_n\|_X. $$

By compactness, there are $f_1 \ge 0$ and a subsequence $(\hat{f}_{n'})$ such that $\hat{f}_{n'} \to f_1$ weak * $\sigma(X, Y)$.
Passing to the limit as $n' \to \infty$ in the above estimate, we find

$$ (2.30) \qquad \langle f_1, \varphi \rangle_{X,Y} = \lim_{n' \to \infty} \langle \hat{f}_{n'}, \varphi \rangle_{X,Y} \ge 1 - \gamma, $$

and in particular $f_1 \ne 0$.

Under the assumption (2.29), modifying slightly the previous argument, we have

$$ \|\hat{f}_n\|_X \le \gamma \|\hat{f}_n\|_X + \langle w_n, \varphi \rangle_{X,Y} + \|v_n\|_X, $$

which, together with

$$ \langle \hat{f}_n, \varphi \rangle_{X,Y} = \langle w_n, \varphi \rangle_{X,Y} + \langle v_n, \varphi \rangle_{X,Y}, $$

implies

$$ \langle \hat{f}_n, \varphi \rangle_{X,Y} \ge 1 - \gamma - \|v_n\|_X + \langle v_n, \varphi \rangle_{X,Y}. $$

By compactness again, there are $f_1 \ge 0$ and a subsequence $(\hat{f}_{n'})$ such that $\hat{f}_{n'} \to f_1$ weak * $\sigma(X, Y)$, and passing to the limit as $n' \to \infty$ in the above estimate, we conclude again to (2.30). $\square$

Let us comment on Lemma 2.19 and in particular the condition (2.29).

In the case when $X = L^\infty(E, \mathcal{E}, \mu) = (L^1(E, \mathcal{E}, \mu))'$, we can relate condition (2.29) to the assumption that there exist $f_0 \in X_+$ and $\varphi \in Y_+ \setminus \{0\}$ such that

$$ (2.31) \qquad \|S_{\mathcal{L}}(t)f_0\|_X \le \langle S_{\mathcal{L}}(t)f_0, \varphi \rangle, \quad \forall t \ge 0. $$

This last condition is reminiscent from conditions that appear in probabilistic inspired methods for the ergodicity of semigroups, see the condition (1b) in Theorem 1.7 but also Assumption (A2) in [102], both in the vein of [130, Condition $\mathcal{Z}$]. Assume indeed (2.31), let $\eta > \kappa_1 - \kappa_0 > 0$ and consider the trivial decomposition $\mathcal{L} = \mathcal{A} + \mathcal{B} = \eta + (\mathcal{L} - \eta)$. Then set $\kappa_B := \kappa_1 - \eta < \kappa_0$, so that for any $\alpha > \kappa_B$, $\mathcal{B} - \alpha = \mathcal{L} - (\eta + \alpha)$ is invertible since $\eta + \alpha > \eta + \kappa_B = \kappa_1$. We thus have for any $\alpha > \kappa_B$

$$ \mathcal{W}(\alpha) := \eta(\alpha - \mathcal{B})^{-1} = \eta \int_0^\infty e^{-(\eta+\alpha)t} S_{\mathcal{L}}(t) dt $$

and (2.31) then ensures that

$$ \|W(\alpha)f_0\|_X \le \langle W(\alpha)f_0, \varphi \rangle. $$

We recover (2.29) with $\gamma = 0$ and the difference that $f_0$ is fixed here.

As a Corollary of Lemma 2.18 or Lemma 2.19 and anticipating on the material of part 3, we present now a situation very classical in stochastic processes theory.

**Corollary 2.20.** We consider a positive semigroup $S = S_{\mathcal{L}}$ defined on a Radon space $X = M_{\psi}^1(E)$ for some positive weight functions $\psi$ on $E$, in particular (H1) holds. We also assume that (H2) holds for some $\kappa_0 \in \mathbb{R}$. We finally assume the Lyapunov condition

$$ (2.32) \qquad \mathcal{L}^*\psi \le \kappa_B\psi + M\chi, $$

with $\kappa_B < \kappa_0$, $M \ge 0$ and $\chi \in C_{\psi,0}(E)$, $0 \le \chi \le \psi$. Then condition (H3) holds true.

Let us emphasize that we may assume some regularity on $\psi$ by considering $\psi \in D(\mathcal{L}^*)$ so that (2.32) makes sense in $X$ or just understand (2.32) in the weak sense:

$$ \langle \mathcal{L}f, \psi \rangle \le \kappa_B \langle f, \psi \rangle + M \langle f, \chi \rangle, \quad \forall f \in D(\mathcal{L}) \cap X_+. $$
We now additionally assume that (H3) holds for the dual problem. As recalled during the proof of Lemma 2.3 and by definition of $\lambda_1$, we have $(\lambda_1, +\infty) \subset \rho(\mathcal{L}) = \rho(\mathcal{L}^*)$ and $\lambda_1 \in \Sigma(\mathcal{L}) = \Sigma(\mathcal{L}^*)$. Taking $\lambda_n \searrow \lambda_1$, we argue as in the proof of Lemma 2.6 and we get

$$ \exists \hat{\phi}_n \ge 0, \lambda_n \hat{\phi}_n - \mathcal{L}^* \hat{\phi}_n \to 0, \|\hat{\phi}_n\| = 1. $$

Thanks to (H3) for the dual problem, we deduce that there exist a subsequence $(\hat{\phi}_{n_k})$ and $\phi_1 \in X'$, $\|\phi_1\| = 1$ such that $\hat{\phi}_{n_k} \to \phi_1$. We thus conclude that $\phi_1$ is a solution to the dual problem (1.2) (for the same eigenvalue $\lambda_1$). $\square$

Let us conclude this section by some remarks.

**Remark 2.22.** (1) - As seen above, the condition (H1)-(H2) for the primal and the dual problems are equivalent, and thus one only has to check (H1)-(H2)-(H3) for the primal problem and (H3) for the dual problem in order to solve the first eigentriplet problem. It is worth emphasizing that condition (H3) on the dual problem is not a consequence of the condition (H3) on the primal problem. However, as presented in Lemma 2.7, Lemma 2.8 and Corollary 2.17, there exist several natural situations where both conditions (H3) for the primal and the dual problems hold together. (2) - Alternatively, one may also assume (H1)-(H2)-(H3) for the dual problem, and then use a more classical fixed point theorem for proving the existence of a steady state for the rescaled semi-group by using for instance the Markov-Kakutani fixed point theorem [226] as in [231, Thm. 5.1], the Tychonov fixed point theorem as in [172] or [153, Thm. 1.2] or a Birkhoff-Von Neumann type Theorem as in [81, Thm. 6.1]. For these last techniques, we also refer to Section 3, where such a dynamical approach is adapted to the present context. One may also use the usual Doblin-Harris theory, see for instance [198, 81] and the references therein, and Sections 8.6 and 12.2 for applications of this approach.

## 2.3. Discussion.

We discuss now the existence results presented in the preceding section.

For further references, let us first recall that when $X$ is a Hilbert space and $\mathcal{L}$ is self-adjoint, the first eigenvalue may be simply obtained thanks to the variational problem

$$ (2.35) \qquad \lambda_1 = \sup_{f \in X_+ \setminus \{0\}} \frac{\langle \mathcal{L}f, f \rangle}{\|f\|^2}. $$

We now explain how Theorem 2.21 is a generalization of the classical Krein-Rutman theorem stated in Theorem 1.2. We thus consider a Banach lattice $X$ such that $X_{++} := \text{int}X_+ \neq \emptyset$ and an operator $\mathcal{L}$ such that, for $\kappa_1 \in \mathbb{R}$ and any $\kappa > \kappa_1$, $\mathcal{R} := (\kappa - \mathcal{L})^{-1} : X \to X$ is compact and $\mathcal{R} : X_+ \setminus \{0\} \to X_{++}$, in particular (H1) holds true. As a first step, we recall the following very classical technical lemma of the KR theory.

**Lemma 2.23.** Assume $X_{++} := \text{int}X_+ \neq \emptyset$. For $g \in X_+$ and $f \in X_{++}$, there exists $C \ge 0$ such that $g \le Cf$.

*Proof of Lemma 2.23.* We argue by contradiction. Otherwise, for any $n \ge 1$, we would have $f - g/n \in X_+^c \subset X_{++}^c$ and that last set is closed. Passing to the limit, we get $f \in X_{++}^c$, which is in contradiction with the assumption $f \in X_{++}$. $\square$

For a given $g_0 \in X_+ \setminus \{0\}$, we set $f_0 := \mathcal{R}g_0 \in X_{++}$. From Lemma 2.23, there exists $C_0 \ge 0$ such that $(\kappa - \mathcal{L})f_0 = g_0 \le C_0f_0$. That implies that Lemma 2.4-(ii) holds with $\kappa_0 := \kappa - C_0$, and thus (H2) also holds. One may then define $\mu_1 := \kappa - \lambda_1$, with

$$ \lambda_1 := \inf\{\lambda \in \mathbb{R}; (\lambda' - \mathcal{L})^{-1} \in \mathcal{B}(X), \forall \lambda' \in [\lambda, \kappa]\} \ge \kappa_0. $$

We recall that because of Lemma 2.6 (or its proof), there exist $(\lambda_n)$, $(\hat{f}_n)$ and $(\varepsilon_n)$ such that (2.20) holds, namely

$$ \lambda_n \searrow \lambda_1, \quad \hat{f}_n \ge 0, \quad \varepsilon_n := \lambda_n \hat{f}_n - \mathcal{L} \hat{f}_n \ge 0, \quad \|\hat{f}_n\| = 1, \quad \|\varepsilon_n\| \to 0. $$

We may rewrite the equation as

$$ \hat{f}_n = \mathcal{R}[\varepsilon_n + (\kappa - \lambda_n)\hat{f}_n], $$
Proof of Corollary 2.20. We introduce the splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ where $\mathcal{A}$ is the bounded multiplier operator $\mathcal{A} := M\chi/\psi$. As a bounded perturbation of $\mathcal{L}$, the operator $\mathcal{B}$ is the generator of a semigroup $S_{\mathcal{B}}$. Defining $\tilde{S}_t := S_{\mathcal{L}}(t)e^{-Mt} \ge 0$ and $\mathcal{A}^c := M(1 - \chi/\psi) \ge 0$, we have the Duhamel formula

$$S_{\mathcal{B}} = \tilde{S} + \tilde{S}\mathcal{A}^c * S_{\mathcal{B}}$$

and iterating infinitely many times, we deduce the Dyson-Philips formula

$$S_{\mathcal{B}} = \sum_{k=0}^{\infty} (\tilde{S}\mathcal{A}^c)^{(*k)} * \tilde{S}.$$

That implies that $S_{\mathcal{B}} \ge 0$ as a combination of positive operators. Alternatively, from the very definition of $\mathcal{B}$, we have $\kappa - \mathcal{B} \le (M + \kappa) - \mathcal{L}$ for any $\kappa \in \mathbb{R}$. Choosing $\kappa > \max(\omega(S_{\mathcal{L}}), \omega(S_{\mathcal{B}}))$ and using the direct implication in Lemma 2.1, we have $\mathcal{R}_{\mathcal{B}}(\kappa) \ge \mathcal{R}_{\mathcal{L}}(M + \kappa) \ge 0$. Using the reciprocal implication in Lemma 2.1, we obtain again that $S_{\mathcal{B}} \ge 0$.

Now, for $0 \le f_0 \in D(\mathcal{B})$ and setting $f_t := S_{\mathcal{B}}(t)f_0$, we may compute

$$\frac{d}{dt} \langle f_t, \psi \rangle = \langle \mathcal{B} f_t, \psi \rangle \le \kappa_{\mathcal{B}} \langle f_t, \psi \rangle,$$

so that

$$\|S_{\mathcal{B}}(t)f_0\|_{M_{\psi}^1} \le e^{\kappa_{\mathcal{B}}t} \|f_0\|_{M_{\psi}^1}.$$

Using (2.13), we immediately and classically deduce

$$\|\mathcal{R}_{\mathcal{B}}(\alpha)\|_{\mathcal{B}(M_{\psi}^1)} \le \frac{1}{\alpha - \kappa_{\mathcal{B}}}, \quad \forall \alpha > \kappa_{\mathcal{B}},$$

so that $\mathcal{R}_{\mathcal{B}}(\alpha)$ is bounded in $\mathcal{B}(M_{\psi}^1)$ and $\mathcal{R}_{\mathcal{B}}(\alpha)\mathcal{A}$ is bounded in $\mathcal{B}(M_{\chi}^1, M_{\psi}^1)$ uniformly for $\alpha \ge \kappa_0$. We apply Lemma 2.18 or Lemma 2.19 ((2.28) with $N = 1$, $\gamma = 0$ and $\varphi = \frac{M}{\alpha - \kappa_{\mathcal{B}}}\chi$) in order to conclude. $\square$

In the proof of Corollary 2.20, we may alternatively use the trivial splitting $\mathcal{L} = \tilde{\mathcal{A}} + \tilde{\mathcal{B}} = \eta + (\mathcal{L} - \eta)$ for some $\eta > \kappa_1 - \kappa_0$, so that $\alpha - \tilde{\mathcal{B}}$ is invertible for any $\alpha \ge \kappa_0$, and reformulate the Lyapunov condition

$$(\alpha - \tilde{\mathcal{B}}^*)\psi \ge (\alpha + \eta - \kappa_{\mathcal{B}})\psi - M\chi,$$

for any $\alpha \ge \kappa_0$. Observing that $\tilde{\mathcal{W}}(\alpha) := \tilde{\mathcal{A}}\tilde{\mathcal{R}}_{\tilde{\mathcal{B}}}(\alpha) = \eta(\alpha - \tilde{\mathcal{B}})^{-1}$, we deduce

$$\tilde{\mathcal{W}}^*(\alpha)\psi \le \frac{\eta}{\eta + \alpha - \kappa_{\mathcal{B}}}\psi + \frac{M}{\eta + \alpha - \kappa_{\mathcal{B}}}\tilde{\mathcal{W}}^*(\alpha)\chi.$$

We equivalently have

$$\|\tilde{\mathcal{W}}(\alpha)f\|_{M_{\psi}^1} \le \gamma \|f\|_{M_{\psi}^1} + \langle \tilde{\mathcal{W}}(\alpha)f, \varphi \rangle,$$

uniformly for any $\alpha \ge \kappa_0$, with $\gamma := \frac{\eta}{\eta + \kappa_0 - \kappa_{\mathcal{B}}} < 1$ and $\varphi := \frac{M}{r + \kappa_0 - \kappa_{\mathcal{B}}}\chi$, which is nothing but condition (2.29). We conclude again thanks to Lemma 2.19.

We emphasize again that in almost all our applications, we will be able to use Lemma 2.8-(2) for establishing (H3), except in very irregular situations (we will use Lemma 2.13 in Section 8.5 and Lemma 2.19 in Section 12.2) and in weak dissipative frameworks (we will directly show (H3) by following Lions' proof in Section 7.3 and we will use Theorem 3.4 in Section 7.4). In Section 9.3 we do not use exactly Lemma 2.8 neither, but we adapt the proof for combining two different splittings.

We finally come to the existence of a solution to the first eigenvalue problem and the first eigentriplet problem.

**Theorem 2.21.** Under conditions (H1)-(H2)-(H3), the first eigenvalue problem (2.14) has a solution $(\lambda_1, f_1)$ with $\lambda_1$ satisfying (2.19). When furthermore (H3) holds for the dual problem, then the first eigentriplet problem (1.1)-(1.2) admits a solution $(\lambda_1, f_1, \phi_1) \in \mathbb{R} \times X \times Y$.
so that $(\hat{f}_n)$ belongs to a compact set of $X$ because of the compactness assumption made on $\mathcal{R}$, so that (H3) holds true.

Because of Theorem 2.21, we deduce that there exists $f_1 \in X_+$ such that $\|f_1\| = 1$ and $\mathcal{L}f_1 = \lambda_1$. That implies $f_1 = \mu_1 \mathcal{R}f_1$, and thus that the existence part of Theorem 1.2 is a consequence of Theorem 2.21 for an operator $\mathcal{R}$ which is the positive resolvent of an operator $\mathcal{L}$.

We would like to emphasize on the fact that our definition of the first eigenvalue by (2.15)-(2.16) bears some strong similarity with the definition of the first eigenvalue for elliptic operators in non divergence form as presented in [47]. Indeed, if $\lambda \in \mathcal{I}$, then

$$ \exists f \in X_+ \setminus \{0\}, \quad \mathcal{L}f \le \lambda f. $$

Assuming now that $X$ is a space of functions (defined on a set $E$) and that $f(x) > 0$ for any $x \in E$, we deduce that

$$ \lambda \ge \sup_E \frac{\mathcal{L}f}{f}, $$

and thus $\lambda_1$ is characterized by

$$ \lambda_1 = \inf_{f>0} \sup_E \frac{\mathcal{L}f}{f}, $$

which is nothing but [47, (1.13)] (with a change of sign because of a different sign convention). We thus see that our formulation is a generalization at a more abstract level and for resolvent positive operators of that classical min-max approach for elliptic operators. Some more or less classical references on that subject are [142, 143], [302], [318], [48] and [46]. In particular in [48], two generalized principal eigenvalues

$$ \lambda_1' := \sup\{\kappa_0 \in \mathbb{R}; \exists g_0 \in C_0 \mathcal{L}g_0 \ge \kappa_0 g_0\} $$

and

$$ \lambda_1'' := \inf\{\kappa_1 \in \mathbb{R}; \exists g_1 \in C_1 \mathcal{L}g_1 \le \kappa_1 g_1\} $$

are defined for appropriate cones $C_i \subset X_+ \setminus \{0\}$ for problems with lack of compactness. The links between the three quantities $\lambda_1$, $\lambda_1'$ and $\lambda_1''$ are discussed as well as the possible non existence of a related principal eigenfunction $f_1$. The non existence of associated principal eigenfunction should not be a surprise since it is the case when one considers $\mathcal{L} = \Delta$ in $X = L^2(\mathbb{R}^d)$ where $\mathcal{L}g_1 = \mathcal{L}^*g_1 = \lambda_1''\psi$ with $0 < \psi = 1 \notin X = X'$ and $\lambda_1'' = 0$, but no associated principal eigenfunction exists in $X$. We also refer to [252] where some examples of such a situation are discussed.

For its own interest and further discussions, we finally state and prove a slightly less general variant of Theorem 1.8.

**Theorem 2.24.** Consider a Banach lattice with positive cone $X_+$ and a linear and bounded operator $\mathcal{R}: X \to X$ such that

(i) $\mathcal{R}: X_+ \to X_+$;

(ii) $\exists g_2 \in X_+ \setminus \{0\}, \exists C_2 > 0$ such that $\mathcal{R}g_2 \le C_2 g_2$.

We define

$$ K_2 := \{g \in X_+; \exists a > 0, g \le ag_2\}, $$

and next

$$ A(g) := \inf\{a > 0; g \le ag_2\}, \quad \text{if } g \in K_2, $$

as well as

$$ \mathcal{J} := \{\mu \ge 0; \exists h \in K_2, h \ge \mu\mathcal{R}h + g_2\}. $$

We further assume

(iii) $\mu_1 := \sup \mathcal{J} < +\infty$.

(iv) Any upper bounded and increasing sequences $(g^n)$ of $X$ is convergent in the weak sense $\sigma(X, Y)$. More precisely, if $g_n \le g_{n+1} \le \bar{g} \in X$ for any $n \ge 1$, there exists $g \in X$, $g \le \bar{g}$, such that $g_n \rightharpoonup g$.

(v) Any sequence $(g^n)$ of normalized almost first eigenvectors is relatively compact. More precisely, for any sequence $(g^n)$ of $K_2$ such that $A(g^n) = 1$, $g^n = \mu^n \mathcal{R}g^n + \varepsilon^n$, $\mu^n \nearrow \mu_1$ and $\varepsilon^n \to 0$, there exists $g \in K_2$ and a subsequence $(g^{n_k})$ such that $g^{n_k} \to g$ and $A(g) = 1$.

Then there exists $f_1 \in X_2$ such that $f_1 = \mu_1 \mathcal{R}f_1$ and $A(f_1) = 1$.
Theorem 2.21 generalizes some known versions of the existence part of the Krein-Rutman Theorem where either $\mathcal{L}$ is assumed additionally to be the generator of a semigroup or to have strongly power compact resolvent or even where some additional conditions are made on the positive cone $X_+$. As mentioned in the introduction, some possible references for these previous results are Krein-Rutman [237], Greiner in [186, Cor 1.2] and in [15, C-IV Thm. 2.1] and Webb [359, Prop. 2.5], see also [74, Thm. 2], [277, Thm. 5.3], [251], [35, Thm. 2.1], the textbook [41, Thm. 12.15] and the references therein.

**Proof of Theorem 2.21.** We first assume (H1)-(H2)-(H3). Because of Lemma 2.6, there exists a sequence $(\hat{f}_n)$ of $X$ such that (2.20) holds, and in particular

$$ (2.33) \qquad \langle \lambda_n \hat{f}_n, \phi \rangle - \langle \hat{f}_n, \mathcal{L}^* \phi \rangle = \langle \varepsilon_n, \phi \rangle, $$

for any $\phi \in D(\mathcal{L}^*)$. Because of condition (H3), we may pass to the limit $n' \to \infty$ in equation (2.33) and we deduce that $(\lambda_1, f_1)$ satisfies (2.14) and (2.19).

We now additionally assume that (H3) holds for the dual problem. As recalled during the proof of Lemma 2.3 and by definition of $\lambda_1$, we have $(\lambda_1, +\infty) \subset \rho(\mathcal{L}) = \rho(\mathcal{L}^*)$ and $\lambda_1 \in \Sigma(\mathcal{L}) = \Sigma(\mathcal{L}^*)$. Taking $\lambda_n \searrow \lambda_1$, we argue as in the proof of Lemma 2.6 and we get

$$ \exists \hat{\phi}_n \ge 0, \lambda_n \hat{\phi}_n - \mathcal{L}^* \hat{\phi}_n \to 0, \|\hat{\phi}_n\| = 1. $$

Thanks to (H3) for the dual problem, we deduce that there exist a subsequence $(\hat{\phi}_{n_k})$ and $\phi_1 \in X'$, $\|\phi_1\| = 1$ such that $\hat{\phi}_{n_k} \to \phi_1$. We thus conclude that $\phi_1$ is a solution to the dual problem (1.2) (for the same eigenvalue $\lambda_1$). $\square$

Let us conclude this section by some remarks.

**Remark 2.22.** (1) - As seen above, the condition (H1)-(H2) for the primal and the dual problems are equivalent, and thus one only has to check (H1)-(H2)-(H3) for the primal problem and (H3) for the dual problem in order to solve the first eigentriplet problem. It is worth emphasizing that condition (H3) on the dual problem is not a consequence of the condition (H3) on the primal problem. However, as presented in Lemma 2.7, Lemma 2.8 and Corollary 2.17, there exist several natural situations where both conditions (H3) for the primal and the dual problems hold together. (2) - Alternatively, one may also assume (H1)-(H2)-(H3) for the dual problem, and then use a more classical fixed point theorem for proving the existence of a steady state for the rescaled semigroup by using for instance the Markov-Kakutani fixed point theorem [225] as in [230, Thm. 5.1], the Tychonov fixed point theorem as in [171] or [152, Thm. 1.2] or a Birkhoff-Von Neumann type Theorem as in [80, Thm. 6.1]. For these last techniques, we also refer to Section 3, where such a dynamical approach is adapted to the present context. One may also use the usual Doblin-Harris theory, see for instance [197, 80] and the references therein, and Sections 8.6 and 12.2 for applications of this approach.

Let us finally emphasize that Theorem 2.21 provides an answer to problem (S1) about the existence of a first eigentriplet. In particular, for a positive semigroup $S_\mathcal{L}$ on $X$ which satisfies (H1), (H2) and which enjoys the splitting structure (1.9), (1.10), (1.11), (1.12) and thus (HS1), we may use Theorem 2.21 and Remark 2.22 for establishing Theorem 1.1-(1), at least in three situations: (1) when $X_1 \subset X_0 := X$ is compact (see Lemma 2.8 and Remark 2.9), (2) when the conditions of Lemma 2.11, Lemma 2.15 or Lemma 2.19 are complete for both the primal and the dual problem and (3) when the assumptions of Corollary 2.17 are met.

## 2.3. Discussion.

We discuss now the existence results presented in the preceding section.

For further references, let us first recall that when $X$ is a Hilbert space and $\mathcal{L}$ is self-adjoint, the first eigenvalue may be simply obtained thanks to the variational problem

$$ (2.34) \qquad \lambda_1 = \sup_{f \in X_+ \setminus \{0\}} \frac{\langle \mathcal{L}f, f \rangle}{\|f\|^2}. $$

We now explain how Theorem 2.21 is a generalization of the classical Krein-Rutman theorem stated in Theorem 1.2. We thus consider a Banach lattice $X$ such that $X_{++} := \text{int}X_+ \neq \emptyset$ and an operator $\mathcal{L}$ such that, for $\kappa_1 \in \mathbb{R}$ and any $\kappa > \kappa_1$, $\mathcal{R} := (\kappa - \mathcal{L})^{-1} : X \to X$ is compact and
Proof of Theorem 2.24. We split the proof into three steps.

Step 1. We first establish that for any $\mu \in \mathcal{J}$, there exists $\tilde{g} = \tilde{g}_{\mu} \in K_2$ such that

$$ (2.36) \qquad \tilde{g} = \mu \mathcal{R} \tilde{g} + g_2. $$

We set $\tilde{g}_0 = 0$, $\tilde{g}_1 = g_2$, and we define $(\tilde{g}_n)$ recursively by $\tilde{g}_{n+1} = \mu \mathcal{R} \tilde{g}_n + g_2$, for any $n \ge 1$. We claim that

$$ 0 \le \tilde{g}_n \le \tilde{g}_{n+1} \le h, \quad \text{for any } n \ge 0, $$

where $h$ enters in the definition of $\mu \in \mathcal{J}$. That is obviously true at order $n = 0$. Assuming that last inequality is proved at order $n-1$ for $n \ge 1$, we compute

$$ \tilde{g}_{n+1} = \mu \mathcal{R} \tilde{g}_n + g_2 \ge \mu \mathcal{R} \tilde{g}_{n-1} + g_2 = \tilde{g}_n $$

and

$$ \tilde{g}_{n+1} = \mu \mathcal{R} \tilde{g}_n + g_2 \le \mu \mathcal{R} h + g_2 \le h, $$

which proves the same inequality at order $n$, and thus for any $n \ge 0$. Using the convergence property (iv) of upper bounded increasing sequences in $X$, we deduce that there exists $\tilde{g} \in X_2$ such that $\tilde{g}_n \to \tilde{g}$ and thus (2.36) holds.

Step 2. We obviously have $0 \in \mathcal{J}$ and $\mathcal{J}$ is an interval because if $(\mu, h)$ satisfies the condition $\mu \in \mathcal{J}$ then so do $(\mu', h)$ for any $\mu' \in [0, \mu]$. We finally claim that $\mathcal{J}$ is open. Take indeed $\mu \in \mathcal{J}$ and $\tilde{g} \in K_2$ such that (2.36) holds, what is possible due to Step 1. By definition, there would exist $A > 0$ such that $\tilde{g} \le Ag_2$. Choosing $0 < \varepsilon \le 1/(2AC_2)$ and $M \ge 2$, we compute

$$ \begin{aligned} (M\tilde{g}) - (\mu + \varepsilon)\mathcal{R}(M\tilde{g}) &= Mg_2 - M\varepsilon\mathcal{R}\tilde{g} \\ &\ge Mg_2 - M\varepsilon A\mathcal{R}g_2 \ge M(1 - \varepsilon AC_2)g_2 \ge g_2, \end{aligned} $$

so that $\mu + \varepsilon \in \mathcal{J}$.

Step 3. We first establish by contradiction that $A(\tilde{g}_{\mu}) \nearrow \infty$ when $\mu \nearrow \mu_1$. If it was not the case, there exists $A \in (0, \infty)$ and a sequence $(\mu^n)$ such that $A(\tilde{g}_{\mu^n}) \le A$ as $\mu^n \nearrow \mu_1$. Choosing $0 < \varepsilon \le 1/(2AC_2)$ and $M \ge 2$ as in Step 2, the same computation gives

$$ (M\tilde{g}_{\mu^n}) - (\mu + \varepsilon)\mathcal{R}(M\tilde{g}_{\mu^n}) \ge g_2, $$

so that $\mu^n + \varepsilon \in \mathcal{J}$. That means that $\mu^n + \varepsilon \le \mu_1$, and a contradiction with the fact $\mu^n \nearrow \mu_1$. We next consider $\mu^n \nearrow \mu_1$ and we define

$$ A^n := A(\tilde{g}_{\mu^n}), \quad \hat{g}^n := \frac{\tilde{g}_{\mu^n}}{A^n}, \quad \varepsilon_n := \frac{g_2}{A^n}, \quad \hat{g}^n = \mu^n T \hat{g}^n + \varepsilon_n. $$

We observe that $\varepsilon_n \to 0$ and $A(\hat{g}^n) = 1$. Because of the compactness assumption (v), we deduce that there exists $f_1 \in K_2$ and a subsequence $(\hat{g}^{n_k})$ such that $\hat{g}^{n_k} \to f_1$ and $A(f_1) = 1$. We conclude by passing to the limit in the above almost first eigenvalue equations. $\square$

We may compare Theorem 2.24 with the results presented in the previous section. When $\mathcal{L}$ satisfies condition (H1), we may set $\mathcal{R} := \mathcal{R}_{\mathcal{L}}(\kappa_1)$ so that $\mathcal{R} \in \mathcal{B}(X)$ and $\mathcal{R}$ satisfies (i). In that case, Theorem 2.24 claims the existence of $f_1 \in K_2$ such that $\mathcal{L}f_1 = \lambda_1 f_1$, with $\lambda_1 := \kappa_1 - \mu_1$. The condition (ii) on $\mathcal{R}$ translates as $\mathcal{L}g_2 \le (\kappa_1 - 1/C_2)g_2$ which may be seen as an equivalent of condition (H1) (when working in the space $X_2 := K_2 - K_2$ with norm $\|g\|_2 := A(|g|)$) and $\mathcal{L}$ generates a semigroup $S$. The hypothesis (iii) is nothing but (H2) and the hypothesis (iv) is very natural: it holds in the space $L^p(E)$ and $M^1(E)$ without additional condition on $\mathcal{R}$ and it holds in a space of continuous functions when some additional uniform continuity assumption is made on the range of $\mathcal{R}$. Assumption (v) has to be compared with condition (H3). It is worth emphasizing that when $X \subset L^p(E)$ and $g_2 > 0$ a.e., we simply have $A(g) = \|g/g_2\|_{L^\infty}$ for any $g \in X_+$. As a conclusion, although Theorem 2.21 and Theorem 2.24 bear some similarities none seems to be a consequence of the other. We believe that Theorem 2.21 is more flexible since it does not impose to work with the normalization associated to the seminorm $g \mapsto A(|g|)$ of $L^\infty$-type. It is also worth emphasizing the very similarity between Step 3 in the proof of Theorem 2.24 and the proof of Lemma 2.6 and, on the other hand, that Theorem 1.2 is a particular case of Theorem 2.24 by essentially exploiting the fundamental Lemma 2.23 as shown in [252]. We finally point out that when $Y = X'$, the condition (iv) is equivalent to a property of Banach lattices known as *order continuous norm*, see for instance [264, Definition 2.4.1], as a consequence of [264,
$\mathcal{R}: X_+ \setminus \{0\} \to X_{++}$, in particular (H1) holds true. As a first step, we recall the following very classical technical lemma of the Krein-Rutman theory.

**Lemma 2.23.** Assume $X_{++} := \text{int} X_+ \neq \emptyset$. For $g \in X_+$ and $f \in X_{++}$, there exists $C \ge 0$ such that $g \le Cf$.

*Proof of Lemma 2.23.* We argue by contradiction. Otherwise, for any $n \ge 1$, we would have $f - g/n \in X_+^c \subset X_{++}^c$ and that last set is closed. Passing to the limit, we get $f \in X_{++}^c$, which is in contradiction with the assumption $f \in X_{++}$. $\square$

For a given $g_0 \in X_+ \setminus \{0\}$, we set $f_0 := \mathcal{R}g_0 \in X_{++}$. From Lemma 2.23, there exists $C_0 \ge 0$ such that $(\kappa - \mathcal{L})f_0 = g_0 \le C_0f_0$. That implies that Lemma 2.4-(ii) holds with $\kappa_0 := \kappa - C_0$, and thus (H2) also holds. One may then define $\mu_1 := \kappa - \lambda_1$, with

$$ \lambda_1 := \inf\{\lambda \in \mathbb{R}; (\lambda' - \mathcal{L})^{-1} \in \mathcal{B}(X), \forall \lambda' \in [\lambda, \kappa]\} \ge \kappa_0. $$

We recall that because of Lemma 2.6 (or its proof), there exist $(\lambda_n)$, $(\hat{f}_n)$ and $(\varepsilon_n)$ such that (2.20) holds, namely

$$ \lambda_n \searrow \lambda_1, \quad \hat{f}_n \ge 0, \quad \varepsilon_n := \lambda_n \hat{f}_n - \mathcal{L} \hat{f}_n \ge 0, \quad \|\hat{f}_n\| = 1, \quad \|\varepsilon_n\| \to 0. $$

We may rewrite the equation as

$$ \hat{f}_n = \mathcal{R}[\varepsilon_n + (\kappa - \lambda_n)\hat{f}_n], $$

so that $(\hat{f}_n)$ belongs to a compact set of $X$ because of the compactness assumption made on $\mathcal{R}$, so that (H3) holds true.

Because of Theorem 2.21, we deduce that there exists $f_1 \in X_+$ such that $\|f_1\| = 1$ and $\mathcal{L}f_1 = \lambda_1$. That implies $f_1 = \mu_1 \mathcal{R}f_1$, and thus that the existence part of Theorem 1.2 is a consequence of Theorem 2.21 for an operator $\mathcal{R}$ which is the positive resolvent of an operator $\mathcal{L}$.

We would like to emphasize on the fact that our definition of the first eigenvalue by (2.15)-(2.16) bears some strong similarity with the definition of the first eigenvalue for elliptic operators in non divergence form as presented in [47]. Indeed, if $\lambda \in \mathcal{I}$, then

$$ \exists f \in X_+ \setminus \{0\}, \quad \mathcal{L}f \le \lambda f. $$

Assuming now that $X$ is a space of functions (defined on a set $E$) and that $f(x) > 0$ for any $x \in E$, we deduce that

$$ \lambda \ge \sup_E \frac{\mathcal{L}f}{f}, $$

and thus $\lambda_1$ is characterized by

$$ \lambda_1 = \inf_{f>0} \sup_E \frac{\mathcal{L}f}{f}, $$

which is nothing but [47, (1.13)] (with a change of sign because of a different sign convention). We thus see that our formulation is a generalization at a more abstract level and for resolvent positive operators of that classical min-max approach for elliptic operators. Some more or less classical references on that subject are [141, 142], [301], [317], [48] and [46]. In particular in [48], two generalized principal eigenvalues

$$ \lambda_1' := \sup\{\kappa_0 \in \mathbb{R}; \exists g_0 \in \mathcal{C}_0 \mathcal{L}g_0 \ge \kappa_0 g_0\} $$

and

$$ \lambda_1'' := \inf\{\kappa_1 \in \mathbb{R}; \exists g_1 \in \mathcal{C}_1 \mathcal{L}g_1 \le \kappa_1 g_1\} $$

are defined for appropriate cones $\mathcal{C}_i \subset X_+ \setminus \{0\}$ for problems with lack of compactness. The links between the three quantities $\lambda_1$, $\lambda_1'$ and $\lambda_1''$ are discussed as well as the possible non existence of a related principal eigenfunction $f_1$. The non existence of associated principal eigenfunction should not be a surprise since it is the case when one considers $\mathcal{L} = \Delta$ in $X = L^2(\mathbb{R}^d)$ where $\mathcal{L}\psi = \mathcal{L}^*\psi = \lambda_1''\psi$ with $0 < \psi = 1 \notin X = X'$ and $\lambda_1'' = 0$, but no associated principal eigenfunction exists in $X$. We also refer to [251] where some examples of such a situation are discussed.

For its own interest and further discussions, we finally state and prove a slightly less general variant of Theorem 1.8.
Thm. 2.4.2 (iii)] along with the fact that weakly convergent increasing sequences in Banach lattices are automatically norm convergent, see e.g. [264, Prop. 1.4.1].

### 3. EXISTENCE THROUGH A DYNAMICAL APPROACH

In this part, we develop a dynamical approach for proving the existence part of the Krein-Rutman Theorem. We thus always consider a positive semigroup $S = S_\mathcal{L}$ on a Banach lattice $X$. We recover Theorem 2.21 under slightly reinforced assumptions. Above all, we are able to extend the existence part of the Krein-Rutman Theorem to a more general framework, namely to the case when $\mathcal{L}$ only enjoys a suitable weakly dissipative condition.

#### 3.1. About dissipativity.

Let us start by recalling some classical definitions and results. We say that an operator $\mathcal{L}$ defined in a Banach space $X$ is dissipative if there is some number $\kappa \in \mathbb{R}$ such that

$$ \forall f \in D(\mathcal{L}), \exists f^* \in J_f, \Re e(f^*, \mathcal{L}f) \le \kappa \|f\|^2, $$

where we define the associated dual set $J_f \subset X'$ of $f$ by

$$ (3.1) \qquad J_f := \{\varphi \in X'; \langle \varphi, f \rangle = \|f\| = \|\varphi\|_{X'}\}. $$

In that situation and in order to be more precise, we should say that $\mathcal{L}-\kappa$ is dissipative. It is worth emphasizing that $J_f \ne \emptyset$ thanks to the corollary (2.2) of the Hahn-Banach dominated extension theorem. We say that an operator $\mathcal{L}$ is hypodissipative in a Banach space $X$ if there exist an equivalent norm $\|\cdot\|$ in $X$ and a number $\kappa \in \mathbb{R}$ such that

$$ (3.2) \qquad \forall f \in D(\mathcal{L}), \exists f^* \in J_{f, \|\cdot\|_\mathcal{L}}, \Re e(f^*, \mathcal{L}f) \le \kappa \|f\|^2, $$

where

$$ (3.3) \qquad J_{f, \|\cdot\|_\mathcal{L}} := \{\varphi \in X'; \langle \varphi, f \rangle = \|f\|^2 = \|\varphi\|_{X'}^2\}. $$

The only difference between the two definitions (3.1) and (3.3) comes from the norms in which the normalization is performed. When $\mathcal{L}$ is the generator of a semigroup $S_\mathcal{L}$, one can show that the growth bound $\omega = \omega(S_\mathcal{L})$ defined in (2.9) also satisfies

$$ \omega = \inf\{\kappa \in \mathbb{R}, (3.2) \text{ holds for some equivalent norm } \|\cdot\|\}, $$

and $S_\mathcal{L}$ is a semigroup of contraction when $\mathcal{L}$ is dissipative with $\kappa = 0$. At least formally, denoting $f_t := S(t)f$, for $f \in D(\mathcal{L})$, we deduce from (3.2) that

$$ \frac{1}{2} \frac{d}{dt} \|f_t\|^2 = \Re e((f_t)^*, \mathcal{L}f_t) \le \kappa \|f_t\|^2, $$

and together with the Grönwall lemma, we deduce

$$ \|S(t)f\| \le e^{\kappa t} \|f\|, \quad \forall t \ge 0, $$

which is nothing but (2.10). That last estimate is actually equivalent to the hypodissipativity estimate (3.2). Quite similarly, when

$$ (3.4) \qquad \exists \psi \in Y_+ \setminus \{0\}, \exists \kappa \in \mathbb{R}, \pm \mathcal{L}^* \psi \le \kappa \psi, $$

we may compute

$$ \pm \frac{d}{dt} \langle f_t, \psi \rangle = \pm \langle \mathcal{L} f_t, \psi \rangle = \pm \langle f_t, \mathcal{L}^* \psi \rangle \le \kappa \langle f_t, \psi \rangle, $$

and together with the Grönwall lemma, we get

$$ (3.5) \qquad \pm \langle S_t f, \psi \rangle \le \pm e^{\pm \kappa t} \langle f, \psi \rangle, \quad \forall t \ge 0. $$

Two important more accurate versions of the previous ones are presented now. They will be of main importance in the sequel. On the one hand, we may assume that $\mathcal{L}$ satisfies a Lyapunov type condition, namely there exists $\psi_i \in Y_+$ and $\kappa \in \mathbb{R}$ such that

$$ (3.6) \qquad \mathcal{L}^* \psi_2 \le \kappa \psi_2 + \psi_0, $$
**Theorem 2.24.** Consider a Banach lattice $X$ and a linear and bounded operator $\mathcal{R}: X \to X$ such that

(i) $\mathcal{R}: X_+ \to X_+$;

(ii) $\exists g_2 \in X_+ \setminus \{0\}, \exists C_2 > 0$ such that $\mathcal{R}g_2 \le C_2g_2$.

We define

$$K_2 := \{g \in X_+; \exists a > 0, g \le ag_2\},$$

and next

$$A(g) := \inf\{a > 0; g \le ag_2\}, \quad \text{if } g \in K_2,$$

as well as

$$\mathcal{J} := \{\mu \ge 0; \exists h \in K_2, h \ge \mu \mathcal{R}h + g_2\}.$$

We further assume

(iii) $\mu_1 := \sup \mathcal{J} < +\infty$.

(iv) Any upper bounded and increasing sequences $(g^n)$ of $X$ is convergent in the weak sense $\sigma(X, Y)$. More precisely, if $g_n \le g_{n+1} \le \bar{g} \in X$ for any $n \ge 1$, there exists $g \in X$, $g \le \bar{g}$, such that $g_n \to g$.

(v) Any sequence $(g^n)$ of normalized almost first eigenvectors is relatively compact. More precisely, for any sequence $(g^n)$ of $K_2$ such that $A(g^n) = 1$, $g^n = \mu^n \mathcal{R}g^n + \varepsilon^n$, $\mu^n \nearrow \mu_1$ and $\varepsilon^n \to 0$, there exists $g \in K_2$ and a subsequence $(g_{n^k})$ such that $g_{n^k} \to g$ and $A(g) = 1$.

Then there exists $f_1 \in X_2$ such that $f_1 = \mu_1 \mathcal{R}f_1$ and $A(f_1) = 1$.

**Proof of Theorem 2.24.** We split the proof into three steps.

**Step 1.** We first establish that for any $\mu \in \mathcal{J}$, there exists $\tilde{g} = \tilde{g}_\mu \in K_2$ such that

$$(2.35) \qquad \tilde{g} = \mu \mathcal{R} \tilde{g} + g_2.$$

We set $\tilde{g}_0 = 0$, $\tilde{g}_1 = g_2$, and we define $(\tilde{g}_n)$ recursively by $\tilde{g}_{n+1} = \mu \mathcal{R} \tilde{g}_n + g_2$, for any $n \ge 1$. We claim that

$$0 \le \tilde{g}_n \le \tilde{g}_{n+1} \le h, \quad \text{for any } n \ge 0,$$

where $h$ enters in the definition of $\mu \in \mathcal{J}$. That is obviously true at order $n = 0$. Assuming that last inequality is proved at order $n-1$ for $n \ge 1$, we compute

$$\tilde{g}_{n+1} = \mu \mathcal{R} \tilde{g}_n + g_2 \ge \mu \mathcal{R} \tilde{g}_{n-1} + g_2 = \tilde{g}_n$$

and

$$\tilde{g}_{n+1} = \mu \mathcal{R}g_n + g_2 \le \mu \mathcal{R}h + g_2 \le h,$$

which proves the same inequality at order $n$, and thus for any $n \ge 0$. Using the convergence property (iv) of upper bounded increasing sequences in $X$, we deduce that there exists $\tilde{g} \in X_2$ such that $\tilde{g}_n \to \tilde{g}$ and thus (2.35) holds.

**Step 2.** We obviously have $0 \in \mathcal{J}$ and $\mathcal{J}$ is an interval because if $(\mu, h)$ satisfies the condition $\mu \in \mathcal{J}$ then so do $(\mu', h)$ for any $\mu' \in [0, \mu]$. We finally claim that $\mathcal{J}$ is open. Take indeed $\mu \in \mathcal{J}$ and $\tilde{g} \in K_2$ such that (2.35) holds, what is possible due to Step 1. By definition, there would exist $a > 0$ such that $\tilde{g} \le ag_2$. Choosing $0 < \varepsilon \le 1/(2aC_2)$ and $M \ge 2$, we compute

$$(M\tilde{g}) - (\mu + \varepsilon)\mathcal{R}(M\tilde{g}) = Mg_2 - M\varepsilon\mathcal{R}\tilde{g} \\ \ge Mg_2 - M\varepsilon a \mathcal{R}g_2 \ge M(1 - \varepsilon a C_2)g_2 \ge g_2,$$

so that $\mu + \varepsilon \in \mathcal{J}$.

**Step 3.** We first establish by contradiction that $A(\tilde{g}_\mu) \nearrow \infty$ when $\mu \nearrow \mu_1$. If it was not the case, there exists $a \in (0, \infty)$ and a sequence $(\mu^n)$ such that $A(\tilde{g}_{\mu^n}) \le a$ as $\mu^n \nearrow \mu_1$. Choosing $0 < \varepsilon \le 1/(2aC_2)$ and $M \ge 2$ as in Step 2, the same computation gives

$$(M\tilde{g}_{\mu^n}) - (\mu + \varepsilon)\mathcal{R}(M\tilde{g}_{\mu^n}) \ge g_2,$$

so that $\mu^n + \varepsilon \in \mathcal{J}$. That means that $\mu^n + \varepsilon \le \mu_1$, and a contradiction with the fact $\mu^n \nearrow \mu_1$. We next consider $\mu^n \nearrow \mu_1$ and we define

$$a^n := A(\tilde{g}_{\mu^n}), \quad \hat{g}^n := \frac{\tilde{g}_{\mu^n}}{a^n}, \quad \varepsilon_n := \frac{g_2}{a^n}, \quad \hat{g}^n = \mu^n \mathcal{R} \hat{g}^n + \varepsilon_n.$$
with $\psi_2 > 0$ and $\psi_0/\psi_2 \to 0$ at infinity. For $f_t = S_L(t)f$, $f \in D(\mathcal{L}) \cap X_+$, a similar computation as above gives

$$ \frac{d}{dt} \langle f_t, \psi_2 \rangle = \langle f_t, \mathcal{L}^* \psi_2 \rangle \le \kappa \langle f_t, \psi_2 \rangle + \langle f_t, \psi_0 \rangle. $$

Denoting $[f]_i := \langle |f|, \psi_i \rangle$ and using the Grönwall lemma, we classically deduce

$$ (3.7) \qquad [S(t)f]_2 \le e^{\kappa t}[f]_2 + \int_0^t e^{\kappa(t-s)}[S(s)f]_0 ds, \quad \forall t \ge 0. $$

The Lyapunov condition (3.6) is particularly relevant and useful in a Radon measures space framework $X = M_{\psi_2}^1(E)$ for some weight function $\psi_2$ on $E$.

On the other hand, we may generalize the above Lyapunov condition by assuming the structure condition

(HS2) there exist a splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ and $\kappa_B \in \mathbb{R}$ such that $\mathcal{A}$ is $\mathcal{B}$-bounded, that means

$$ \exists C \ge 0, \forall f \in X, \quad \|\mathcal{A}f\| \le C(\|f\| + \|\mathcal{B}f\|), $$

the operator $\mathcal{B}$ generates a semigroup $S_B$ and

$$ (3.8) \qquad \|(S_B \mathcal{A})^{(*\ell)} * S_B(t)\|_{\mathcal{B}(X)} = \mathcal{O}(e^{\alpha t}), \quad \forall t > 0, $$

for any $\ell \ge 0$ and $\alpha > \kappa_B$.

Here and below, for two functions $U : \mathbb{R}_+ \to \mathcal{B}(X_0, X_1)$ and $V : \mathbb{R}_+ \to \mathcal{B}(X_1, X_2)$, we define the convolution function

$$ (V * U)(t) := \int_0^t V(t-s)U(s) ds, $$

when the integral is well-defined. For $U : \mathbb{R}_+ \to \mathcal{B}(X)$, we also recursively define $U^{(*0)} = I$ and $U^{(*(\ell+1))} = U^{(*\ell)} * U$. Using this convolution notation, the Duhamel formula writes

$$ S_{\mathcal{L}} = S_{\mathcal{B}} + S_{\mathcal{B}} \mathcal{A} * S_{\mathcal{L}}, $$

and iterating this formula, for any $N \ge 1$, we get the following iterated Duhamel formula

$$ (3.9) \qquad S_{\mathcal{L}} = S_{\mathcal{B}} + \dots + (S_{\mathcal{B}} \mathcal{A})^{*(N-1)} * S_{\mathcal{B}} + (S_{\mathcal{B}} \mathcal{A})^{*(N)} * S_{\mathcal{L}}. $$

When $S_{\mathcal{L}}$ is well defined in another space $X_0 \supset X$ and the last iterated convolution term enjoys the regularity property $\|(S_{\mathcal{B}} \mathcal{A})^{(*N)}(t)\|_{\mathcal{B}(X_0, X)} = \mathcal{O}(e^{\alpha t})$ for all $t > 0$ and $\alpha > \kappa_B$, we deduce from the above iterated Duhamel formula, the estimate

$$ (3.10) \qquad \|S(t)f\| \le C_0 e^{\alpha t} \|f\| + C_1 \int_0^t e^{\alpha(t-s)} \|S(s)f\|_0 ds, \quad \forall t \ge 0, \alpha > \kappa_B, \forall f \in X, $$

for some constants $C_i \ge 1$ and where $\| \cdot \|_0$ stands for the norm in $X_0$. We may observe that the estimate (3.7) in the case of a Lyapunov condition is a particular case of (3.10) corresponding to the norms $\| \cdot \| = [\cdot]_2$ and $\| \cdot \|_0 = [\cdot]_0$. More specifically, in a Radon measures space framework, the splitting condition (HS2) is obtained by introducing the bounded operator $\mathcal{A}f := f\psi_0$ and the generator $\mathcal{B} := \mathcal{L} - \mathcal{A}$. Because of (3.6), we have $\mathcal{B}^*\psi_2 \le \kappa\psi_2$, and arguing as for establishing (3.7), we have $[S_{\mathcal{B}}(t)f]_2 \le e^{\kappa t}[f]_2$ for any $t \ge 0$ and $f \in X$. That last growth condition is equivalent to assuming that $\mathcal{B} - \kappa$ is dissipative for the norm $[\cdot]_2$, so that we have established that $\mathcal{L}$ enjoys the splitting condition (HS2).

## 3.2. Existence in the dissipative case.

In this section, we give an existence result for a positive semigroup $S_{\mathcal{L}}$ on a Banach lattice $X$ satisfying a kind of regularity/compactness assumption in the spirit of the structure condition (HS2) discussed above.

**Theorem 3.1.** On a Banach lattice $X = Y'$, with $Y$ separable Banach lattice, consider a positive semigroup $S = S_{\mathcal{L}}$ satisfying the growth bound (2.10), and set $\kappa_1 := \omega' + \log M$ for some $\omega' > \omega(S_{\mathcal{L}})$.

We assume

$$ (1) \quad \exists \phi_0 \in Y_+ \setminus \{0\}, \exists \kappa_0 \in \mathbb{R} \text{ such that } [S(t)f]_0 \ge e^{\kappa_0 t}[f]_0 \text{ for any } t \ge 0 \text{ and } f \in X_+, \\ \text{where we denote } [f]_0 := \langle |f|, \phi_0 \rangle; $$
We observe that $\varepsilon_n \to 0$ and $A(\hat{g}^n) = 1$. Because of the compactness assumption (v), we deduce that there exists $f_1 \in K_2$ and a subsequence $(\hat{g}^{n_k})$ such that $\hat{g}^{n_k} \to f_1$ and $A(f_1) = 1$. We conclude by passing to the limit in the above almost first eigenvalue equations. $\square$

We may compare Theorem 2.24 with the results presented in the previous section. When $\mathcal{L}$ satisfies condition (H1), we may set $\mathcal{R} := \mathcal{R}_\mathcal{L}(\kappa_1)$ so that $\mathcal{R} \in \mathcal{B}(X)$ and $\mathcal{R}$ satisfies (i). In that case, Theorem 2.24 claims the existence of $f_1 \in K_2$ such that $\mathcal{L}f_1 = \lambda_1 f_1$, with $\lambda_1 := \kappa_1 - \mu_1$. The condition (ii) on $\mathcal{R}$ translates as $\mathcal{L}g_2 \le (\kappa_1 - 1/C_2)g_2$ which may be seen as an equivalent of condition (H1) (when working in the space $X_2 := K_2 - K_2$ with norm $\|g\|_2 := A(|g|)$ and $\mathcal{L}$ generates a semigroup $S$. The hypothesis (iii) is nothing but (H2) and the hypothesis (iv) is very natural: it holds in the space $L^p(E)$ and $M^1(E)$ without additional condition on $\mathcal{R}$ and it holds in a space of continuous functions when some additional uniform continuity assumption is made on the range of $\mathcal{R}$. Assumption (v) has to be compared with condition (H3). It is worth emphasizing that when $X \subset L^p(E)$ and $g_2 > 0$ a.e., we simply have $A(g) = \|g/g_2\|_{L^\infty}$ for any $g \in X_+$. As a conclusion, although Theorem 2.21 and Theorem 2.24 bear some similarities none seems to be a consequence of the other. We believe that Theorem 2.21 is more flexible since it does not impose to work with the normalization associated to the seminorm $g \mapsto A(|g|)$ of $L^\infty$-type. In a sense, above Lions' proof is dual to the probabilistic approaches in [110, 111, 34, 35], these last ones being more in the spirit of Lemma 2.18, Lemma 2.19 and Corollary 2.20. It is also worth emphasizing the very similarity between Step 3 in the proof of Theorem 2.24 and the proof of Lemma 2.6 and, on the other hand, that Theorem 1.2 is a particular case of Theorem 2.24 by essentially exploiting the fundamental Lemma 2.23 as shown in [251]. We finally point out that when $Y = X'$, the condition (iv) is equivalent to a property of Banach lattices known as order continuous norm, see for instance [263, Definition 2.4.1], as a consequence of [263, Thm. 2.4.2 (iii)] along with the fact that weakly convergent increasing sequences in Banach lattices are automatically norm convergent, see e.g. [263, Prop. 1.4.1].

### 3. EXISTENCE THROUGH A DYNAMICAL APPROACH

In this part, we develop a dynamical approach for proving the existence part of the Krein-Rutman Theorem. We thus always consider a positive semigroup $S = S_\mathcal{L}$ on a Banach lattice $X$ and we give an alternative proof of the existence of $(\lambda_1, f_1)$ under conditions implying (H1)-(H2) and some structure conditions in the spirit of those in the previous Section which imply (H3), see in particular (HS2) and (HS3). Above all, we are able to extend the existence part of the Krein-Rutman Theorem to a more general framework, namely to the case when $\mathcal{L}$ only enjoys a suitable weakly dissipative condition.

#### 3.1. About dissipativity.

Let us start by recalling some classical definitions and results. We say that an operator $\mathcal{L}$ defined in a Banach space $X$ is dissipative if there is some number $\kappa \in \mathbb{R}$ such that

$$ (3.1) \qquad \forall f \in D(\mathcal{L}), \exists f^* \in J_f, \quad \Re e \langle f^*, \mathcal{L} f \rangle \le \kappa \|f\|^2, $$

where we define the associated dual set $J_f \subset X'$ of $f$ by

$$ (3.2) \qquad J_f := \{\varphi \in X'; \langle \varphi, f \rangle = \|f\| = \|\varphi\|_{X'}\}. $$

In that situation and in order to be more precise, we should say that $\mathcal{L}-\kappa$ is dissipative. It is worth emphasizing that $J_f \ne \emptyset$ thanks to the corollary (2.2) of the Hahn-Banach dominated extension theorem. At least formally, denoting $f_t := S(t)f$, for $f \in D(\mathcal{L})$, we deduce from (3.1) that

$$ \frac{1}{2} \frac{d}{dt} \|f_t\|^2 = \Re e \langle (f_t)^*, \mathcal{L} f_t \rangle \le \kappa \|f_t\|^2, $$

and together with the Grönwall lemma, we deduce

$$ \|S(t)f\| \le e^{\kappa t} \|f\|, \quad \forall t \ge 0, $$

which is nothing but (2.10). Quite similarly, when

$$ (3.3) \qquad \exists \psi \in Y_+ \setminus \{0\}, \exists \kappa \in \mathbb{R}, \pm \mathcal{L}^* \psi \le \kappa \psi, $$
(2) there exist $\kappa, C_0, C_1 \in \mathbb{R}$ with $\kappa < \kappa_0$, $C_0 \ge 1$ and $C_1 \ge 0$, such that

$$ (3.11) \qquad \|S(t)f\| \le C_0 e^{\kappa t} \|f\| + C_1 \int_0^t e^{\kappa(t-s)} [S(s)f]_0 ds, \quad \forall t \ge 0, \forall f \in X. $$

Then there exist $\lambda_1 \in [\kappa_0, \kappa_1]$ and $f_1 \in X_+ \setminus \{0\}$ such that $\mathcal{L}f_1 = \lambda_1 f_1$.

Let us mention that this result shares similarities with [260, Cor. 2.7] and [111, Thm. 4.2], see also [259, 94] for earlier works in that direction.

**Remark 3.2.** (1) Assumption (2) in the statement of Theorem 3.1 holds when there exist $V, W$ such that

$$ (3.12) \qquad S = V + W * S, \quad W \ge 0, $$

and there exist $\kappa, C_V, C_W \in \mathbb{R}$, $\kappa < \kappa_0$, $C_V \ge 1$, $C_W > 0$ such that

$$ (3.13) \qquad \|V(t)\|_{\mathcal{B}(X)} \le C_V e^{\kappa t}, \quad \|W(t)\|_{\mathcal{B}(X_0, X)} \le C_W e^{\kappa t}. $$

(2) Under the structural condition (HS2) together with some regularization effect on the semigroup of the type

$$ \|(S_B A)^{(N)}(t)\|_{\mathcal{B}(X_0, X)} = \mathcal{O}(e^{\kappa t}), \quad \forall t > 0, \kappa \in (\kappa_B, \kappa_0), $$

we recover the above condition (3.12)-(3.13) with

$$ (3.14) \qquad V := S_B + \dots + (S_B A)^{(N-1)} * S_B, \quad W := (S_B A)^{(N)}, $$

because of the iterated Duhamel formula (3.9). In that case, the representation formula (2.13) holds true for any $z > \lambda_1$ from Lemma 2.2-(ii) and we easily compute

$$ \mathcal{R}_L(z) = \mathcal{V}(z) + \mathcal{W}(z)\mathcal{R}_L(z), \quad \forall z > \lambda_1, $$

with

$$ \mathcal{V}(z) := \int_0^\infty e^{-\lambda t} V(t) dt, \quad \mathcal{W}(z) := \int_0^\infty e^{-\lambda t} W(t) dt, \quad \forall z > \kappa. $$

We observe that $\mathcal{W}$ satisfies (2.28) in Lemma 2.15 if $\mathcal{W}$ satisfies (3.13) and the set $\mathcal{C}$ defined by (2.26) satisfies the same compactness properties as required in the statement of Lemma 2.11. We may thus apply Lemma 2.15 (see also Remark 2.16) and deduce that (H3) holds for the primal problem. We finally obtain the same conclusion as in Theorem 3.1 thanks to Theorem 2.21.

(3) Under the same structural condition (HS2) as above, but assuming now that

$$ \|W(t)\|_{\mathcal{B}(X, X_1)} = \mathcal{O}(e^{\kappa t}), \quad \forall t > 0, \kappa \in (\kappa_B, \kappa_0), $$

with $W := (S_B A)^{(N)}$ and $X_1 \subset X$ with strongly compact embedding, we observe that $S$ does not necessary satisfies the assumptions of Theorem 3.1, but it rather satisfies the assumptions of Lemma 2.7 with $K_T := (W * S)(T)$ and $T > 0$ large enough. In that situation, we also obtain the same conclusion as in Theorem 3.1 thanks to Lemma 2.7 and Theorem 2.21.

**Proof of Theorem 3.1.** We split the proof into two steps.

**Step 1.** We define the set

$$ \mathcal{C} := \{f \in X_+, [f]_0 = 1, \|f\| \le R\}, $$

for a convenient constant $R > 0$ to be fixed later. For any fixed $t > 0$, we next define the nonlinear weakly $\sigma(X, Y)$ continuous mapping

$$ \Phi_t : \mathcal{C} \to X, \quad f \mapsto \frac{S_t f}{[S_t f]_0}. $$

Thanks to assumption (1), we may observe that it is well defined because

$$ (3.15) \qquad [S_t f]_0 \ge e^{\kappa_0 t} [f]_0 = e^{\kappa_0 t} > 0. $$

For any $f \in \mathcal{C}$, we thus immediately have $\Phi_t f \ge 0$ and $[\Phi_t f]_0 = 1$. On the other hand, from assumption (1) again and the semigroup property, we have

$$ (3.16) \qquad [S(t)f]_0 \ge e^{\kappa_0(t-s)} [S(s)f]_0. $$
we may compute

$$ \pm \frac{d}{dt} \langle f_t, \psi \rangle = \pm \langle \mathcal{L} f_t, \psi \rangle = \pm \langle f_t, \mathcal{L}^* \psi \rangle \le \kappa \langle f_t, \psi \rangle, $$

and together with the Grönwall lemma, we get

$$ (3.4) \qquad \pm \langle S_t f, \psi \rangle \le \pm e^{\pm \kappa t} \langle f, \psi \rangle, \quad \forall t \ge 0. $$

Two important more accurate versions of the previous ones are presented now. They will be of main importance in the sequel. On the one hand, we may assume that $\mathcal{L}$ satisfies a Lyapunov type condition, namely there exists $\psi_i \in Y_+$ and $\kappa \in \mathbb{R}$ such that

$$ (3.5) \qquad \mathcal{L}^* \psi_2 \le \kappa \psi_2 + \psi_0, $$

with $\psi_2 > 0$ and $\psi_0/\psi_2 \to 0$ at infinity. For $f_t = S_{\mathcal{L}}(t)f$, $f \in D(\mathcal{L}) \cap X_+$, a similar computation as above gives

$$ \frac{d}{dt} \langle f_t, \psi_2 \rangle = \langle f_t, \mathcal{L}^* \psi_2 \rangle \le \kappa \langle f_t, \psi_2 \rangle + \langle f_t, \psi_0 \rangle. $$

Denoting $[f]_i := \langle |f|, \psi_i \rangle$ and using the Grönwall lemma, we classically deduce

$$ (3.6) \qquad [S(t)f]_2 \le e^{\kappa t}[f]_2 + \int_0^t e^{\kappa(t-s)} [S(s)f]_0 ds, \quad \forall t \ge 0. $$

The Lyapunov condition (3.5) is particularly relevant and useful in a Radon measures space framework $X = M_{\psi_2}^1(E)$ for some weight function $\psi_2$ on $E$.

On the other hand, we may generalize the above Lyapunov condition by assuming the structure condition

(HS2) there exist a splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$ and $\kappa_B \in \mathbb{R}$ such that $\mathcal{A}$ is $\mathcal{B}$-bounded, that means

$$ \exists C \ge 0, \forall f \in X, \quad \|\mathcal{A}f\| \le C(\|f\| + \|\mathcal{B}f\|), $$

the operator $\mathcal{B}$ generates a semigroup $S_B$ and

$$ (3.7) \qquad \|(S_B \mathcal{A})^{(*\ell)} * S_B(t)\|_{\mathcal{B}(X)} = \mathcal{O}(e^{\alpha t}), \quad \forall t > 0, $$

for any $\ell \ge 0$ and $\alpha > \kappa_B$.

Here and below, for two functions $U : \mathbb{R}_+ \to \mathcal{B}(X_0, X_1)$ and $V : \mathbb{R}_+ \to \mathcal{B}(X_1, X_2)$, we define the convolution function

$$ (V * U)(t) := \int_0^t V(t-s)U(s)ds, $$

when the integral is well-defined. For $U : \mathbb{R}_+ \to \mathcal{B}(X)$, we also recursively define $U^{(*0)} = I$ and $U^{(*(\ell+1))} = U^{(*\ell)} * U$. Using this convolution notation, the Duhamel formula writes

$$ S_{\mathcal{L}} = S_{\mathcal{B}} + S_{\mathcal{B}} \mathcal{A} * S_{\mathcal{L}}, $$

and iterating this formula, for any $N \ge 1$, we get the following iterated Duhamel formula

$$ (3.8) \qquad S_{\mathcal{L}} = S_{\mathcal{B}} + \dots + (S_{\mathcal{B}} \mathcal{A})^{*(N-1)} * S_{\mathcal{B}} + (S_{\mathcal{B}} \mathcal{A})^{*(N)} * S_{\mathcal{L}}. $$

When $S_{\mathcal{L}}$ is well defined in another space $X_0 \supset X$ and the last iterated convolution term enjoys the regularity property $\|(S_{\mathcal{B}} \mathcal{A})^{(*N)}(t)\|_{\mathcal{B}(X_0, X)} = \mathcal{O}(e^{\alpha t})$ for all $t > 0$ and $\alpha > \kappa_B$, we deduce from the above iterated Duhamel formula, the estimate

$$ (3.9) \qquad \|S(t)f\| \le C_0 e^{\alpha t} \|f\| + C_1 \int_0^t e^{\alpha(t-s)} \|S(s)f\|_0 ds, \quad \forall t \ge 0, \alpha > \kappa_B, \forall f \in X, $$

for some constants $C_i \ge 1$ and where $\| \cdot \|_0$ stands for the norm in $X_0$. We may observe that the estimate (3.6) in the case of a Lyapunov condition is a particular case of (3.9) corresponding to the norms $\| \cdot \| = [\cdot]_2$ and $\| \cdot \|_0 = [\cdot]_0$. More specifically, in a Radon measures space framework, the splitting condition (HS2) is obtained by introducing the bounded operator $\mathcal{A}f := f\psi_0$ and the generator $\mathcal{B} := \mathcal{L} - \mathcal{A}$. Because of (3.5), we have $\mathcal{B}^* \psi_2 \le \kappa \psi_2$, and arguing as for establishing (3.6), we have $[S_{\mathcal{B}}(t)f]_2 \le e^{\kappa t}[f]_2$ for any $t \ge 0$ and $f \in X$. That last growth condition is equivalent to assuming that $\mathcal{B} - \kappa$ is dissipative for the norm $[\cdot]_2$, so that we have established that $\mathcal{L}$ enjoys the splitting condition (HS2).
For $f \in C$ and $t \ge 0$, we next compute

$$ \begin{aligned} \|\Phi_t f\| & \le C_0 e^{-\alpha t} \|f\| + C_1 \int_0^t e^{-\alpha(t-s)} ds \\ & \le C_0 e^{-\alpha t} R + \frac{C_1}{\alpha}, \end{aligned} $$

where we have set $\alpha := \kappa_0 - \kappa_B > 0$. Fixing $T_0$ such that $C_0 e^{-\alpha T_0} \le 1/2$ and next $R \ge 2C_1/\alpha$, we have thus $\Phi_{T_0} : C \to C$. Thanks to the Tykonov fixed point Theorem, there exists $f_{T_0} \in C$ such that $\Phi_{T_0} f_{T_0} = f_{T_0}$. In other words, we have established the existence of $f_{T_0} \in X$ such that

$$ (3.17) \qquad f_{T_0} \ge 0, \quad [f_{T_0}]_0 = 1, \quad S_{T_0} f_{T_0} = e^{\lambda_1 T_0} f_{T_0}, $$

with $\lambda_1 := (1/T_0) \log[S_{T_0} f_{T_0}]_0 \in [\kappa_0, \kappa_1]$.

**Step 2.** Rewriting equation (3.17) as

$$ 0 = e^{-\lambda_1 T_0} S_{T_0} f_{T_0} - f_{T_0} = (\mathcal{L} - \lambda_1) \int_0^{T_0} e^{-\lambda_1 t} S_t f_{T_0} dt $$

and defining

$$ f_1 := \int_0^{T_0} e^{-\lambda_1 t} S_t f_{T_0} dt, $$

we get that $f_1 \in X_+ \setminus \{0\}$ satisfies $\mathcal{L}f_1 = \lambda_1 f_1$. $\square$

We present now a second proof based on a large times dynamical argument which is classical in the mean ergodicity theory of Von Neumann and Birkhoff introduced in [356, 60] and which will be adapted in the weak dissipativity case in Section 3.5 below.

**Alternative Step 2.** We define $\tilde{S}_t := S_t e^{-\lambda_1 t}$, so that $f_{T_0}$ becomes a periodic state for $\tilde{S}_t$ from (3.17), namely

$$ \tilde{S}_t f_{T_0} = \tilde{S}_{t-kT_0} f_{T_0}, \quad k := [t/T_0], \quad \forall t > 0. $$

Using (3.15) and the above relation, we have

$$ \begin{aligned} [\tilde{S}_t f_{T_0}]_0 &= [\tilde{S}_{t-kT_0} f_{T_0}]_0 \\ &\ge e^{(\kappa_0 - \lambda_1)(t-kT_0)} [f_{T_0}]_0 \ge e^{(\kappa_0 - \lambda_1)T_0} =: r_* > 0, \end{aligned} $$

for any $t \ge 0$. On the other hand, thanks to the growth bound (2.10), we have

$$ \begin{aligned} \|\tilde{S}_t f_{T_0}\| &= \|\tilde{S}_{t-kT_0} f_{T_0}\| \\ &\le Me^{(\kappa-\lambda_1)(t-kT_0)} \|f_{T_0}\| \le Me^{(\kappa-\lambda_1)T_0} R =: R^* < \infty, \end{aligned} $$

for any $t \ge 0$. We finally define

$$ u_T := \frac{1}{T} \int_0^T \tilde{S}_t f_{T_0} dt. $$

From the previous estimates, both sequences $(\tilde{S}_t f_{T_0})$ and $(u_T)$ are bounded in

$$ \mathbb{K} := \{f \in X; f \ge 0, [f]_0 \ge r_*, \|f\| \le R^*\}. $$

By compactness, there exists a subsequence $(u_{T_k})$ and $f_1 \in \mathbb{K}$ such that $u_{T_k} \to f_1$ in a weak sense as $k \to \infty$. For any fixed $t > 0$, we observe that

$$ \begin{aligned} \tilde{S}_t f_1 - f_1 &= \lim_{k \to \infty} \left\{ \frac{1}{T_k} \int_0^{T_k} \tilde{S}_t \tilde{S}_s f_{T_0} ds - \frac{1}{T_k} \int_0^{T_k} \tilde{S}_s f_{T_0} ds \right\} \\ &= \lim_{k \to \infty} \left\{ \frac{1}{T_k} \int_{T_k}^{T_k+t} \tilde{S}_s f_{T_0} ds - \frac{1}{T_k} \int_0^t \tilde{S}_s f_{T_0} ds \right\} = 0, \end{aligned} $$

where we have used that $(\tilde{S}_s f_{T_0})$ is uniformly bounded in the last line. As a consequence, $f_1$ is a stationary state for the rescaled semigroup $\tilde{S}_t$, and thus an eigenfunction associated to the eigenvalue $\lambda_1$ for the operator $\mathcal{L}$. $\square$
### 3.2. Existence in the dissipative case.

In this section, we give an existence result for a positive semigroup $S_L$ on a Banach lattice $X$ satisfying a kind of regularity/compactness assumption in the spirit of the structure condition (HS2) discussed above. Here and below, for an element $\psi \in Y_+$, we define the (semi)norm

$$[f]_{\psi} := \langle \psi, |f| \rangle, \quad \forall f \in X.$$

We will sometime use the shorthand $[\cdot]_i$ for $[\cdot]_{\psi_i}$.

**Theorem 3.1.** On a Banach lattice $X = Y'$, with $Y$ separable Banach lattice, consider a positive semigroup $S = S_L$ satisfying the growth bound (2.10), and set $\kappa_1 := \omega' + \log M$ for some $\omega' > \omega(S_L)$.

We assume

(1) $\exists \phi_0 \in Y_+ \setminus \{0\}, \exists \kappa_0 \in \mathbb{R}$ such that $[S(t)f]_{\phi_0} \ge e^{\kappa_0 t} [f]_{\phi_0}$ for any $t \ge 0$ and $f \in X_+$;

(2) there exist $\kappa, C_0, C_1 \in \mathbb{R}$ with $\kappa < \kappa_0, C_0 \ge 1$ and $C_1 \ge 0$, such that

$$(3.10) \qquad \|S(t)f\| \le C_0 e^{\kappa t} \|f\| + C_1 \int_0^t e^{\kappa(t-s)} [S(s)f]_{\phi_0} ds, \quad \forall t \ge 0, \forall f \in X.$$

Then there exist $\lambda_1 \in [\kappa_0, \kappa_1]$ and $f_1 \in X_+ \setminus \{0\}$ such that $\mathcal{L}f_1 = \lambda_1 f_1$.

Let us mention that this result shares similarities with [259, Cor. 2.7] and [110, Thm. 4.2], see also [258, 93] for earlier works in that direction.

**Remark 3.2.** (1) Assumption (2) in the statement of Theorem 3.1 holds when there exist $V, W$ such that

$$(3.11) \qquad S = V + W * S, \quad W \ge 0,$$

and there exist $\kappa, C_V, C_W \in \mathbb{R}$, $\kappa < \kappa_0$, $C_V \ge 1$, $C_W > 0$ such that

$$(3.12) \qquad \|V(t)\|_{\mathcal{B}(X)} \le C_V e^{\kappa t}, \quad \|W(t)\|_{\mathcal{B}(X_0, X)} \le C_W e^{\kappa t}.$$

(2) Under the structural condition (HS2) together with some regularization effect on the semigroup of the type

$$\|(S_B A)^{(N)}(t)\|_{\mathcal{B}(X_0, X)} = \mathcal{O}(e^{\kappa t}), \quad \forall t > 0, \kappa \in (\kappa_B, \kappa_0),$$

we recover the above condition (3.11)-(3.12) with

$$(3.13) \qquad V := S_B + \dots + (S_B A)^{(N-1)} * S_B, \quad W := (S_B A)^{(N)},$$

because of the iterated Duhamel formula (3.8). In that case, the representation formula (2.13) holds true for any $z > \lambda_1$ from Lemma 2.2-(ii) and we easily compute

$$\mathcal{R}_L(z) = \mathcal{V}(z) + \mathcal{W}(z)\mathcal{R}_L(z), \quad \forall z > \lambda_1,$$

with

$$\mathcal{V}(z) := \int_0^\infty e^{-\lambda t} V(t) dt, \quad \mathcal{W}(z) := \int_0^\infty e^{-\lambda t} W(t) dt, \quad \forall z > \kappa.$$

We observe that $\mathcal{W}$ satisfies (2.27) in Lemma 2.15 if $\mathcal{W}$ satisfies (3.12) and the set $C$ defined by (2.25) satisfies the same compactness properties as required in the statement of Lemma 2.11. We may thus apply Lemma 2.15 (see also Remark 2.16) and deduce that (H3) holds for the primal problem. We finally obtain the same conclusion as in Theorem 3.1 thanks to Theorem 2.21.

(3) Under the same structural condition (HS2) as above, but assuming now that

$$\|W(t)\|_{\mathcal{B}(X, X_1)} = \mathcal{O}(e^{\kappa t}), \quad \forall t > 0, \kappa \in (\kappa_B, \kappa_0),$$

with $W := (S_B A)^{(N)}$ and $X_1 \subset X$ with strongly compact embedding, we observe that $S$ does not necessary satisfies the assumptions of Theorem 3.1, but it rather satisfies the assumptions of Lemma 2.7 with $K_T := (W * S)(T)$ and $T > 0$ large enough. In that situation, we also obtain the same conclusion as in Theorem 3.1 thanks to Lemma 2.7 and Theorem 2.21.

(4) The condition (H1) follows from Lemma 2.2 and the above assumption (1) implies (H2) thanks to Lemma 2.4.
### 3.3. About weak dissipativity.

In this section, we recall some definitions and results about the weak dissipativity. We say that the generator $\mathcal{B}$ of a semigroup $S_{\mathcal{B}}$ is weakly dissipative in a Banach space $X_i$ if there exist a second Banach space $X_{i-1} \supset X_i$ and some numbers $\kappa \in \mathbb{R}$ and $\sigma > 0$ such that

$$ \forall f \in D(\mathcal{B}|_{X_i}), \exists f^* \in J_{f,X_i}, \langle f^*, \mathcal{B}f \rangle \le \kappa \|f\|_{X_i}^2 - \sigma \|f\|_{X_{i-1}}^2, $$

where we define the associated dual set $J_{f,X_i} \subset X_i'$ of $f$ (for the norm $\| \cdot \|_{X_i}$) by

$$ (3.18) \qquad J_{f,X_i} := \{ \varphi \in X_i' ; \langle \varphi, f \rangle = \|f\|_{X_i}^2 = \|\varphi\|_{X_i'}^2 \}. $$

By translation, we may assume that $\kappa = 0$, an hypothesis we will always make in the sequel of this section. We will furthermore assume the splitting structure $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{A}$ a $\mathcal{B}$-bounded operator and $\mathcal{B}$ weakly dissipative generator.

More precisely, we assume that there exists one more Banach lattice $X_0 \supset X_1 \supset X_2 := X$, with norm denoted by $\| \cdot \|_k := \| \cdot \|_{X_k}$, such that $\mathcal{B}$ generates a semigroup and is weakly dissipative in each $X_k$: for any $k = 1, 2$

$$ (3.19) \qquad \forall f \in D(\mathcal{B}|_{X_k}), \exists f^* \in J_{f,X_k}, \langle f^*, \mathcal{B}f \rangle_{X_k', X_k} \le -\sigma \|f\|_{k-1}^2. $$

This classically implies (or we can take the next inequality as a definition of the weak dissipativity condition) that

$$ (3.20) \qquad \frac{d}{dt} \|S_{\mathcal{B}}(t)f\|_k + \sigma \|S_{\mathcal{B}}(t)f\|_{k-1} \le 0, \quad \forall t \ge 0, \forall f \in X_k, \forall k = 1, 2. $$

We assume that $X_k$ is dense into $X_{k-1}$ for $k = 1, 2$ and that $X_1$ is an interpolated space between $X_0$ and $X_2$ in the sense that there exists a continuous and strictly decreasing function $\eta : (0, 1] \to [0, \infty)$, $\eta(\varepsilon) \to \infty$ when $\varepsilon \to 0$, $\eta(1) = 0$, such that

$$ (3.21) \qquad \|f\|_1 \le \varepsilon \|f\|_2 + \eta(\varepsilon) \|f\|_0, \quad \forall \varepsilon \in (0, 1], \forall f \in X_2. $$

From (3.20) with $k = 2$, we deduce

$$ (3.22) \qquad \|S_{\mathcal{B}}(t)f\|_2 \le \|f\|_2, \quad \forall t \ge 0, \forall f \in X_2. $$

Next, for $k = 1$, gathering the weak dissipativity condition (3.20), the interpolation condition (3.21) and the non expansion property (3.22) in the space $X_2$, we get

$$ \begin{aligned} \frac{d}{dt} \|S_{\mathcal{B}}(t)f\|_1 + \frac{\sigma}{\eta(\varepsilon)} \|S_{\mathcal{B}}(t)f\|_1 &\le \frac{\sigma\varepsilon}{\eta(\varepsilon)} \|S_{\mathcal{B}}(t)f\|_2 \\ &\le \frac{\sigma\varepsilon}{\eta(\varepsilon)} \|f\|_2, \end{aligned} $$

for any $t \ge 0$, $\varepsilon \in (0, 1)$ and $f \in X_2$. We deduce

$$ \frac{d}{dt} \left( \|S_{\mathcal{B}}(t)f\|_1 e^{\frac{\sigma}{\eta(\varepsilon)}t} \right) \le \frac{\sigma\varepsilon}{\eta(\varepsilon)} e^{\frac{\sigma}{\eta(\varepsilon)}t} \|f\|_2, $$

and thanks to the Grönwall lemma, we obtain

$$ (3.23) \qquad \|S_{\mathcal{B}}(t)f\|_1 \le \Theta(t) \|f\|_2, $$

for any $t \ge 0$ and $f \in X_2$, with

$$ (3.24) \qquad \Theta(t) := \inf_{\varepsilon \in (0,1)} \left( e^{-\frac{\sigma}{\eta(\varepsilon)}t} + \varepsilon \right) \to 0 \quad \text{as} \quad t \to +\infty. $$

On the other hand, using the representation formula

$$ \mathcal{R}_{\mathcal{B}}(z) f = \int_{0}^{\infty} e^{-zt} S_{\mathcal{B}}(t) f \, dt, \quad \forall z \in \Delta_0, \forall f \in X_2, $$

together with (3.20), we get

$$ \sigma \|\mathcal{R}_{\mathcal{B}}(z) f\|_1 \le \int_0^\infty \sigma \|S_{\mathcal{B}}(t) f\|_1 dt \le \|f\|_2, $$

for any $z \in \bar{\Delta}_0$ and $f \in X_2$. We next assume that

$$ (3.25) \qquad \Theta(t)^{-1} \|A S_{\mathcal{B}}(t) f\|_1 + \int_0^\infty \|A S_{\mathcal{B}}(t) f\|_1 dt \lesssim \|f\|_1, $$
*Proof of Theorem 3.1.* We split the proof into two steps. During the proof, we will use the short-hand $[·]_0 := [·]_{\phi_0}$.

**Step 1.** We define the set

$$C := \{f \in X_+, [f]_0 = 1, \|f\| \le R\},$$

for a convenient constant $R > 0$ to be fixed later. For any fixed $t > 0$, we next define the nonlinear weakly $\sigma(X, Y)$ continuous mapping

$$\Phi_t : C \to X, \quad f \mapsto \frac{S_t f}{[S_t f]_0}.$$

Thanks to assumption (1), we may observe that it is well defined because

$$([S_t f]_0) \ge e^{\kappa_0 t} [f]_0 = e^{\kappa_0 t} > 0. \quad (3.14)$$

For any $f \in C$, we thus immediately have $\Phi_t f \ge 0$ and $[\Phi_t f]_0 = 1$. On the other hand, from assumption (1) again and the semigroup property, we have

$$([S(t)f]_0) \ge e^{\kappa_0(t-s)} [S(s)f]_0. \quad (3.15)$$

Then, for $f \in C$ and $t \ge 0$, taking advantage of (3.10) and (3.15), we compute

$$\begin{aligned} \|\Phi_t f\| & \le C_0 e^{-\alpha t} \|f\| + C_1 \int_0^t e^{-\alpha(t-s)} ds \\ & \le C_0 e^{-\alpha t} R + \frac{C_1}{\alpha}, \end{aligned}$$

where we have set $\alpha := \kappa_0 - \kappa > 0$. Fixing $T_0$ such that $C_0 e^{-\alpha T_0} \le 1/2$ and next $R \ge 2C_1/\alpha$, we have thus $\Phi_{T_0} : C \to C$. Thanks to the Tykonov fixed point Theorem, there exists $f_{T_0} \in C$ such that $\Phi_{T_0} f_{T_0} = f_{T_0}$. In other words, we have established the existence of $f_{T_0} \in X$ such that

$$([S_{T_0}f_{T_0}]_0) \ge 0, \quad ([f_{T_0}]_0 = 1, \quad S_{T_0}f_{T_0} = e^{\lambda_1 T_0} f_{T_0}, \quad (3.16)$$

$\text{with } \lambda_1 := (1/T_0) \log [S_{T_0} f_{T_0}]_0 \in [\kappa_0, \kappa_1].$

**Step 2.** Rewriting equation (3.16) as

$$0 = e^{-\lambda_1 T_0} S_{T_0} f_{T_0} - f_{T_0} = (\mathcal{L} - \lambda_1) \int_0^{T_0} e^{-\lambda_1 t} S_t f_{T_0} dt$$

and defining

$$f_1 := \int_0^{T_0} e^{-\lambda_1 t} S_t f_{T_0} dt,$$

we get that $f_1 \in X_+ \setminus \{0\}$ satisfies $\mathcal{L}f_1 = \lambda_1 f_1$. $\square$

We present now a second proof based on a large times dynamical argument which is classical in the mean ergodicity theory of Von Neumann and Birkhoff introduced in [355, 60] and which will be adapted in the weak dissipativity case in Section 3.5 below.

*Alternative Step 2.* We define $\tilde{S}_t := S_t e^{-\lambda_1 t}$, so that $f_{T_0}$ becomes a periodic state for $\tilde{S}_t$ from (3.16), namely

$$\tilde{S}_t f_{T_0} = \tilde{S}_{t-kT_0} f_{T_0}, \quad k := [t/T_0], \quad \forall t > 0.$$

Using (3.14) and the above relation, we have

$$\begin{aligned} [\tilde{S}_t f_{T_0}]_0 &= [\tilde{S}_{t-kT_0} f_{T_0}]_0 \\ &\ge e^{(\kappa_0 - \lambda_1)(t-kT_0)} [f_{T_0}]_0 \ge e^{(\kappa_0 - \lambda_1)T_0} =: r_* > 0, \end{aligned}$$

for any $t \ge 0$. On the other hand, thanks to the growth bound (2.10), we have

$$\begin{aligned} \|\tilde{S}_t f_{T_0}\| &= \|\tilde{S}_{t-kT_0} f_{T_0}\| \\ &\le Me^{(\kappa-\lambda_1)(t-kT_0)} \|f_{T_0}\| \le Me^{(\kappa-\lambda_1)T_0} R =: R^* < \infty, \end{aligned}$$

for any $t \ge 0$. We finally define

$$u_T := \frac{1}{T} \int_0^T \tilde{S}_t f_{T_0} dt.$$
that there exist $\alpha > 1$, $N \ge 1$, $C \ge 1$ such that

$$ (3.26) \quad \sup_{x+iy \in \Delta_0} \| \mathcal{A} \mathcal{R}_B^{1+\epsilon_1} (x+iy) \dots \mathcal{A} \mathcal{R}_B^{1+\epsilon_N} (x+iy) f \|_2 \le \frac{C}{\langle y \rangle^\alpha} \|f\|_2, $$

for any $\epsilon \in \{0, 1\}^N$, $\epsilon_1 + \dots + \epsilon_N \le 1$, and that

$$ (3.27) \quad \sup_{z \in \Delta_0} \|(\mathcal{R}_B(z)\mathcal{A})^N f\|_{X_1} \le \|f\|_1, $$

with $X_1$ compactly imbedded in $X_1$. The necessity to add $(\epsilon_i)$ in (3.26) is probably purely technical and not restrictive for applications. In examples, we can take $N = 2N'$, when

$$ (3.28) \quad \sup_{x+iy \in \Delta_0} \|(\mathcal{A} \mathcal{R}_B)^{N'}(x+iy)f\|_3 \le \frac{C}{\langle y \rangle^\alpha} \|f\|_2, $$

for some convenient space $X_3$ such that $\mathcal{A}: X_1 \to X_3$ and $\sup_{z \in \Delta_0} \|\mathcal{R}_B(z)\|_{\mathcal{B}(X_3, X_2)} < \infty$. At the level of the semigroup, (3.28) is typically a consequence of

$$ \|(\mathcal{A} \mathcal{S}_B)^{(*N'')}(t)\|_{\mathcal{B}(X_2, X_3^\zeta)} \in L^1(\mathbb{R}_+), $$

with $\zeta > 0$, where $X_3^\zeta := \{f \in X_3, \mathcal{L}^\zeta f \in X_3\}$ stands for the (possibly fractional) domain for the operator defined in $X_3$. However, (3.26) is a bit more general than that last estimate. We refer to [278, 273, 280, 277] for precise definition, examples and discussion. For further references, we observe that (3.23) and (3.25) together imply

$$ \begin{aligned} \frac{1}{T} \int_0^T \|(S_B * \mathcal{A} S_B)(t) f\|_1 dt &\le \frac{1}{T} \int_0^T \int_0^T \|S_B(t-s) \mathcal{A} S_B(s) f\|_1 ds dt \\ &\le \frac{1}{T} \int_0^T \int_0^T \|S_B(u)\|_{\mathcal{B}(X_2, X_1)} \|\mathcal{A} S_B(s) f\|_1 duds \\ &\lesssim \frac{1}{T} \int_0^T \Theta(u) du \|f\|_2. \end{aligned} $$

Arguing in a similar way for any $\ell \ge 1$, we establish

$$ (3.29) \quad \frac{1}{T} \int_0^T \|(S_B * (\mathcal{A} S_B)^{(*\ell)})(t) f\|_1 dt \lesssim \frac{1}{T} \int_0^T \Theta du \|f\|_2 \to 0 \quad \text{as } T \to \infty. $$

For synthesizing and for further references, let us now bring out some possible general framework for semigroup enjoying weak dissipativity. We introduce the following structure condition on a semigroup $S_\mathcal{L}$ and its generator $\mathcal{L}$ by assuming

(HS3) there exist a splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$, some Banach lattices $X_2 \subset X_1$, an integer $N \ge 1$ 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 $\mathcal{A}$ is positive, $\mathcal{B}$ generates a positive semigroup $S_B$ and the following estimates hold

$$ (3.30) \quad \|(S_B \mathcal{A})^{(*\ell)} * S_B \|_{\mathcal{B}(X_2, X_1)} = \mathcal{O}(\Theta_1), \quad \forall \ell \in \{0, \dots, N-1\}, $$

$$ (3.31) \quad \|(S_B \mathcal{A})^{(*N)}\|_{\mathcal{B}(X_1, X_2)} = \mathcal{O}(\Theta_2). $$

We now particularize our discussion to a Radon measures framework. We assume that there exist some weight functions $\psi_i$ on $E$, $\psi_0 \lesssim \psi_1 \le \psi_2$, with $\psi_2(x)/\psi_1(x) \to \infty$ as $x \to \infty$ so that $M_{\psi_2}^1 \subset \subset M_{\psi_1}^1$ (compact imbedding for the weak convergence), a function $\chi \in C_c(E)$, $0 \le \chi \le 1$, and a constant $M \ge 0$ such that

(i) $\mathcal{L}^*\psi_1 \le -\psi_0 + M\chi$;

(ii) $\mathcal{L}^*\psi_2 \le M\chi$;

(iii) $\psi_1 \le \varepsilon\psi_2 + \eta(\varepsilon)\psi_0$ for any $\varepsilon > 0$,

for a function $\eta: (0, 1] \to (0, \infty)$ such that $\eta(1) = 0$, $\eta(\varepsilon) \to \infty$ when $\varepsilon \to 0$, and

$$ (3.32) \quad t \mapsto \Theta(t) := \inf_{\varepsilon \in (0,1)} \left( e^{-\frac{t}{\eta(\varepsilon)}} + \varepsilon \right) \in L^1(0, \infty). $$

It is worth emphasizing that from the very definition, we have automatically that $\Theta$ is positive and decreasing, $\Theta(0) = 1$ and $\Theta(t) \to 0$ as $t \to \infty$. Arguing similarly as we did during the proof of Corollary 2.20 and the end of Section 3.1, we introduce the splitting

$$ \mathcal{A} := M\chi, \quad \mathcal{B} := \mathcal{L} - \mathcal{A}, $$
From the previous estimates, both sequences $(\tilde{S}_t f_{T_0})$ and $(u_T)$ are bounded in

$$ \mathbb{K} := \{ f \in X; f \ge 0, [f]_0 \ge r_*, \|f\| \le R^* \}. $$

By compactness, there exists a subsequence $(u_{T_k})$ and $f_1 \in \mathbb{K}$ such that $u_{T_k} \to f_1$ in a weak sense as $k \to \infty$. For any fixed $t > 0$, we observe that

$$ \begin{aligned} \tilde{S}_t f_1 - f_1 &= \lim_{k \to \infty} \left\{ \frac{1}{T_k} \int_0^{T_k} \tilde{S}_t \tilde{S}_s f_{T_0} ds - \frac{1}{T_k} \int_0^{T_k} \tilde{S}_s f_{T_0} ds \right\} \\ &= \lim_{k \to \infty} \left\{ \frac{1}{T_k} \int_{T_k}^{T_k+t} \tilde{S}_s f_{T_0} ds - \frac{1}{T_k} \int_0^t \tilde{S}_s f_{T_0} ds \right\} = 0, \end{aligned} $$

where we have used that $(\tilde{S}_s f_{T_0})$ is uniformly bounded in the last line. As a consequence, $f_1$ is a stationary state for the rescaled semigroup $\tilde{S}_t$, and thus an eigenfunction associated to the eigenvalue $\lambda_1$ for the operator $\mathcal{L}$. $\square$

### 3.3. About weak dissipativity.

In this section, we recall some definitions and results about the weak dissipativity. We say that the generator $\mathcal{B}$ of a semigroup $S_{\mathcal{B}}$ is weakly dissipative in a Banach space $X_i$ if there exist a second Banach space $X_{i-1} \supset X_i$ and some numbers $\kappa \in \mathbb{R}$ and $\sigma > 0$ such that

$$ \forall f \in D(\mathcal{B}|_{X_i}), \exists f^* \in J_{f,X_i}, \quad \langle f^*, \mathcal{B}f \rangle \le \kappa \|f\|_{X_i}^2 - \sigma \|f\|_{X_{i-1}}^2, $$

where we define the associated dual set $J_{f,X_i} \subset X_i'$ of $f$ (for the norm $\| \cdot \|_{X_i}$) by

$$ (3.17) \qquad J_{f,X_i} := \{ \varphi \in X_i' ; \langle \varphi, f \rangle = \|f\|_{X_i}^2 = \|\varphi\|_{X_i'}^2 \}. $$

By translation, we may assume that $\kappa = 0$, an hypothesis we will always make in the sequel of this section. We will furthermore assume the splitting structure $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{A}$ a $\mathcal{B}$-bounded operator and $\mathcal{B}$ a weakly dissipative generator.

More precisely, we assume that there exists one more Banach lattice $X_0 \supset X_1 \supset X_2 := X$, with norm denoted by $\| \cdot \|_k := \| \cdot \|_{X_k}$, such that $\mathcal{B}$ generates a semigroup and is weakly dissipative in each $X_k$: for any $k = 1, 2$

$$ (3.18) \qquad \forall f \in D(\mathcal{B}|_{X_k}), \exists f^* \in J_{f,X_k}, \quad \langle f^*, \mathcal{B}f \rangle_{X_k', X_k} \le -\sigma \|f\|_{k-1}^2. $$

This classically implies (or we can take the next inequality as a definition of the weak dissipativity condition) that

$$ (3.19) \qquad \frac{d}{dt} \|S_{\mathcal{B}}(t)f\|_k + \sigma \|S_{\mathcal{B}}(t)f\|_{k-1} \le 0, \quad \forall t \ge 0, \forall f \in X_k, \forall k = 1, 2. $$

We assume that $X_k$ is dense into $X_{k-1}$ for $k = 1, 2$ and that $X_1$ is an interpolated space between $X_0$ and $X_2$ in the sense that there exists a continuous and strictly decreasing function $\eta : (0, 1] \to [0, \infty)$, $\eta(\varepsilon) \to \infty$ when $\varepsilon \to 0$, $\eta(1) = 0$, such that

$$ (3.20) \qquad \|f\|_1 \le \varepsilon \|f\|_2 + \eta(\varepsilon) \|f\|_0, \quad \forall \varepsilon \in (0, 1], \forall f \in X_2. $$

From (3.19) with $k = 2$, we deduce

$$ (3.21) \qquad \|S_{\mathcal{B}}(t)f\|_2 \le \|f\|_2, \quad \forall t \ge 0, \forall f \in X_2. $$

Next, for $k = 1$, gathering the weak dissipativity condition (3.19), the interpolation condition (3.20) and the non expansion property (3.21) in the space $X_2$, we get

$$ \begin{aligned} \frac{d}{dt} \|S_{\mathcal{B}}(t)f\|_1 + \frac{\sigma}{\eta(\varepsilon)} \|S_{\mathcal{B}}(t)f\|_1 &\le \frac{\sigma\varepsilon}{\eta(\varepsilon)} \|S_{\mathcal{B}}(t)f\|_2 \\ &\le \frac{\sigma\varepsilon}{\eta(\varepsilon)} \|f\|_2, \end{aligned} $$

for any $t \ge 0$, $\varepsilon \in (0, 1)$ and $f \in X_2$. We deduce

$$ \frac{d}{dt} \left( \|S_{\mathcal{B}}(t)f\|_1 e^{\frac{\sigma}{\eta(\varepsilon)} t} \right) \le \frac{\sigma\varepsilon}{\eta(\varepsilon)} e^{\frac{\sigma}{\eta(\varepsilon)} t} \|f\|_2, $$

and thanks to the Grönwall lemma, we obtain

$$ (3.22) \qquad \|S_{\mathcal{B}}(t)f\|_1 \le \Theta(t) \|f\|_2, $$
and we establish that $S_B$ is a positive semigroup on $X = M_{\psi_2}^1(E)$. More precisely, for $0 \le f_0 \in D(\mathcal{B})$ in the domain of $S_B$ and denoting $f_t := S_B(t)f_0$, we may compute

$$ \frac{d}{dt} \int f_t \psi_2 \le \int f_t \mathcal{B}^* \psi_2 \le 0 $$

and similarly

$$ \frac{d}{dt} \int f_t \psi_1 \le \int f_t \mathcal{B}^* \psi_1 \le - \int f_t \psi_0. $$

Integrating both differential inequalities, we deduce $S_B \in L_t^\infty(\mathcal{B}(M_{\psi_i}^1))$, $i=1,2$ and

$$ \int_0^\infty \|S_B(t)f_0\|_{M_{\psi_0}^1} dt \le \|f_0\|_{M_{\psi_1}^1}, \quad \forall f_0 \in M_{\psi_1}^1. $$

We may make a slight (but important) improvement of the previous estimate by proceeding similarly as we did for proving (3.23). Using the same notations as in the above computation, we indeed have

$$ \frac{d}{dt} \int f_t \psi_1 + \frac{1}{\eta(\varepsilon)} \int f_t \psi_1 \le \frac{\varepsilon}{\eta(\varepsilon)} \int f_t \psi_2 \le \frac{\varepsilon}{\eta(\varepsilon)} \int f_0 \psi_2, $$

where we have used (i) and (iii) in the first inequality and the previous $L_t^\infty(\mathcal{B}(M_{\psi_2}^1))$ bound in the second inequality. Integrating in time, we deduce

$$ \|S_B(t)f\|_{M_{\psi_1}^1} \le \Theta(t) \|f\|_{M_{\psi_2}^1}, \quad \forall t > 0. $$

Taking $X_i := M_{\psi_i}^1$ and $N=1$, we see that $\mathcal{L}$ then satisfies (HS3) with $\Theta_i = \Theta$.

**3.4. First existence result in the weakly dissipative case.** We first come back to the proof of Theorem 2.21 and explain what goes wrong when we try to adapt it to a weak dissipativity context. More precisely, we assume that $S_\mathcal{L}$ is a positive semigroup (so that (H1) holds) satisfying $\mathcal{L}^*\phi_0 \ge 0$ for some $\phi_0 \in X' \setminus \{0\}$ (so that (H2) holds) and the splitting structure (HS3) for some bounded operator $\mathcal{A}$ and some weakly dissipative operator $\mathcal{B}$, in the sense that (3.19) holds. In such a situation, we may define

$$ \lambda_1 := \inf\{\lambda \in \mathbb{R}; \mathcal{R}_\mathcal{L}(\kappa) \in \mathcal{B}(X), \forall \kappa \ge \lambda\} \ge 0, $$

and there exist sequences $(\lambda_n)$ of $\mathbb{R}$ and $(\hat{f}_n)$ of $X_+$ such that

$$ \lambda_n \searrow \lambda_1 \ge 0, \quad \|\hat{f}_n\| = 1, \quad \varepsilon_n := \lambda_n \hat{f}_n - \mathcal{L} \hat{f}_n \to 0 \text{ in } X, $$

thanks to Lemma 2.6. In the simplest situation, we may further assume that $\mathcal{R}_B(\kappa) : X_1 \to X_0$ is uniformly bounded in $\kappa \ge \lambda_1$ and $\mathcal{A} : X_0 \to X_1$ with $X = X_1 \subset X_0$. The issue is that even in that case, we may write

$$ \hat{f}_n = \mathcal{R}_B(\lambda_n) \mathcal{A} \hat{f}_n + \mathcal{R}_B(\lambda_n) \varepsilon_n, $$

but it is not clear how to conclude that $(\hat{f}_n)$ belongs to a compact set in $X$ because it is not clear that $\mathcal{R}_B(\lambda_n)\varepsilon_n \to 0$ in $X$.

The next result aims to establish that last convergence under suitable quite strong (although natural and true in some examples) assumptions on the operator $\mathcal{L}$. The proof is adapted from [231, Sec. 6.3] and mixes some dynamical argument together with the stationary approach developed in Section 2.2.

**Theorem 3.3.** Consider a positive semigroup $S_\mathcal{L}$ in a Banach lattice $X = X_2 \subset X_1 \subset X_0$ such that its generator $\mathcal{L}$ satisfies

(1) there exists $\phi_0 \in D(\mathcal{L}^*)$, $\phi_0 \ge 0$, $\phi_0 \ne 0$, such that $\mathcal{L}^*\phi_0 \ge 0$;

(2) $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{A}$ and $\mathcal{B}$ satisfying (3.23), (3.25), (3.26) and (3.27).

Then, there exist $\lambda_1 \ge 0$ and $f_1 \in X_1$ such that

(3.33) \qquad $\|f_1\|_{X_1} = 1, \quad f_1 \ge 0, \quad \mathcal{L}f_1 = \lambda_1 f_1.$

**Proof of Theorem 3.3.** We split the proof into four steps.

**Step 1.** We know from Lemma 2.2 and Lemma 2.4-(i) that (H1) and (H2) hold. We may then define $\lambda_1 \ge 0$ with the help of (2.16). If $\lambda_1 > 0$, we see that $\mathcal{V}(\alpha)$ defined in (2.22) is bounded in $\mathcal{B}(X)$ uniformly on $\alpha \ge \kappa_0 := \lambda_1/2$ because of (3.23) and (3.25), and that $\mathcal{W}(\alpha)$ also defined in
for any $t \ge 0$ and $f \in X_2$, with

$$ (3.23) \qquad \Theta(t) := \inf_{\varepsilon \in (0,1)} \left(e^{-\frac{\sigma}{n(\varepsilon)}t} + \varepsilon\right) \to 0 \quad \text{as} \quad t \to +\infty. $$

On the other hand, using the representation formula

$$ \mathcal{R}_B(z)f = \int_0^\infty e^{-zt} S_B(t) f dt, \quad \forall z \in \Delta_0, \forall f \in X_2, $$

together with (3.19), we get

$$ \sigma \|\mathcal{R}_B(z)f\|_1 \le \int_0^\infty \sigma \|S_B(t)f\|_1 dt \le \|f\|_2, $$

for any $z \in \bar{\Delta}_0$ and $f \in X_2$. We next assume that

$$ (3.24) \qquad \Theta(t)^{-1} \|AS_B(t)f\|_1 + \int_0^\infty \|AS_B(t)f\|_1 dt \lesssim \|f\|_1, $$

that there exist $\alpha > 1$, $N \ge 1$, $C \ge 1$ such that

$$ (3.25) \qquad \sup_{x+iy \in \Delta_0} \|AR_B^{1+\varepsilon_1}(x+iy) \dots AR_B^{1+\varepsilon_N}(x+iy)f\|_2 \le \frac{C}{\langle y \rangle^\alpha} \|f\|_2, $$

for any $\varepsilon \in \{0, 1\}^N$, $\varepsilon_1 + \dots + \varepsilon_N \le 1$, and that

$$ (3.26) \qquad \sup_{z \in \Delta_0} \|(\mathcal{R}_B(z)\mathcal{A})^N f\|_{X_1} \le \|f\|_1, $$

with $X_1$ compactly imbedded in $X_1$. The necessity to add $(\varepsilon_i)$ in (3.25) is probably purely technical and not restrictive for applications. In examples, we can take $N = 2N'$, when

$$ (3.27) \qquad \sup_{x+iy \in \Delta_0} \|(AR_B)^{N'}(x+iy)f\|_3 \le \frac{C}{\langle y \rangle^\alpha} \|f\|_2, $$

for some convenient space $X_3$ such that $\mathcal{A}: X_1 \to X_3$ and $\sup_{z \in \Delta_0} \|\mathcal{R}_B(z)\|_{\mathcal{B}(X_3, X_2)} < \infty$. At the level of the semigroup, (3.27) is typically a consequence of

$$ \|(\mathcal{A}S_B)^{(N'')}(t)\|_{\mathcal{B}(X_2, X_3^\zeta)} \in L^1(\mathbb{R}_+), $$

with $\zeta > 0$, where $X_3^\zeta := \{f \in X_3, \mathcal{L}^\zeta f \in X_3\}$ stands for the (possibly fractional) domain for the operator defined in $X_3$. However, (3.25) is a bit more general than that last estimate. We refer to [277, 272, 279, 276] for precise definition, examples and discussion. For further references, we observe that (3.22) and (3.24) together imply

$$ \begin{align*} \frac{1}{T} \int_0^T \|(S_B * \mathcal{A}S_B)(t)f\|_1 dt &\le \frac{1}{T} \int_0^T \int_0^t \|S_B(t-s)\mathcal{A}S_B(s)f\|_1 dsdt \\ &\le \frac{1}{T} \int_0^T \int_0^T \|S_B(u)\|_{\mathcal{B}(X_2, X_1)} \|\mathcal{A}S_B(s)f\|_1 duds \\ &\lesssim \frac{1}{T} \int_0^T \Theta(u) du \|f\|_2. \end{align*} $$

Arguing in a similar way for any $\ell \ge 1$, we establish

$$ (3.28) \qquad \frac{1}{T} \int_0^T \|(S_B * (\mathcal{A}S_B)^{(*\ell}))(t)f\|_1 dt \lesssim \frac{1}{T} \int_0^T \Theta(u) du \|f\|_2 \to 0 \quad \text{as} \quad T \to \infty. $$

For synthesizing and for further references, let us now bring out some possible general framework for semigroup enjoying weak dissipativity. We introduce the following structure condition on a semigroup $S_\mathcal{L}$ and its generator $\mathcal{L}$ by assuming

(HS3) there exist a splitting $\mathcal{L} = \mathcal{A} + \mathcal{B}$, some Banach lattices $X_2 \subset X_1$, an integer $N \ge 1$ 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 $\mathcal{A}$ is positive, $\mathcal{B}$ generates a positive semigroup $S_B$ and the following estimates hold

$$ (3.29) \qquad \|(S_B \mathcal{A})^{(*\ell)} * S_B \|_{\mathcal{B}(X_2, X_1)} = \mathcal{O}(\Theta_1), \quad \forall \ell \in \{0, \dots, N-1\}, $$

$$ (3.30) \qquad \|(S_B \mathcal{A})^{(*N)}\|_{\mathcal{B}(X_1, X_2)} = \mathcal{O}(\Theta_2). $$
(2.22) satisfies (2.23) because of (3.25) and Remark 2.9-(1). Using Lemma 2.8, we get that (H3) holds, and we conclude thanks to Theorem 2.21 in that case.

In the sequel, we always assume $\lambda_1 = 0$.

Step 2. Let us fix $f_0 \in D(\mathcal{L})$ such that $f_0 \ge 0$ and $C_0 := \langle f_0, \phi_0 \rangle > 0$, which exists by definition of $\phi_0$. Denoting $f(t) := S_{\mathcal{L}}(t)f_0$, we have

$$ \frac{d}{dt} \langle f(t), \phi_0 \rangle = \langle \mathcal{L} f(t), \phi_0 \rangle = \langle f(t), \mathcal{L}^* \phi_0 \rangle \ge 0, $$

which in turns implies

$$ \langle f(t), \phi_0 \rangle \ge C_0, \quad \forall t \ge 0. $$

Step 3. We claim that $\|\mathcal{R}_{\mathcal{L}}(0)\|_{\mathcal{B}(X_2, X_1)} = +\infty$. That in particular implies $\|\mathcal{R}_{\mathcal{L}}(0)\|_{\mathcal{B}(X)} = +\infty$ and thus $0 \in \Sigma(\mathcal{L})$. We assume by contradiction that $K_{2,1} := \|\mathcal{R}_{\mathcal{L}}(0)\|_{\mathcal{B}(X_2, X_1)} < +\infty$. First, because $S_{\mathcal{L}}$ is positive, we have

$$ |\mathcal{R}_{\mathcal{L}}(z)f| \le \int_0^{\infty} e^{-t\Re z} S_{\mathcal{L}}(t)|f| dt = |\mathcal{R}_{\mathcal{L}}(\Re z)|f|, $$

from which we deduce

$$ \|\mathcal{R}_{\mathcal{L}}(z)\|_{\mathcal{B}(X_2, X_1)} \le \|\mathcal{R}_{\mathcal{L}}(\Re z)\|_{\mathcal{B}(X_2, X_1)}, \quad \forall z \in \Delta_0. $$

As a consequence, we have

$$ (3.34) \qquad \sup_{y \in \mathbb{R}} \|\mathcal{R}_{\mathcal{L}}(iy)\|_{\mathcal{B}(X_2, X_1)} \le K_{2,1}. $$

We write the representation formulas (taken from [278, (2.21)])

$$ S_{\mathcal{L}}(t)f = \mathcal{T}_0(t) + \lim_{M \to \infty} \mathcal{T}_{1,M}(t) $$

with

$$ \mathcal{T}_0(t) := \sum_{\ell=0}^{N-1} S_B * (A S_B)^{(*\ell)}(t) f $$

and

$$ \mathcal{T}_{1,M}(t) := \frac{i}{2\pi} \int_{a-iM}^{a+iM} e^{zt} \mathcal{R}_{\mathcal{L}}(z) (\mathcal{A}\mathcal{R}_{\mathcal{B}}(z))^N f dz, $$

for any $f \in D(\mathcal{L})$, $t \ge 0$ and $a > 0$. On the one hand, from (3.29), we have the Cesàro mean convergence

$$ (3.35) \qquad \frac{1}{T} \int_0^T \mathcal{T}_0(t) dt \to 0 \text{ in } X_1, \text{ as } T \to \infty. $$

On the other hand, we estimate the contribution of $\mathcal{T}_{1,M}$. Integrating by part, we have

$$ \mathcal{T}_{1,M}(t) = \frac{i}{2\pi} \int_{a-iM}^{a+iM} e^{zt} \frac{d}{dz} [\mathcal{R}_{\mathcal{L}}(z) (\mathcal{A}\mathcal{R}_{\mathcal{B}}(z))^N] f dz, $$

with

$$ \frac{d}{dz} [\mathcal{R}_{\mathcal{L}}(z) (\mathcal{A}\mathcal{R}_{\mathcal{B}}(z))^N] = \sum_{\substack{\epsilon \in \mathbb{N}^{N+1}, \\ |\epsilon| = 1}} \mathcal{R}_{\mathcal{L}}(z)^{1+\epsilon_0} \mathcal{A}\mathcal{R}_{\mathcal{B}}^{1+\epsilon_1}(z) \dots \mathcal{A}\mathcal{R}_{\mathcal{B}}^{1+\epsilon_N}(z). $$

Together with condition (3.26) and estimate (3.34), we get

$$ \begin{aligned} & \left\| \frac{d}{dz} [\mathcal{R}_{\mathcal{L}}(z) (\mathcal{A}\mathcal{R}_{\mathcal{B}}(z))^N] f \right\|_1 \\ & \le (K_{2,1} + K_{2,1}^2) N \sup_{\epsilon \in \mathbb{N}^N, |\epsilon| \le 1} \|\mathcal{A}\mathcal{R}_{\mathcal{B}}^{1+\epsilon_1}(z) \dots \mathcal{A}\mathcal{R}_{\mathcal{B}}^{1+\epsilon_N}(z) f\|_2 \\ & \le \frac{C_1}{\langle y \rangle^\alpha} \|f\|_2, \end{aligned} $$

uniformly for any $z = x + iy \in \Delta_0$, for some constant $C_1 > 0$. We deduce

$$ (3.36) \qquad \| \lim_{M \to \infty} \mathcal{T}_{1,M}(t) \|_1 \le \frac{1}{t} \frac{1}{2\pi} \int_{\mathbb{R}} \frac{C_1}{\langle y \rangle^\alpha} dy \|f\|_2 \to 0, $$
We now particularize our discussion to a Radon measures framework. We assume that there exist some weight functions $\psi_i$ on $E$, $\psi_0 \lesssim \psi_1 \le \psi_2$, with $\psi_2(x)/\psi_1(x) \to \infty$ as $x \to \infty$ so that $M_{\psi_2}^1 \subset M_{\psi_1}^1$ (compact imbedding for the weak convergence), a function $\chi \in C_c(E)$, $0 \le \chi \le 1$, and a constant $M \ge 0$ such that

(i) $\mathcal{L}^*\psi_1 \le -\psi_0 + M\chi$;

(ii) $\mathcal{L}^*\psi_2 \le M\chi$;

(iii) $\psi_1 \le \varepsilon\psi_2 + \eta(\varepsilon)\psi_0$ for any $\varepsilon > 0$,

for a function $\eta: (0, 1] \to (0, \infty)$ such that $\eta(1) = 0$, $\eta(\varepsilon) \to \infty$ when $\varepsilon \to 0$, and

$$ (3.31) \qquad t \mapsto \Theta(t) := \inf_{\varepsilon \in (0,1)} \left( e^{-\frac{t}{\eta(\varepsilon)}} + \varepsilon \right) \in L^1(0, \infty). $$

It is worth emphasizing that from the very definition, we have automatically that $\Theta$ is positive and decreasing, $\Theta(0) = 1$ and $\Theta(t) \to 0$ as $t \to \infty$. Arguing similarly as we did during the proof of Corollary 2.20 and the end of Section 3.1, we introduce the splitting

$$ \mathcal{A} := M\chi, \quad \mathcal{B} := \mathcal{L} - \mathcal{A}, $$

and we establish that $S_B$ is a positive semigroup on $X = M_{\psi_2}^1(E)$. More precisely, for $0 \le f_0 \in D(\mathcal{B})$ in the domain of $S_B$ and denoting $f_t := S_B(t)f_0$, we may compute

$$ \frac{d}{dt} \int f_t \psi_2 \le \int f_t \mathcal{B}^* \psi_2 \le 0 $$

and similarly

$$ \frac{d}{dt} \int f_t \psi_1 \le \int f_t \mathcal{B}^* \psi_1 \le - \int f_t \psi_0. $$

Integrating both differential inequalities, we deduce $S_B \in L_t^\infty(\mathcal{B}(M_{\psi_i}^1))$, $i = 1, 2$ and

$$ \int_0^\infty \|S_B(t)f_0\|_{M_{\psi_0}^1} dt \le \|f_0\|_{M_{\psi_1}^1}, \quad \forall f_0 \in M_{\psi_1}^1. $$

We may make a slight (but important) improvement of the previous estimate by proceeding similarly as we did for proving (3.22). Using the same notations as in the above computation, we indeed have

$$ \frac{d}{dt} \int f_t \psi_1 + \frac{1}{\eta(\varepsilon)} \int f_t \psi_1 \le \frac{\varepsilon}{\eta(\varepsilon)} \int f_t \psi_2 \le \frac{\varepsilon}{\eta(\varepsilon)} \int f_0 \psi_2, $$

where we have used (i) and (iii) in the first inequality and the previous $L_t^\infty(\mathcal{B}(M_{\psi_2}^1))$ bound in the second inequality. Integrating in time, we deduce

$$ \|S_B(t)f\|_{M_{\psi_1}^1} \le \Theta(t) \|f\|_{M_{\psi_2}^1}, \quad \forall t > 0. $$

Taking $X_i := M_{\psi_i}^1$ and $N = 1$, we see that $\mathcal{L}$ then satisfies (HS3) with $\Theta_i = \Theta$.

### 3.4. First existence result in the weakly dissipative case.
We first come back to the proof of Theorem 2.21 and explain what goes wrong when we try to adapt it to a weak dissipativity context. More precisely, we assume that $S_\mathcal{L}$ is a positive semigroup (so that (H1) holds) satisfying $\mathcal{L}^*\phi_0 \ge 0$ for some $\phi_0 \in X' \setminus \{0\}$ (so that (H2) holds) and the splitting structure (HS3) for some bounded operator $\mathcal{A}$ and some weakly dissipative operator $\mathcal{B}$, in the sense that (3.18) holds. In such a situation, we may define

$$ \lambda_1 := \inf\{\lambda \in \mathbb{R}; \mathcal{R}_\mathcal{L}(\kappa) \in \mathcal{B}(X), \forall \kappa \ge \lambda\} \ge 0, $$

and there exist sequences $(\lambda_n)$ of $\mathbb{R}$ and $(\hat{f}_n)$ of $X_+$ such that

$$ \lambda_n \searrow \lambda_1 \ge 0, \quad \|\hat{f}_n\| = 1, \quad \varepsilon_n := \lambda_n \hat{f}_n - \mathcal{L} \hat{f}_n \to 0 \text{ in } X, $$

thanks to Lemma 2.6. In the simplest situation, we may further assume that $\mathcal{R}_B(\kappa): X_1 \to X_0$ is uniformly bounded in $\kappa \ge \lambda_1$ and $\mathcal{A}: X_0 \to X_1$ with $X = X_1 \subset X_0$. The issue is that even in that case, we may write

$$ \hat{f}_n = \mathcal{R}_B(\lambda_n)\mathcal{A}\hat{f}_n + \mathcal{R}_B(\lambda_n)\varepsilon_n, $$

but it is not clear how to conclude that $(\hat{f}_n)$ belongs to a compact set in $X$ because it is not clear that $\mathcal{R}_B(\lambda_n)\varepsilon_n \to 0$ in $X$.
as $t \to \infty$. Gathering (3.35) and (3.36), we conclude in particular that

$$ \frac{1}{T} \int_{0}^{T} S_{\mathcal{L}}(t) f_{0} dt \to 0 \text{ in } X_{1}, \text{ as } T \to \infty, $$

which is in contradiction with the estimate of Step 2.

**Step 4. Conclusion.** Taking advantage of the convenient blow up of $\mathcal{R}_{\mathcal{L}}(\lambda)$ as $\lambda \searrow 0$ established in the previous Step 3, we may now argue similarly as in the proof of Theorem 2.21. From Step 2, there exists a sequence $(\lambda_n)$ such that $\lambda_n \to 0$ and

$$ \|\mathcal{R}_{\mathcal{L}}(\lambda_n)\|_{\mathcal{B}(X_2, X_1)} \to \infty. $$

That means that there exist $(f_n)$ and $(g_n)$ such that

$$ \|f_n\|_{X_1} \to \infty, \quad \|g_n\|_{X_2} = 1, \quad f_n = \mathcal{R}_{\mathcal{L}}(\lambda_n) g_n, $$

or equivalently that there exist $(\hat{f}_n)$ and $(\varepsilon_n)$ (by defining $\hat{f}_n := f_{n\pm}/\|f_{n\pm}\|_{X_1}$, $\varepsilon_n := g_{n\pm}/\|f_{n\pm}\|_{X_1}$) satisfying

$$ (3.37) \qquad \|\hat{f}_n\|_{X_1} = 1, \quad \hat{f}_n \ge 0, \quad \|\varepsilon_n\|_{X_2} \to 0, \quad \varepsilon_n = (\lambda_n - \mathcal{L}) \hat{f}_n. $$

As in the proof of Lemma 2.8, we deduce that (2.25) holds, that is

$$ (3.38) \qquad \hat{f}_n = \sum_{\ell=0}^{N-1} (\mathcal{R}_B(\lambda_n)A)^\ell \mathcal{R}_B(\lambda_n)\varepsilon_n + (\mathcal{R}_B(\lambda_n)A)^N \hat{f}_n. $$

Using the uniform boundedness

$$ (\mathcal{R}_B(\lambda_n)A)^\ell \mathcal{R}_B(\lambda_n) \in \mathcal{B}(X_2, X_1), \quad (\mathcal{R}_B(\lambda_n)A)^N \in \mathcal{B}(X_1, X_1), \quad X_1 \subset \subset X_1, $$

we deduce that $(\hat{f}_n)$ belongs to a compact set of $X_1$, or in other words, that there exist a subsequence of $(\hat{f}_n)$ (not relabeled) and $f_1 \in X_1$ such that $\hat{f}_n \to f_1$ in $X_1$. We may pass to the limit in (3.37), and we get (3.33). $\square$

**3.5. Second existence result in the weakly dissipative case.** Using a pure dynamical approach adapted from the second proof of Theorem 3.1 and from [81, Thm. 6.1], we establish a second existence result which is less demanding in terms of conditions on the semigroup $S_{\mathcal{L}}$.

**Theorem 3.4.** Consider a positive semigroup $S = S_{\mathcal{L}}$ on a Banach lattice $X = Y'$ for a separable Banach lattice $Y$. We assume

(i) there exists $\phi_0 \in Y_+ \setminus \{0\}$ such that $[S_t f]_0 \ge [f]_0$ for any $f \in X_+$ and $f \mapsto [f]_0 := \langle |f|, \phi_0 \rangle$ is a norm on $X$. We then denotes $\mathcal{X}$ the vector space $X$ endowed with this norm $[\cdot]_0$;

(ii) there exist $v \in L^\infty(\mathbb{R}_+; \mathcal{B}(X))$ and $0 \le w \in L^1(\mathbb{R}_+; \mathcal{B}(X, X))$ such that

$$ (3.39) \qquad S = v + w * S, $$

and we set

$$ (3.40) \qquad M := \sup_{t \ge 0} \|v(t)\|_{\mathcal{B}(X)} < \infty, \quad \Theta(t) := \|w(t)\|_{\mathcal{B}(X, X)} \in L^1(\mathbb{R}_+). $$

Then there exists a pair $(\lambda_1, f_1) \in \mathbb{R}_+ \times X_+ \setminus \{0\}$ such that $\mathcal{L}f_1 = \lambda_1 f_1$.

**Remark 3.5.** (1) When $S_{\mathcal{L}}$ satisfies (HS3) then (3.39) holds with

$$ (3.41) \qquad v := \sum_{\ell=0}^{N-1} S_B * (AS_B)^{(*\ell)}, \quad w := (S_B A)^{(*N)}. $$

(2) By definition of the norm $[\cdot]_0$ of $\mathcal{X}$, we see that $\mathcal{X}$ is a weighted $L^1$ space or a weighted Radon measures space. In many applications, when both $\mathcal{X}$ and $X$ are Radon measures spaces, one can choose $N=1$. On the other hand, when $X$ is for instance a (possibly weighted) $L^p$ space with $p > 1$, one must take $N \ge 2$ in most of the applications. In condition (ii), the first bound is not really demanding and almost automatic in view of the estimates exhibited in Section 3.3. The second bound is a kind of regularity estimate reminiscent of the enlarging and shrinkage technique developed in [294, 190, 274].
The next result aim precisely to establish that last convergence under suitable quite strong (al-
though natural and true in some examples) assumptions on the operator $\mathcal{L}$. The proof is adapted
from [230, Sec. 6.3] and mixes some dynamical argument together with the stationary approach
developed in Section 2.2.

**Theorem 3.3.** Consider a positive semigroup $S_{\mathcal{L}}$ in a Banach lattice $X = X_2 \subset X_1 \subset X_0$ such that its generator $\mathcal{L}$ satisfies

(1) there exists $\phi_0 \in D(\mathcal{L}^*)$, $\phi_0 \ge 0$, $\phi_0 \ne 0$, such that $\mathcal{L}^*\phi_0 \ge 0$;

(2) $\mathcal{L} = \mathcal{A} + \mathcal{B}$ with $\mathcal{A}$ and $\mathcal{B}$ satisfying (3.22), (3.24), (3.25) and (3.26).
Then, there exist $\lambda_1 \ge 0$ and $f_1 \in X_1$ such that

$$
(3.32) \qquad \|f_1\|_{X_1} = 1, \quad f_1 \ge 0, \quad \mathcal{L}f_1 = \lambda_1 f_1.
$$

*Proof of Theorem 3.3.* We split the proof into four steps.

Step 1. We know from Lemma 2.2 and Lemma 2.4-(i) that (H1) and (H2) hold. We may then define $\lambda_1 \ge 0$ with the help of (2.16). If $\lambda_1 > 0$, we see that $\mathcal{V}(\alpha)$ defined in (2.21) is bounded in $\mathcal{B}(X)$ uniformly on $\alpha \ge \kappa_0 := \lambda_1/2$ because of (3.22) and (3.24), and that $\mathcal{W}(\alpha)$ also defined in (2.21) satisfies (2.22) because of (3.24) and Remark 2.9-(1). Using Lemma 2.8, we get that (H3) holds, and we conclude thanks to Theorem 2.21 in that case.

In the sequel, we always assume $\lambda_1 = 0$.

Step 2. Let us fix $f_0 \in D(\mathcal{L})$ such that $f_0 \ge 0$ and $C_0 := \langle f_0, \phi_0 \rangle > 0$, which exists by definition of $\phi_0$. Denoting $f(t) := S_{\mathcal{L}}(t)f_0$, we have

$$
\frac{d}{dt} \langle f(t), \phi_0 \rangle = \langle \mathcal{L} f(t), \phi_0 \rangle = \langle f(t), \mathcal{L}^* \phi_0 \rangle \ge 0,
$$
which in turns implies

$$
\langle f(t), \phi_0 \rangle \geq C_0, \quad \forall t \geq 0.
$$

Step 3. We claim that $\|\mathcal{R}_\mathcal{L}(0)\|_{\mathcal{B}(X_2, X_1)} = +\infty$. That in particular implies $\|\mathcal{R}_\mathcal{L}(0)\|_{\mathcal{B}(X)} = +\infty$ and thus $0 \in \Sigma(\mathcal{L})$. We assume by contradiction that $\mathcal{K}_{2,1} := \|\mathcal{R}_\mathcal{L}(0)\|_{\mathcal{B}(X_2, X_1)} < +\infty$. First, because $S_\mathcal{L}$ is positive, we have

$$
|\mathcal{R}_\mathcal{L}(z)f| \le \int_0^\infty e^{-t\Re z} S_\mathcal{L}(t)|f| dt = |\mathcal{R}_\mathcal{L}(\Re z)|f|,
$$
from which we deduce

$$
\| \mathcal{R}_{\mathcal{L}}(z) \|_{\mathcal{B}(X_2, X_1)} \le \| \mathcal{R}_{\mathcal{L}}(\mathcal{R}ez) \|_{\mathcal{B}(X_2, X_1)}, \quad \forall z \in \Delta_0.
$$

As a consequence, we have

$$
(3.33) \qquad \sup_{y \in \mathbb{R}} \| \mathcal{R}_{\mathcal{L}}(iy) \|_{\mathcal{B}(X_2, X_1)} \le \mathcal{K}_{2,1}.
$$

We write the representation formulas (taken from [277, (2.21)])

$$
S_{\mathcal{L}}(t) f = \mathcal{T}_0(t) + \lim_{M \to \infty} \mathcal{T}_{1,M}(t)
$$
with

$$
\mathcal{T}_0(t) := \sum_{\ell=0}^{N-1} S_B * (\mathcal{A}S_B)^{(*\ell)}(t)f
$$
and

$$
\mathcal{T}_{1,M}(t) := \frac{i}{2\pi} \int_{a-iM}^{a+iM} e^{zt} \mathcal{R}_{\mathcal{L}}(z) (\mathcal{A}\mathcal{R}_{\mathcal{B}}(z))^N f \, dz,
$$

for any $f \in D(\mathcal{L})$, $t \ge 0$ and $a > 0$. On the one hand, from (3.28), we have the Cesàro mean
convergence

$$
(3.34) \qquad \frac{1}{T} \int_{0}^{T} \mathcal{T}_{0}(t) \, dt \to 0 \text{ in } X_{1}, \text{ as } T \to \infty.
$$

On the other hand, we estimate the contribution of $\mathcal{T}_{1,M}$. Integrating by part, we have

$$
\mathcal{T}_{1,M}(t) = \frac{1}{t} \frac{i}{2\pi} \int_{a-iM}^{a+iM} e^{zt} \frac{d}{dz} [\mathcal{R}_{\mathcal{L}}(z) (\mathcal{A}\mathcal{R}_{\mathcal{B}}(z))^N] f \, dz,
$$
Proof of Theorem 3.4. We split the proof into three steps.

Step 1. We define

$$R := \max(2\|\Theta\|_{L^1}, \|g_0\|),$$

for some $g_0 \in X_+$ such that $[g_0]_0 = 1$, and next the nonempty convex and compact (in the weak $\ast \sigma(X, Y)$ sense) set

$$C := \{f \in X_+; [f]_0 = 1, \|f\| \le R\},$$

as well as the increasing function

$$\lambda(t) := \inf_{f \in C} [S(t)f]_0, \quad \forall t \ge 0.$$

We have the alternative

• (1) $\sup \lambda > 2M$,

• (2) $\sup \lambda \le 2M$.

Step 2. We assume that the first term (1) of the alternative holds true, or in other words, there exists $T_0 > 0$ such that

$$(3.42) \qquad \forall f \in C, \quad [S_{T_0}f]_0 \ge 2M.$$

We define as before

$$\Phi_{T_0}f := \frac{S_{T_0}f}{[S_{T_0}f]_0}, \quad \forall f \in C.$$

By construction, for any $f \in C$, we have $\Phi_{T_0}f \ge 0$ and $[\Phi_{T_0}f]_0 = 1$. On the other hand, using the splitting structure (3.39) and the estimates (3.40), we have

$$\|S(t)f\| \le M\|f\| + \int_0^t \Theta(t-s)[S(s)f]_0 ds.$$

From hypothesis (i) and the semigroup property, we also have

$$[S_t f]_0 \ge [S_s f]_0, \quad \forall t \ge s \ge 0.$$

The two above estimates together imply

$$\begin{aligned} \|\Phi_{T_0}f\| & \le \frac{M\|f\|}{[S_{T_0}f]_0} + \int_0^{T_0} \Theta(T_0-s) \frac{[S_s f]_0}{[S_{T_0}f]_0} ds \\ & \le \frac{1}{2}\|f\| + \|\Theta\|_{L^1} \le R, \end{aligned}$$

for any $f \in C$. We have thus proved $\Phi_{T_0} : C \to C$. Thanks to the Tykonov fixed point Theorem, there exists $f_{T_0} \in C$ such that $\Phi_{T_0}f_{T_0} = f_{T_0}$. In other words, we have built a pair of “almost eigenvalue and eigenfunction”

$$f_{T_0} \ge 0, \quad [f_{T_0}]_0 = 1, \quad S_{T_0}f_{T_0} = e^{\lambda_1 T_0} f_{T_0},$$

with $e^{\lambda_1 T_0} = [S_{T_0}f]_0$ and thus $\lambda_1 \in [0, \kappa_1]$. We conclude to the existence of $f_1 \in C$ such that $\mathcal{L}f_1 = \lambda_1 f_1$ really similarly as in Step 2 of the Second proof of Theorem 3.1.

Step 3. We assume that the second term (2) of the alternative holds true. In that case, for any $n \ge 1$, there exists $f_n \in C$ such that $[S(n)f_n]_0 \le 2M$. By compactness, there exists $f_0 \in C$ and a subsequence $(f_{n_k})$ such that $f_{n_k} \to f_0 \in C$ and

$$\forall t \ge 0, \forall k (n_k \ge t), \quad [S(t)f_{n_k}]_0 \le [S(n_k)f_{n_k}]_0 \le 2M,$$

so that

$$(3.43) \qquad \forall t \ge 0, \quad [S(t)f_0]_0 \le 2M.$$

Using this particular initial datum, we argue similarly as in [81, proof of Theorem 6.1], and we conclude to the existence of a stationary state. More precisely, we come back to the splitting structure (3.39) of the semigroup $S$ and we introduce the associated Cesàro means

$$(3.44) \qquad U_T := \frac{1}{T} \int_0^T S(t) dt, \quad V_T := \frac{1}{T} \int_0^T v(t) dt, \quad K_T := \frac{1}{T} \int_0^T (w * S)(t) dt,$$
with

$$ \frac{d}{dz}[\mathcal{R}_L(z) (\mathcal{A}\mathcal{R}_B(z))^N] = \sum_{\substack{\varepsilon \in \mathbb{N}^{N+1}, \\ |\varepsilon|=1}} \mathcal{R}_L(z)^{1+\varepsilon_0} \mathcal{A}\mathcal{R}_B^{1+\varepsilon_1}(z) \dots \mathcal{A}\mathcal{R}_B^{1+\varepsilon_N}(z). $$

Together with condition (3.25) and estimate (3.33), we get

$$ \begin{aligned} & \left\| \frac{d}{dz} [\mathcal{R}_L(z) (\mathcal{A}\mathcal{R}_B(z))^N] f \right\|_1 \\ & \le (K_{2,1} + K_{2,1}^2) N \sup_{\varepsilon \in \mathbb{N}^N, |\varepsilon| \le 1} \| \mathcal{A}\mathcal{R}_B^{1+\varepsilon_1}(z) \dots \mathcal{A}\mathcal{R}_B^{1+\varepsilon_N}(z) f \|_2 \\ & \le \frac{C_1}{\langle y \rangle^\alpha} \|f\|_2, \end{aligned} $$

uniformly for any $z = x + iy \in \Delta_0$, for some constant $C_1 > 0$. We deduce

$$ (3.35) \qquad \left\| \lim_{M \to \infty} \mathcal{T}_{1,M}(t) \right\|_1 \le \frac{1}{t} \frac{1}{2\pi} \int_{\mathbb{R}} \frac{C_1}{\langle y \rangle^\alpha} dy \|f\|_2 \to 0, $$

as $t \to \infty$. Gathering (3.34) and (3.35), we conclude in particular that

$$ \frac{1}{T} \int_{0}^{T} S_{\mathcal{L}}(t) f_{0} dt \to 0 \text{ in } X_{1}, \text{ as } T \to \infty, $$

which is in contradiction with the estimate of Step 2.

**Step 4. Conclusion.** Taking advantage of the convenient blow up of $\mathcal{R}_L(\lambda)$ as $\lambda \searrow 0$ established in the previous step, we may now argue similarly as in the proof of Theorem 2.21. More precisely, from Step 3, there exists a sequence $(\lambda_n)$ such that $\lambda_n \to 0$ and

$$ \|\mathcal{R}_L(\lambda_n)\|_{\mathcal{B}(X_2, X_1)} \to \infty. $$

That means that there exist $(f_n)$ and $(g_n)$ such that

$$ \|f_n\|_{X_1} \to \infty, \quad \|g_n\|_{X_2} = 1, \quad f_n = \mathcal{R}_L(\lambda_n) g_n, $$

or equivalently that there exist $(\hat{f}_n)$ and $(\varepsilon_n)$ (by defining $\hat{f}_n := f_{n\pm}/\|f_{n\pm}\|_{X_1}$, $\varepsilon_n := g_{n\pm}/\|f_{n\pm}\|_{X_1}$) satisfying

$$ (3.36) \qquad \|\hat{f}_n\|_{X_1} = 1, \quad \hat{f}_n \ge 0, \quad \|\varepsilon_n\|_{X_2} \to 0, \quad \varepsilon_n = (\lambda_n - \mathcal{L}) \hat{f}_n. $$

As in the proof of Lemma 2.8, we deduce that (2.24) holds, that is

$$ (3.37) \qquad \hat{f}_n = \sum_{\ell=0}^{N-1} (\mathcal{R}_B(\lambda_n)\mathcal{A})^\ell \mathcal{R}_B(\lambda_n)\varepsilon_n + (\mathcal{R}_B(\lambda_n)\mathcal{A})^N \hat{f}_n. $$

Using the uniform boundedness

$$ (\mathcal{R}_B(\lambda_n)\mathcal{A})^\ell \mathcal{R}_B(\lambda_n) \in \mathcal{B}(X_2, X_1), \quad (\mathcal{R}_B(\lambda_n)\mathcal{A})^N \in \mathcal{B}(X_1, X_1), \quad X_1 \subset \subset X_1, $$

we deduce that $(\hat{f}_n)$ belongs to a compact set of $X_1$, or in other words, that there exist a subsequence of $(\hat{f}_n)$ (not relabeled) and $f_1 \in X_1$ such that $\hat{f}_n \to f_1$ in $X_1$. We may pass to the limit in (3.36), and we get (3.32). $\square$

**3.5. Second existence result in the weakly dissipative case.** Using a pure dynamical approach adapted from the second proof of Theorem 3.1 and from [80, Thm. 6.1], we establish a second existence result which is less demanding in terms of conditions on the semigroup $S_L$.

**Theorem 3.4.** Consider a positive semigroup $S = S_L$ on a Banach lattice $X = Y'$ for a separable Banach lattice $Y$. We assume

(i) there exists $\phi_0 \in Y_+$ such that $[S_t f]_{\phi_0} \ge [f]_{\phi_0}$ for any $f \in X_+$ and such that $f \mapsto [f]_{\phi_0}$ is a norm on $X$. We then denote $\mathcal{X}_0$ the vector space $X$ endowed with the norm $[\cdot]_{\phi_0}$;

(ii) there exist $v \in L^\infty(\mathbb{R}_+; \mathcal{B}(X))$ and $0 \le w \in L^1(\mathbb{R}_+; \mathcal{B}(\mathcal{X}_0, X))$ such that

$$ (3.38) \qquad S = v + w * S, $$

and we set

$$ (3.39) \qquad M := \sup_{t \ge 0} \|v(t)\|_{\mathcal{B}(X)} < \infty, \quad \Theta(t) := \|w(t)\|_{\mathcal{B}(\mathcal{X}_0, X)} \in L^1(\mathbb{R}_+). $$
for any $T > 0$. We obviously have

$$ \|V_T\|_{\mathcal{B}(X)} \le \frac{1}{T} \int_0^T \|v(t)\|_{\mathcal{B}(X)} dt \le M. $$

On the other hand, we have

$$ \int_0^T (w * S)(t) dt = \int_0^T \int_s^T w(t-s) dt S(s) ds \le \int_0^T w(\tau) d\tau \int_0^T S(s) ds, $$

thanks to the Fubini theorem and the positivity of the two operators involved in this integral formula. We deduce

$$ \begin{aligned} \|K_T f_0\| & \le \left\| \int_0^T w(\tau) d\tau \frac{1}{T} \int_0^T S(s) f_0 ds \right\| \\ & \le \int_0^\infty \|w(\tau)\|_{\mathcal{B}(X,X)} d\tau \left[ \frac{1}{T} \int_0^T S(s) f_0 ds \right]_0 = \|\Theta\|_{L^1} [U_T f_0]_0, \end{aligned} $$

thanks to assumption (ii), so that $K_T f_0$ is uniformly bounded in $X$ thanks to (3.43) and the elementary estimate $[U_T f_0]_0 \le [S_T f_0]_0$. We then deduce that $U_T = V_T + K_T$ satisfies

$$ \|U_T f_0\| \le M \|f_0\| + 2M \|\Theta\|_{L^1} \quad \text{and} \quad 1 \le [S_T f_0]_0 \le 2M, $$

for any $T > 0$. By compactness, there exists $T_k \to +\infty$ and $f_1 \in X_+$ such that $U_{T_k} f \to f_1$ weakly * in $X$. Thanks to the second inequality, we have $[f_1]_0 \ge 1$. We then argue thanks to the usual mean ergodic theorem trick. For any fixed $s > 0$, we observe that

$$ \begin{aligned} S(s)f_1 - f_1 &= \lim_{k \to \infty} \left\{ \frac{1}{T_k} \int_0^{T_k} S(s)S(t)f_0 dt - \frac{1}{T_k} \int_0^{T_k} S(t)f_0 dt \right\} \\ &= \lim_{k \to \infty} \left\{ \frac{1}{T_k} \int_{T_k}^{T_k+s} S(t)f_0 dt - \frac{1}{T_k} \int_0^s S(t)f_0 dt \right\} \end{aligned} $$

weakly * in $X$. By the lower semicontinuous property of the norm $[\cdot]_0$, we deduce

$$ [S(s)f_1 - f_1]_0 \le \liminf_{k \to \infty} \left\{ \frac{1}{T_k} \int_{T_k}^{T_k+s} [S(t)f_0]_0 dt + \frac{1}{T_k} \int_0^s [S(t)f_0]_0 dt \right\} = 0, $$

so that $f_1$ is a stationary solution, and thus $f_1$ is an eigenfunction associated to the eigenvalue $\lambda_1 = 0$. $\square$

## 4. IRREDUCIBILITY AND GEOMETRY OF THE FIRST EIGENVALUE

In this section, we are concerned with the geometric part of the Krein-Rutman theorem for an unbounded operator $\mathcal{L}$ on a Banach lattice $X$. We assume that the conclusions of the existence part are achieved, namely

(C1) the first primal and dual eigenvalue problem has a solution $(\lambda_1, f_1, \phi_1)$: there exist $\lambda_1 \in \mathbb{R}$, $f_1 \in X_+ \cap D(\mathcal{L}) \setminus \{0\}$, $\phi_1 \in Y_+ \cap D(\mathcal{L}^*) \setminus \{0\}$ such that

$$ (4.1) \qquad \mathcal{L}f_1 = \lambda_1 f_1 \quad \text{and} \quad \mathcal{L}^*\phi_1 = \lambda_1 \phi_1. $$

By construction, we also have $\Sigma(\mathcal{L}) \subset \{z \in \mathbb{C}, \Re e(z) \le \lambda_1\}$.

Assuming that $S$ is positive as for the existence part and an additional strong maximum principle property, we analyze the first eigenvalue problem.
Then there exists a pair $(\lambda_1, f_1) \in \mathbb{R}_+ \times X_+ \setminus \{0\}$ such that $\mathcal{L}f_1 = \lambda_1 f_1$.

**Remark 3.5.** (1) When $S_{\mathcal{L}}$ satisfies (HS3) then (3.38) holds with

$$ (3.40) \qquad v := \sum_{\ell=0}^{N-1} S_B * (\mathcal{A}S_B)^{(*\ell)}, \quad w := (S_B \mathcal{A})^{(*N)}. $$

(2) By definition of this norm $[\cdot]_{\phi_0}$, we see that $\mathcal{X}_0$ is a weighted $L^1$ space or a weighted Radon measures space. In many applications, when both $\mathcal{X}_0$ and $X$ are Radon measures spaces, one can choose $N=1$. On the other hand, when $X$ is for instance a (possibly weighted) $L^p$ space with $p>1$, one must take $N \ge 2$ in most of the applications. In condition (ii), the first bound is not really demanding and almost automatic in view of the estimates exhibited in Section 3.3. The second bound is a kind of regularity estimate reminiscent of the enlarging and shrinkage technique developed in [293, 189, 273].

**Proof of Theorem 3.4.** We split the proof into three steps. During the proof, we will use the shorthand $[\cdot]_0 := [\cdot]_{\phi_0}$.

**Step 1.** We define

$$ R := \max(2\|\Theta\|_{L^1}, \|g_0\|), $$

for some $g_0 \in X_+$ such that $[g_0]_0 = 1$, and next the nonempty convex and compact (in the weak $\ast\sigma(X, Y)$ sense) set

$$ C := \{f \in X_+; [f]_0 = 1, \|f\| \le R\}, $$

as well as the increasing function

$$ \lambda(t) := \inf_{f \in C} [S(t)f]_0, \quad \forall t \ge 0. $$

We have the alternative

• (1) $\sup \lambda > 2M$,

• (2) $\sup \lambda \le 2M$.

**Step 2.** We assume that the first term (1) of the alternative holds true, or in other words, there exists $T_0 > 0$ such that

$$ (3.41) \qquad \forall f \in C, \quad [S_{T_0}f]_0 \ge 2M. $$

We define as before

$$ \Phi_{T_0} f := \frac{S_{T_0} f}{[S_{T_0} f]_0}, \quad \forall f \in C. $$

By construction, for any $f \in C$, we have $\Phi_{T_0} f \ge 0$ and $[\Phi_{T_0} f]_0 = 1$. On the other hand, using the splitting structure (3.38) and the estimates (3.39), we have

$$ \|S(t)f\| \le M\|f\| + \int_0^t \Theta(t-s)[S(s)f]_0 ds. $$

From hypothesis (i) and the semigroup property, we also have

$$ [S_t f]_0 \ge [S_s f]_0, \quad \forall t \ge s \ge 0. $$

The two above estimates together imply

$$ \begin{aligned} \|\Phi_{T_0} f\| & \le \frac{M\|f\|}{[S_{T_0}f]_0} + \int_0^{T_0} \Theta(T_0-s) \frac{[S_s f]_0}{[S_{T_0}f]_0} ds \\ & \le \frac{1}{2}\|f\| + \|\Theta\|_{L^1} \le R, \end{aligned} $$

for any $f \in C$. We have thus proved $\Phi_{T_0} : C \to C$. Thanks to the Tykonov fixed point Theorem, there exists $f_{T_0} \in C$ such that $\Phi_{T_0} f_{T_0} = f_{T_0}$. In other words, we have built a pair of “almost eigenvalue and eigenfunction”

$$ f_{T_0} \ge 0, \quad [f_{T_0}]_0 = 1, \quad S_{T_0} f_{T_0} = e^{\lambda_1 T_0} f_{T_0}, $$

with $e^{\lambda_1 T_0} = [S_{T_0} f]_0$ and thus $\lambda_1 \in [0, \kappa_1]$. We conclude to the existence of $f_1 \in C$ such that $\mathcal{L}f_1 = \lambda_1 f_1$ really similarly as in Step 2 of the Second proof of Theorem 3.1.
4.1. **More about positivity.** For further references, we introduce several notions which are strongly related to the positivity property for semigroups.

**The signum operator** sign. In a real Banach lattice X, we say that sign $f \in \mathcal{B}(X, X'')$ is a signum operator for $f \in X$, if it satisfies the following properties

$$ (4.2) \qquad (\operatorname{sign} f) f = |f|, $$

$$ (4.3) \qquad (\operatorname{sign} f) g \le |g|, \quad \forall g \in X. $$

In the sequel, we will always assume that such an operator exists. We refer to [15, Sec. C.I & C.II] for a general introduction to the topic. In practice, we will only need a weak formulation of the sign operator (see below) which may be defined only in some subspace $\mathcal{X} \subset X$. We always additionally assume that the signum operator satisfies

$$ (\operatorname{sign}(-f))(-g) = (\operatorname{sign} f)g, \quad \forall g \in X, $$

$$ (\operatorname{sign} f) g = g, \quad \forall g \in X, \text{ if } f \in X_+, $$

We also define

$$ \operatorname{sign}_+ f := \frac{1}{2} (I + \operatorname{sign} f). $$

• When $X$ is a space of functions, the sign operator sign $f$ associated to $f \in X$ corresponds to the multiplication by the function sign $f := \mathbf{1}_{f>0} - \mathbf{1}_{f<0}$. When $X := L^p(E)$, we obviously see that sign $f \in \mathcal{B}(L^p(E))$ for any $f \in L^p(E)$. On the other hand, when $X := C_0(E)$, we only have sign $f \in \mathcal{B}(C_0(E); M^\infty(E))$, where $M^\infty(E)$ denotes the space of uniformly bounded measurable functions, so that $M^\infty(E) \subset (C_0(E))''$. In a space of bounded measures $X = M^1(E)$, we may define the sign operator by means of the Radon-Nikodym theorem. For a given $f \in M^1(E)$, using Hahn decomposition, there exists indeed a measurable function $\alpha : E \to \{-1, 1\}$ such that $f = \alpha|f|$, and we then define $(\operatorname{sign} f)g = \alpha g$ for any $g \in M^1(E)$.

• When $X$ is $\sigma$-order complete, in the sense that any increasing and upper bounded sequence has a supremum (a common least upper bound), the operator sign exists and is more regular, namely sign $f \in \mathcal{B}(X)$ for any $f \in X$, see [296] and also [15, Sec. C.I.8]. We recover in particular that sign $f \in \mathcal{B}(L^p(E))$ for any $f \in L^p(E)$.

**Weak principle maximum and Kato's inequality.** We introduce now two definitions formulated on an operator $\mathcal{L}$ which are almost equivalent to the positivity property of the semigroup $S$ when $\mathcal{L}$ is the generator of $S$.

• We say that the operator $\mathcal{L}$ satisfies the weak maximum principle when

$$ (4.4) \qquad \kappa \in \mathbb{R}, f \in D(\mathcal{L}) \text{ and } (\kappa - \mathcal{L})f \ge 0 \text{ imply } f \ge 0; $$

• We say that the operator $\mathcal{L}$ satisfies Kato's inequality when

$$ (4.5) \qquad \forall f \in D(\mathcal{L}), \quad \mathcal{L}|f| \ge (\operatorname{sign} f)\mathcal{L}f. $$

Since $|f|$ does not necessarily belong to $D(\mathcal{L})$, the correct way to understand Kato's inequality is

$$ (4.6) \qquad \forall f \in D(\mathcal{L}), \forall \psi \in D(\mathcal{L}^*) \cap Y_+, \quad \langle |f|, \mathcal{L}^*\psi \rangle \ge (\operatorname{sign} f)\mathcal{L}f, \psi. $$

We immediately see from the definitions that (4.5) is equivalent to assume

$$ (4.7) \qquad \forall f \in D(\mathcal{L}), \quad \mathcal{L}f_+ \ge (\operatorname{sign}_+ f)\mathcal{L}f. $$

**Remark 4.1.** We complement Lemma 2.1, by claiming that for a semigroup $S = S_\mathcal{L}$ on a Banach lattice $X$, there is equivalence between the fact that $S$ is positive and $\kappa - \mathcal{L}$ satisfies the weak maximum principle for any $\kappa > \omega(\mathcal{L})$, what is straightforward using that these properties are equivalent to the fact that $\mathcal{R}_\mathcal{L}(\kappa) \ge 0$ for any $\kappa > \omega(\mathcal{L})$. These properties also imply that Kato's inequality holds true, see [296, 12], [13, Prop. 1.1], [11, Rk. 3.10] and the textbook [15, Thms C.II.2.4, C.II.2.6 and Rk. C-II.3.12]. When for instance $f, |f| \in D(\mathcal{L})$, we may indeed compute

$$ (\operatorname{sign} f)\mathcal{L}f = (\operatorname{sign} f) \lim_{t \to 0} \frac{S_t f - f}{t} \le \lim_{t \to 0} \frac{S_t |f| - |f|}{t} = \mathcal{L}|f|, $$

where we have used the very definition of the generator $\mathcal{L}$ and the properties (4.2)-(4.3) of sign $f$ in the inequality.
Step 3. We assume that the second term (2) of the alternative holds true. In that case, for any $n \ge 1$, there exists $f_n \in C$ such that $[S(n)f_n]_0 \le 2M$. By compactness, there exists $f_0 \in C$ and a subsequence $(f_{n_k})$ such that $f_{n_k} \to f_0 \in C$ and

$$ \forall t \ge 0, \forall k (n_k \ge t), \quad [S(t)f_{n_k}]_0 \le [S(n_k)f_{n_k}]_0 \le 2M, $$

so that

$$ (3.42) \qquad \forall t \ge 0, \quad [S(t)f_0]_0 \le 2M. $$

Using this particular initial datum, we argue similarly as in [80, proof of Thm. 6.1], and we conclude to the existence of a stationary state. More precisely, we come back to the splitting structure (3.38) of the semigroup S and we introduce the associated Cesàro means

$$ (3.43) \qquad U_T := \frac{1}{T} \int_0^T S(t) \, dt, \quad V_T := \frac{1}{T} \int_0^T v(t) \, dt, \quad K_T := \frac{1}{T} \int_0^T (w * S)(t) \, dt, $$

for any $T > 0$. We obviously have

$$ \|V_T\|_{\mathcal{B}(X)} \le \frac{1}{T} \int_0^T \|v(t)\|_{\mathcal{B}(X)} dt \le M. $$

On the other hand, we have

$$ \int_0^T (w * S)(t) \, dt = \int_0^T \int_s^T w(t-s) \, dt S(s) \, ds \le \int_0^T w(\tau) \, d\tau \int_0^T S(s) \, ds, $$

thanks to the Fubini theorem and the positivity of the two operators involved in this integral formula. We deduce

$$ \begin{aligned} \|K_T f_0\| & \le \left\| \int_0^T w(\tau) d\tau \frac{1}{T} \int_0^T S(s) f_0 \, ds \right\| \\ & \le \int_0^\infty \|w(\tau)\|_{\mathcal{B}(X_0, X)} d\tau \left[ \frac{1}{T} \int_0^T S(s) f_0 \, ds \right]_0 = \|\Theta\|_{L^1} [U_T f_0]_0, \end{aligned} $$

thanks to assumption (ii), so that $K_T f_0$ is uniformly bounded in $X$ thanks to (3.42) and the elementary estimate $[U_T f_0]_0 \le [S_T f_0]_0$. We then deduce that $U_T = V_T + K_T$ satisfies

$$ \|U_T f_0\| \le M \|f_0\| + 2M \|\Theta\|_{L^1} \quad \text{and} \quad 1 \le [S_T f_0]_0 \le 2M, $$

for any $T > 0$. By compactness, there exists $T_k \to +\infty$ and $f_1 \in X_+$ such that $U_{T_k} f \to f_1$ weakly $*$ in $X$. Thanks to the second inequality, we have $[f_1]_0 \ge 1$. We then argue thanks to the usual mean ergodic theorem trick. For any fixed $s > 0$, we observe that

$$ \begin{aligned} S(s)f_1 - f_1 &= \lim_{k \to \infty} \left\{ \frac{1}{T_k} \int_0^{T_k} S(s)S(t)f_0 dt - \frac{1}{T_k} \int_0^{T_k} S(t)f_0 dt \right\} \\ &= \lim_{k \to \infty} \left\{ \frac{1}{T_k} \int_{T_k}^{T_k+s} S(t)f_0 dt - \frac{1}{T_k} \int_0^s S(t)f_0 dt \right\} \end{aligned} $$

weakly $*$ in $X$. By the lower semicontinuous property of the norm $[\cdot]_0$, we deduce

$$ [S(s)f_1 - f_1]_0 \le \liminf_{k \to \infty} \left\{ \frac{1}{T_k} \int_{T_k}^{T_k+s} [S(t)f_0]_0 dt + \frac{1}{T_k} \int_0^s [S(t)f_0]_0 dt \right\} = 0, $$

so that $f_1$ is a stationary solution, and thus $f_1$ is an eigenfunction associated to the eigenvalue $\lambda_1 = 0$. $\square$

Let us emphasize on the fact that Theorem 3.4 does not implies (S1), but only provides an existence result of the direct eigenvalue problem. In this more involved weakly dissipative framework, in order to tackle the eigentriplet problem, we may proceed in the following way. Assuming for instance that both $S_L$ and $S_{L^*}$ satisfy the conditions of Theorem 3.4 (an alternative argument combining Theorem 3.4 with results of Section 2 will be used in Section 7.4), we have

(C1) the first primal and dual eigenvalue problems have a solution: there exist $\lambda_1, \lambda_1^* \in \mathbb{R}$, $f_1 \in X_+ \cap D(\mathcal{L}) \setminus \{0\}$, $\phi_1 \in Y_+ \cap D(\mathcal{L}^*) \setminus \{0\}$ such that

$$ (3.44) \qquad \mathcal{L}f_1 = \lambda_1 f_1 \quad \text{and} \quad \mathcal{L}^*\phi_1 = \lambda_1^* \phi_1. $$
We end this section by introducing other notions of positivity which strengthen the previously defined positivity condition.

**Strict order.** We may define a first stronger order $>$(or $<$) on $X$ by writing for $f \in X$

$$f > 0 \quad \text{if} \quad f \in X_+ \setminus \{0\}$$

and similarly a stronger order $>$(or $<$) on $X'$ by writing for $\phi \in X'$

$$\phi > 0 \quad \text{if} \quad \phi \in X'_+ \setminus \{0\}.$$

We may next define the strict (and stronger) order $\gg$(or $\ll$) on $X$ by writing for $f \in X$

$$(4.8) \qquad f \gg 0 \text{ or } f \in X_{++} \quad \text{iff} \quad \forall \psi \in X'_+ \setminus \{0\}, \quad \langle \psi, f \rangle > 0,$$

and similarly the strict order $\gg$(or $\ll$) on $X'$ by writing for $\phi \in X'$

$$(4.9) \qquad \phi \gg 0 \text{ or } \phi \in X'_{++} \quad \text{iff} \quad \forall g \in X_+ \setminus \{0\}, \quad \langle \phi, g \rangle > 0.$$

On the two Banach lattices $X$ and $Y$, we thus have three positivity notions with $\gg$(associated to $X_{++}$ and $Y_{++}$) stronger than $>$ (associated to $X_+ \setminus \{0\}$ and $Y_+ \setminus \{0\}$) which itself is stronger than $\ge$ (associated to $X_+$ and $Y_+$).

Let us comment on the notion of strict positivity.

**Remark 4.2.** When $X = Y'$ for instance, there are two possible strict positivity notions on $X$ given by (4.8) for the space $X$ (namely $\phi \in X$ is strictly positive when $\langle \xi, \psi \rangle > 0$ for any $\xi \in X'_+ \setminus \{0\}$) and by (4.9) for the space $Y$ (namely $\phi \in X$ is strictly positive when $\langle \psi, g \rangle > 0$ for any $g \in Y_+ \setminus \{0\}$). They clearly coincide when $X$ is reflexive, but in general the first one is stronger than the second one. In that situation, we will always consider that $X$ is endowed with the weakest “dual” strict order (4.9).

**Examples 4.3.** In the space $C_0(E)$, the strict order is defined by $f \gg 0$ iff $f(x) > 0$ for any $x \in E$. In a space $L^p(E, \mathcal{E}, \mu)$, $1 \le p \le \infty$, the strict order is defined by $f \gg 0$ iff $f(x) > 0$ for $\mu$-a.e. $x \in E$. In the space $M^1(E) = C_0(E)'$, the strict order is defined by duality by $f \gg 0$ iff $\langle f, \varphi \rangle > 0$ for any $\varphi \in C_0(E)$, $\varphi \ge 0$, $\varphi \ne 0$.

**Remark 4.4.** In a Banach lattice $X$ such that $\text{int} X_+ \ne \emptyset$, the common definition of the strict order is $X_{++} := \text{int} X_+$. In particular, in the case when $E$ is compact and $X = C_0(E) = C(E)$, we have $\text{int} X_+ \ne \emptyset$ and the definition of $X_{++}$ introduced in Examples 4.3 coincides with $\text{int} X_+$. In all the other examples considered, we have $\text{int} X_+ = \emptyset$, and thus our definition of the strict order does not coincide with the one defined through the set $\text{int} X_+$.

**Remark 4.5.** Another notion of strict order can be defined through the notions of ideals and quasi-interior points as briefly explained now, see [15] or [41, Chap. 10] and the references therein for details. Defining the segment $[g_1, g_2]$ and the set $I_f$ for $g_1, g_2 \in X$ and $f \in X_+ \setminus \{0\}$ by

$$[g_1, g_2] := \{g \in X; g_1 \le g \le g_2\}, \quad I_f := \bigcup_{k \ge 0} [-kf, kf] = \text{Span}[0, f],$$

one shows that $I_f$ is an ideal in the sense that $g \in I_f$ implies $|g| \in I_f$ and $0 \le g \le f$ implies $g \in I_f$. We say that $f$ is an order unit if $I_f = X$. When $\text{int} X_+ \ne \emptyset$, we find that $f$ is an order unit iff $f \in \text{int} X_+$ from Lemma 2.23, so that we recover the notion of strict positivity defined above. On the other hand, we say that $f$ is a quasi-interior point if $\bar{I}_f = X$. It can be shown that $f$ is a quasi-interior point iff $f$ is strictly positive in the sense of the direct strict order (4.8), see for instance [337, Thm. II.6.3]. These two notions of strict positivity and quasi interior point thus coincide when $X$ is reflexive or when $X = L^p(E, \mathcal{E}, \mu)$, $1 \le p < \infty$, see also [41, Examples 10.16] when $\mu$ is a $\sigma$-finite diffuse (or atomless) measure. On the other hand, it is important to point out again that the “dual” strict order (4.9) considered here is a weaker notion than the quasi-interior point notion. For instance, in $X = C_0(E)' = M^1(E)$, there is no quasi-interior point but $X_{++} \ne \emptyset$.

We finally point out the following result. For a semigroup $S = S_\mathcal{L}$ in a Banach lattice, under the mild assumption that there exists a strictly positive subeigenvector for the dual problem, namely

$$\exists \phi \in X'_{++}, \exists b \in \mathbb{R}, \quad \mathcal{L}^* \phi \le b \phi,$$

then Kato's inequality (4.5) implies that $S$ is positive, see [13, Thm. 1.6].
This conclusion is weaker than the conclusion (S1) obtained in Section 2, which additionally ensures that $\lambda_1^* = \lambda_1$ and $\Sigma(\mathcal{L}) \subset \{z \in \mathbb{C}, \Re(e(z)) \le \lambda_1\}$. It is however the fundamental existence step. The additional mild geometric properties of the conclusion (S1) may be obtained thanks to the material developed in the next Sections 4 and 5. Let us emphasize that in the previous section, $\lambda_1$ is defined thanks to (2.16), which because of $\mathcal{I} = \mathcal{I}_2$ in Lemma 2.3, automatically implies the inclusion $\Sigma(\mathcal{L}) \subset \{z \in \mathbb{C}, \Re(e(z)) \le \lambda_1\}$. Here, the construction of $\lambda_1$ is different, and the last inclusion is no longer guaranteed.

### 4. IRREDUCIBILITY AND GEOMETRY OF THE FIRST EIGENVALUE

In this section, we start the study of the geometric and asymptotic behavior parts of the Krein-Rutman theorem. More precisely, we prove the geometric part (S2) and the mean ergodicity part (E1) under the partial conclusion (C1) of (S1) and some additional conditions. In particular we introduce the structure condition (X1) on the Banach lattice, the variant condition (H1') of (H1) and the strong positivity condition (H4) which is mainly based on a strong maximum principle (for a positive semigroup, that last condition being nothing but the irreducibility).

#### 4.1. More about positivity.
For further references, we introduce several notions which are strongly related to the positivity property for semigroups.

**The signum operator sign.** In a real Banach lattice $X$, we say that $\text{sign} f \in \mathcal{B}(X, X'')$ is a signum operator for $f \in X$, if it satisfies the following properties

$$ (4.1) \qquad (\text{sign} f) f = |f|, $$

$$ (4.2) \qquad (\text{sign} f) g \le |g|, \quad \forall g \in X. $$

In the sequel, we will always assume that such an operator exists. We refer to [15, Sec. C.I & C.II] for a general introduction to the topic. In practice, we will only need a weak formulation of the sign operator (see below) which may be defined only in some subspace $\mathcal{X} \subset X$. We always additionally assume that the signum operator satisfies

$$ (\text{sign}(-f))(-g) = (\text{sign} f)g, \quad \forall g \in X, $$

$$ (\text{sign} f) g = g, \quad \forall g \in X, \text{ if } f \in X_+, $$

We also define

$$ \text{sign}_+ f := \frac{1}{2}(I + \text{sign} f). $$

* When $X$ is a space of functions, the sign operator $\text{sign} f$ associated to $f \in X$ corresponds to the multiplication by the function $\text{sign} f := \mathbf{1}_{f>0} - \mathbf{1}_{f<0}$. When $X := L^p(E)$, we obviously see that $\text{sign} f \in \mathcal{B}(L^p(E))$ for any $f \in L^p(E)$. On the other hand, when $X := C_0(E)$, we only have $\text{sign} f \in \mathcal{B}(C_0(E); M^\infty(E))$, where $M^\infty(E)$ denotes the space of uniformly bounded measurable functions, so that $M^\infty(E) \subset (C_0(E))''$. In a space of bounded measures $X = M^1(E)$, we may define the sign operator by means of the Radon-Nikodym theorem. For a given $f \in M^1(E)$, using Hahn decomposition, there exists indeed a measurable function $\alpha : E \to \{-1, 1\}$ such that $f = \alpha|f|$, and we then define $(\text{sign} f)g = \alpha g$ for any $g \in M^1(E)$.

* When $X$ is $\sigma$-order complete, in the sense that any increasing and upper bounded sequence has a supremum (a common least upper bound), the operator $\text{sign} f$ exists and is more regular, namely $\text{sign} f \in \mathcal{B}(X)$ for any $f \in X$, see [295] and also [15, Sec. C.I.8]. We recover in particular that $\text{sign} f \in \mathcal{B}(L^p(E))$ for any $f \in L^p(E)$.

**Weak maximum principle and Kato's inequality.** We introduce now two definitions formulated on an operator $\mathcal{L}$ which are almost equivalent to the positivity property of the semigroup $S$ when $\mathcal{L}$ is the generator of $S$.

* We say that the operator $\mathcal{L}$ satisfies the weak maximum principle when

$$ (4.3) \qquad \kappa \in \mathbb{R}, f \in D(\mathcal{L}) \text{ and } (\kappa - \mathcal{L})f \ge 0 \text{ imply } f \ge 0; $$

* We say that the operator $\mathcal{L}$ satisfies Kato's inequality when

$$ (4.4) \qquad \forall f \in D(\mathcal{L}), \quad \mathcal{L}|f| \ge (\text{sign} f)\mathcal{L}f. $$
**4.2. Irreducibility and strong maximum principle.** We present some other material involving the strict positivity. When satisfied by $\xi \in X$ or $Y$, we will in particular make use of the property

$$ (4.10) \qquad \xi_+ \gg 0 \text{ implies } \xi \gg 0 \text{ (or equivalently } \xi_- = 0). $$

For further references, we introduce some general framework for the couple of Banach lattices we will use in the sequel:

$$ (\mathbf{X}1) \left\{ \begin{array}{ll} (i) & X_{++} \neq \emptyset, Y_{++} \neq \emptyset \text{ and the signum operator is well defined in both } X \text{ and } Y; \\ (ii) & \text{the property (4.10) holds in both spaces } X \text{ and } Y. \end{array} \right. $$

While $(\mathbf{X}1)-(i)$ is always satisfied in the applications, it is not the case for $(\mathbf{X}1)-(ii)$.

**Lemma 4.6.** The property (4.10) holds true in a space $X$ endowed with the direct strict order (4.8), in particular in $X = L^p$, $p \in [1, \infty)$, and $X = C_0$, and also in the space $L^\infty = (L^1)'$ endowed with the dual strict order (4.9).

*Proof of Lemma 4.6.* We start recalling the proof of (4.10) in a general space $X$ endowed with the strict order (4.8). Consider an element $f$ of the Banach lattice $X$ and assume that $f_+ \gg 0$. The vectors $f_+$ and $f_-$ are disjoint, i.e. $f_+ \wedge f_- = 0$, see for instance [264, Thm. 1.1.1 iv)]. On the one hand, since $f_+ \gg 0$, we have that

$$ (nf_+) \wedge f_- \to f_- $$

with respect to the norm as $n \to \infty$, see [337, Thm. II.6.3]. On the other hand, we have

$$ 0 \le (nf_+) \wedge f_- \le (nf_+) \wedge (nf_-) = n(f_+ \wedge f_-) = 0, $$

for every integer $n \ge 1$, where the last equality follows from the fact that $f_+$ and $f_-$ are disjoint. We deduce by passing to the limit $n \to \infty$ that $f_- = 0$. Thus, $f = f_+ \gg 0$.

We now establish (4.10) when $X = L^\infty$ (notice that exactly the same arguments may be used when $X = L^p$ and $X = C_0$, what provides an elementary proof in these cases too). Take $f \in L^\infty$ such that $f_+ \gg 0$. From the definition of the strict order made explicit in Examples 4.3, we have $f_+(x) = \max(f(x), 0) > 0$ a.e., so that $f(x) > 0$ a.e. and finally $f_-(x) = 0$ a.e.. $\square$

We give now a counter-example in the Radon measures space case.

**Example 4.7.** Consider $M^1([0, 1]) = C([0, 1])'$ endowed with the dual strict order (4.9). Let $(q_n)$ be an enumeration of the rational numbers in $[0, 1]$ and let $r$ be an irrational number in $[0, 1]$. The functional $\phi$ given by $\langle \phi, f \rangle := \sum_{n=1}^\infty 1/2^n f(q_n) - f(r)$ satisfies $\phi_+ \gg 0$, but $\phi$ itself is not positive.

For an operator $A \in \mathcal{B}(X)$, we have yet formalized a positivity condition in section 2.1, by (P1) $A \ge 0$ if $A : X_+ \to X_+$.

Other possible definition of positivity may be

$$ (P2) \quad A : X_+ \setminus \{0\} \to X_+ \setminus \{0\}; $$

$$ (P3) \quad A : X_{++} \to X_{++}. $$

We now define a stronger notion of positivity, named as strong positivity condition, by

$$ (P4) \quad A > 0 \text{ if } A : X_+ \setminus \{0\} \to X_{++}. $$

We list without proof some elementary properties about these different notions and also refer to Section 6.2 for further discussion. We have (P2) $\Rightarrow$ (P1), (P3) $\Rightarrow$ (P1) as well as (P4) $\Rightarrow$ ((P3), (P2)). We also have $A : X_+ \to X_+$ iff $A^* : Y_+ \to Y_+$; $A : X_{++} \to X_{++}$ iff $A^* : Y_{++} \to Y_{++}$; $A : X_+ \setminus \{0\} \to X_{++}$ iff $A^* : Y_+ \setminus \{0\} \to Y_{++}$.

We say that $\lambda - \mathcal{L}$ satisfies the strong maximum principle if

$$ (4.11) \qquad f \in X_+ \cap D(\mathcal{L}), (\lambda - \mathcal{L})f \ge 0 \text{ imply } f \gg 0 \text{ or } f = 0. $$

It is worth emphasizing that if $\lambda - \mathcal{L}$ satisfies the strong maximum principle for some $\lambda \in \mathbb{R}$ then $\lambda' - \mathcal{L}$ satisfies the strong maximum principle for any $\lambda' \le \lambda$.

We say that a positive semigroup $S$ is irreducible if

$$ (4.12) \qquad \forall f \in X_+ \setminus \{0\}, \forall \phi \in Y_+ \setminus \{0\}, \exists \tau > 0 \quad \langle S_\tau f, \phi \rangle > 0. $$

A semigroup $S$ is classically said to be irreducible and aperiodic if the above positivity condition holds for all sufficiently large times, namely

$$ (4.13) \qquad \forall f \in X_+ \setminus \{0\}, \forall \phi \in Y_+ \setminus \{0\}, \exists T > 0, \forall \tau \ge T \quad \langle S_\tau f, \phi \rangle > 0. $$
Since $|f|$ does not necessarily belong to $D(\mathcal{L})$, the correct way to understand Kato's inequality is
(4.5) \qquad $\forall f \in D(\mathcal{L}), \forall \psi \in D(\mathcal{L}^*) \cap Y_+, \langle |f|, \mathcal{L}^*\psi \rangle \ge \langle (\operatorname{sign} f)\mathcal{L}f, \psi \rangle.$

We immediately see from the definitions that (4.4) is equivalent to assume

$$
(4.6) \qquad \forall f \in D(\mathcal{L}), \quad \mathcal{L}f_+ \ge (\operatorname{sign}_+ f) \mathcal{L}f.
$$

**Remark 4.1.** We complement Lemma 2.1, by claiming that for a semigroup $S = S_{\mathcal{L}}$ on a Banach lattice $X$, there is equivalence between the fact that $S$ is positive and $\kappa - \mathcal{L}$ satisfies the weak maximum principle for any $\kappa > \omega(\mathcal{L})$, what is straightforward using that these properties are equivalent to the fact that $\mathcal{R}_{\mathcal{L}}(\kappa) \ge 0$ for any $\kappa > \omega(\mathcal{L})$. These properties also imply that Kato's inequality holds true, see [295, 12], [13, Prop. 1.1], [11, Rk. 3.10] and the textbook [15, Thms C.II.2.4, C.II.2.6 and Rk. C-II.3.12]. When for instance $f, |f| \in D(\mathcal{L})$, we may indeed compute

$$
(\operatorname{sign} f) \mathcal{L} f = (\operatorname{sign} f) \lim_{t \to 0} \frac{S_t f - f}{t} \le \lim_{t \to 0} \frac{S_t |f| - |f|}{t} = \mathcal{L} |f|,
$$

where we have used the very definition of the generator $\mathcal{L}$ and the properties (4.1)-(4.2) of sign $f$
in the inequality.

We end this section by introducing other notions of positivity which strengthen the previously
defined positivity condition.

**Strict order.** We may define a first stronger order $>^-$ (or $<^-$) on $X$ by writing for $f \in X$
$$f > 0 \quad \text{if} \quad f \in X_+ \setminus \{0\}$$

and similarly a stronger order $>^-$ (or $<^-$) on $X'$ by writing for $\phi \in X'$

$$
\phi > 0 \quad \text{if} \quad \phi \in X'_+ \setminus \{0\}.
$$

We may next define the strict (and stronger) order $\gg$ (or $\ll$) on $X$ by writing for $f \in X$

$$
(4.7) \qquad f \gg 0 \text{ or } f \in X_{++} \quad \text{iff} \quad \forall \psi \in X'_+ \setminus \{0\}, \quad \langle \psi, f \rangle > 0,
$$

and similarly the strict order $\gg$ (or $\ll$) on $X'$ by writing for $\phi \in X'$

$$
(4.8) \qquad \phi \gg 0 \text{ or } \phi \in X_{++}' \quad \text{iff} \quad \forall g \in X_{+} \setminus \{0\}, \quad \langle \phi, g \rangle > 0.
$$

On the two Banach lattices $X$ and $Y$, we thus have three positivity notions with $\gg$ (associated to $X_{++}$ and $Y_{++}$) stronger than $>^-$ (associated to $X_+ \setminus \{0\}$ and $Y_+ \setminus \{0\}$) which itself is stronger than $\ge^-$ (associated to $X_+$ and $Y_+$). We also emphasize that $\psi \in Y_{++}$ if, and only if, $[\cdot]_{\psi}$ is a norm on $X$.
Let us comment on the notion of strict positivity.

**Remark 4.2.** When $X = Y'$ for instance, there are two possible strict positivity notions on $X$ given by (4.7) for the space $X$ (namely $\phi \in X$ is strictly positive when $\langle \xi, \psi \rangle > 0$ for any $\xi \in X'_+ \setminus \{0\}$) and by (4.8) for the space $Y$ (namely $\phi \in X$ is strictly positive when $\langle \psi, g \rangle > 0$ for any $g \in Y_+ \setminus \{0\}$). They clearly coincide when $X$ is reflexive, but in general the first one is stronger than the second one. In that situation, we will always consider that $X$ is endowed with the weakest “dual” strict order (4.8).

**Examples 4.3.** In the space $C_0(E)$, the strict order is defined by $f \gg 0$ iff $f(x) > 0$ for any $x \in E$. In a space $L^p(E, \mathcal{E}, \mu)$, $1 \le p \le \infty$, the strict order is defined by $f \gg 0$ iff $f(x) > 0$ for $\mu$-a.e. $x \in E$. In the space $M^1(E) = C_0(E)'$, the strict order is defined by duality by $f \gg 0$ iff $\langle f, \varphi \rangle > 0$ for any $\varphi \in C_0(E)$, $\varphi \ge 0$, $\varphi \ne 0$.

**Remark 4.4.** In a Banach lattice X such that int $X_+ \ne \emptyset$, the common definition of the strict order is $X_{++} := \text{int} X_+$. In particular, in the case when E is compact and $X = C_0(E) = C(E)$, we have int $X_+ \ne \emptyset$ and the definition of $X_{++}$ introduced in Examples 4.3 coincides with int $X_+$. In all the other examples considered, we have int $X_+ = \emptyset$, and thus our definition of the strict order does not coincide with the one defined through the set int $X_+$.

**Remark 4.5.** Another notion of strict order can be defined through the notions of ideals and quasi-interior points as briefly explained now, see [15] or [41, Chap. 10] and the references therein for details. Defining the segment $[g_1, g_2]$ and the set $I_f$ for $g_1, g_2 \in X$ and $f \in X_+ \setminus \{0\}$ by

$$
[g_1, g_2] := \{g \in X; g_1 \le g \le g_2\}, \quad I_f := \bigcup_{k \ge 0} [-kf, kf] = \text{Span}[0, f],
$$
Other notions of strong positivity for the semigroup $S$ are

$$ (4.14) \qquad \exists T > 0, \quad S_T : X_+ \setminus \{0\} \to X_+, $$

$$ (4.15) \qquad \exists T > 0, \quad \int_0^T S(t)dt : X_+ \setminus \{0\} \to X_+. $$

We summarize some possible implications between the previous positivity notions.

**Lemma 4.8.** For a positive semigroup $S$, the following hold:

(1) The pointwise strong positivity condition (4.14) implies the condition (4.15);

(2) The integral strong positivity condition (4.15) implies the irreducibility condition (4.12), but the reverse implication is false. Similarly, the irreducibility and aperiodicity condition (4.13) implies the irreducibility condition (4.12), but the reverse implication is false;

(3) The irreducibility condition (4.12) is equivalent to the fact that $\mathcal{R}_L(\lambda) : X_+ \setminus \{0\} \to X_+$, for any $\lambda > \lambda_1$, as well as to the fact that $\lambda - \mathcal{L}$ satisfies the strong maximum principle (4.11) for any $\lambda \in \mathbb{R}$.

The result is very classic, at least for strongly positive semigroup, see e.g. [15, Definition C.3.1] or [41, Prop. 14.10]. For the sake of completeness, we however present a short proof.

*Proof of Lemma 4.8.* We prove (1). We assume (4.14) and we fix $g \in X_+ \setminus \{0\}$, $\phi \in Y_+ \setminus \{0\}$, so that $\langle S(T)g, \phi \rangle > 0$. Observing that the function $t \mapsto \langle S(t)g, \phi \rangle$ is continuous, there exists $\varepsilon > 0$ such that $\langle S(t)g, \phi \rangle > 0$ for any $t \in [T - \varepsilon, T]$, so that

$$ \langle \int_0^T S(t)dt, \phi \rangle = \int_0^T \langle S(t)g, \phi \rangle dt > 0. $$

Because $\phi \in Y_+ \setminus \{0\}$ may be chosen arbitrary, we deduce (4.15).

We prove (2). We assume now (4.15) and we fix $g \in X_+ \setminus \{0\}$, $\phi \in Y_+ \setminus \{0\}$, so that

$$ \int_0^T \langle S(t)g, \phi \rangle dt = \langle \int_0^T S(t)dtg, \phi \rangle > 0, $$

by assumption. We get (4.12) by observing that the function $t \mapsto \langle S(t)g, \phi \rangle$ must be positive somewhere on $[0, T]$. For the reverse implication we refer to [50, 169], where is studied an example of growth-fragmentation operator associated to mitosis satisfying the irreducibility condition (4.12) but not the integral strong positivity condition (4.15) nor the irreducibility and aperiodicity condition (4.13), see also Section 9.

We prove (3). We finally assume (4.12). From the above continuity argument, for any $g \in X_+ \setminus \{0\}$, $\phi \in Y_+ \setminus \{0\}$ there exist $\tau > \varepsilon > 0$ such that $\langle S(t)g, \phi \rangle > 0$ for any $t \in [\tau - \varepsilon, \tau + \varepsilon]$. As a 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

$$ \langle \mathcal{R}_L(\lambda)g, \phi \rangle = \langle \int_0^\infty e^{-\lambda t} S(t)dtg, \phi \rangle > 0. $$

Because $\phi \in Y_+ \setminus \{0\}$ is arbitrary, we have established that $\mathcal{R}_L(\lambda)g \in X_+$ 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 $f = \mathcal{R}_L(\lambda)g \in X_+$, what is the strong maximum principle. This one is obviously equivalent to the strong positivity property $\mathcal{R}_L(\lambda) : X_+ \setminus \{0\} \to X_+$. On the other way round, writing the above identity as

$$ \int_0^\infty e^{-\lambda t} \langle S(t)g, \phi \rangle dt = \langle \mathcal{R}_L(\lambda)g, \phi \rangle, $$

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 (4.12).

□

We present two other elementary results about the strong maximum principle.
one shows that $I_f$ is an ideal in the sense that $g \in I_f$ implies $|g| \in I_f$ and $0 \le g \le f$ implies $g \in I_f$. We say that $f$ is an order unit if $I_f = X$. When $\text{int } X_+ \ne \emptyset$, we find that $f$ is an order unit iff $f \in \text{int } X_+$ from Lemma 2.23, so that we recover the notion of strict positivity defined above. On the other hand, we say that $f$ is a quasi-interior point if $\bar{I}_f = X$. It can be shown that $f$ is a quasi-interior point iff $f$ is strictly positive in the sense of the direct strict order (4.7), see for instance [336, Thm. II.6.3]. These two notions of strict positivity and quasi interior point thus coincide when $X$ is reflexive or when $X = L^p(E, \mathcal{E}, \mu)$, $1 \le p < \infty$, see also [41, Examples 10.16] when $\mu$ is a $\sigma$-finite diffuse (or atomless) measure. On the other hand, it is important to point out again that the “dual” strict order (4.8) considered here is a weaker notion than the quasi-interior point notion. For instance, in $X = C_0(E)' = M^1(E)$, there is no quasi-interior point but $X_{++} \ne \emptyset$.

We finally point out the following result. For a semigroup $S = S_\mathcal{L}$ in a Banach lattice, under the mild assumption that there exists a strictly positive subeigenvector for the dual problem, namely

$$ \exists \phi \in X'_{++}, \exists b \in \mathbb{R}, \mathcal{L}^*\phi \le b\phi, $$

then Kato's inequality (4.4) implies that $S$ is positive, see [13, Thm. 1.6].

**4.2. Irreducibility and strong maximum principle.** We present some other material involving the strict positivity. When satisfied by $\mathcal{E} \subset X$ or $\mathcal{E} \subset Y$, we will in particular make use of the property

$$ (4.9) \qquad (\xi \in \mathcal{E}, \xi_+ \gg 0) \text{ implies } \xi \gg 0 \text{ (or equivalently } \xi_- = 0). $$

The property (4.9) does not always hold in $\mathcal{E} = X$, but it holds in many usual cases.

**Lemma 4.6.** The property (4.9) holds true in a space $X$ endowed with the direct strict order (4.7), in particular in $X = L_m^p$, $p \in [1, \infty)$, and $X = C_{0,m}$. The property (4.9) also holds in a set $\mathcal{E} \subset X \subset M_{\text{loc}}^1$ endowed with the dual strict order (4.8) when $\mathcal{E} \subset L_{\text{loc}}^1$, and in particular in the space $L_{m-1}^\infty = (L_m^1)'$.

**Proof of Lemma 4.6.** We start recalling the proof of (4.9) in a general space $X$ endowed with the strict order (4.7). Consider an element $f$ of the Banach lattice $X$ and assume that $f_+ \gg 0$. The vectors $f_+$ and $f_-$ are disjoint, i.e. $f_+ \wedge f_- = 0$, see for instance [263, Thm. 1.1.1 iv)]. On the one hand, since $f_+ \gg 0$, we have

$$ (nf_+) \wedge f_- \to f_- $$

with respect to the norm as $n \to \infty$, see [336, Thm. II.6.3]. On the other hand, we have

$$ 0 \le (nf_+) \wedge f_- \le (nf_+) \wedge (nf_-) = n(f_+ \wedge f_-) = 0, $$

for every integer $n \ge 1$, where the last equality follows from the fact that $f_+$ and $f_-$ are disjoint. We deduce by passing to the limit $n \to \infty$ that $f_- = 0$. Thus, $f = f_+ \gg 0$.

We now establish (4.9) in the second case. Take thus $f \in \mathcal{E} \subset L_{\text{loc}}^1 \cap X$, with $X \subset M_{\text{loc}}^1$ endowed with the dual order, so that $\langle f_+, \varphi \rangle > 0$ for any $\varphi \in C_c$, $\varphi \ge 0$, $\varphi \ne 0$. Because $f$ (and thus $f_+$) is a measurable function, we have $f_+(x) = \max(f(x), 0) > 0$ for a.e. $x \in E$, so that $f > 0$ a.e. on $E$ and finally $f_- = 0$ a.e. on $E$. $\square$

We give now a counter-example in the Radon measures space case.

**Example 4.7.** Consider $M^1([0, 1]) = C([0, 1])'$ endowed with the dual strict order (4.8). Let $(q_n)$ be an enumeration of the rational numbers in $[0, 1]$ and let $r$ be an irrational number in $[0, 1]$. The functional $\phi$ given by $\langle \phi, f \rangle := \sum_{n=1}^\infty 1/2^n f(q_n) - f(r)$ satisfies $\phi_+ \gg 0$, but $\phi$ itself is not positive.

For an operator $A \in \mathcal{B}(X)$, we have yet formalized a positivity condition in section 2.1, by (P1) $A \ge 0$ if $A : X_+ \to X_+$.

Other possible definition of positivity may be

$$ (P2) A : X_+ \setminus \{0\} \to X_+ \setminus \{0\}; $$

$$ (P3) A : X_{++} \to X_{++}. $$

We now define a stronger notion of positivity, named as strong positivity condition, by

$$ (P4) A > 0 \text{ if } A : X_+ \setminus \{0\} \to X_{++}. $$
**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 (4.7), 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).

**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

$$ \mathcal{L}f_0 = \kappa_1f_0 - h_0 \ge (\kappa_1 - \varepsilon^{-1})f_0, $$

so that condition (H2) holds true with $\kappa_0 := \kappa_1 - \varepsilon^{-1}$ thanks to Lemma 2.4-(ii). $\square$
We list without proof some elementary properties about these different notions and also refer to Section 6.1 for further discussion. We have (P2) ⇒ (P1), (P3) ⇒ (P1) as well as (P4) ⇒ ((P3), (P2)). We also have $A: X_+ \to X_+$ iff $A^*: Y_+ \to Y_+$; $A: X_{++} \to X_{++}$ iff $A^*: Y_{++} \to Y_{++}$; $A: X_+\{0\} \to X_{++}$ iff $A^*: Y_+\{0\} \to Y_{++}$.

We say that $\lambda - \mathcal{L}$ satisfies the strong maximum principle if

$$ (4.10) \qquad f \in X_+ \cap D(\mathcal{L}), (\lambda - \mathcal{L})f \ge 0 \quad \text{imply} \quad f \gg 0 \text{ or } f = 0. $$

It is worth emphasizing that if $\lambda - \mathcal{L}$ satisfies the strong maximum principle for some $\lambda \in \mathbb{R}$ then $\lambda' - \mathcal{L}$ satisfies the strong maximum principle for any $\lambda' \le \lambda$.

We say that a positive semigroup $S$ is irreducible if

$$ (4.11) \qquad \forall f \in X_+\{0\}, \forall \phi \in Y_+\{0\}, \exists \tau > 0 \quad \langle S_\tau f, \phi \rangle > 0. $$

A semigroup $S$ is classically said to be irreducible and aperiodic if the above positivity condition holds for all sufficiently large times, namely

$$ (4.12) \qquad \forall f \in X_+\{0\}, \forall \phi \in Y_+\{0\}, \exists T > 0, \forall \tau \ge T \quad \langle S_\tau f, \phi \rangle > 0. $$

Other notions of strong positivity for the semigroup $S$ are

$$ (4.13) \qquad \exists T > 0, \quad S_T: X_+\{0\} \to X_{++}, $$

$$ (4.14) \qquad \exists T > 0, \quad \int_0^T S(t)dt: X_+\{0\} \to X_{++}. $$

We summarize some possible implications between the previous positivity notions.

**Lemma 4.8.** For a positive semigroup $S$, the following hold:

(1) The given time strong positivity condition (4.13) implies the integral in time strong positivity condition (4.14);

(2) The integral strong positivity condition (4.14) implies the irreducibility condition (4.11), but the reverse implication is false. Similarly, the irreducibility and aperiodicity condition (4.12) implies the irreducibility condition (4.11), but the reverse implication is false;

(3) The irreducibility condition (4.11) is equivalent to the fact that $\mathcal{R}_\mathcal{L}(\lambda): X_+\{0\} \to X_{++}$, for any $\lambda > \lambda_1$, as well as to the fact that $\lambda - \mathcal{L}$ satisfies the strong maximum principle (4.10) for any $\lambda \in \mathbb{R}$.

The result is very classic, at least for strongly positive semigroup, see e.g. [15, Definition C.3.1] or [41, Prop. 14.10]. For the sake of completeness, we however present a short proof.

*Proof of Lemma 4.8.* We prove (1). We assume (4.13) and we fix $g \in X_+\{0\}$, $\phi \in Y_+\{0\}$, so that $\langle S(T)g, \phi \rangle > 0$. Observing that the function $t \mapsto \langle S(t)g, \phi \rangle$ is continuous, there exists $\varepsilon > 0$ such that $\langle S(t)g, \phi \rangle > 0$ for any $t \in [T - \varepsilon, T]$, so that

$$ \left\langle \int_0^T S(t)dt, \phi \right\rangle = \int_0^T \langle S(t)g, \phi \rangle dt > 0. $$

Because $\phi \in Y_+\{0\}$ may be chosen arbitrary, we deduce (4.14).

We prove (2). We assume now (4.14) and we fix $g \in X_+\{0\}$, $\phi \in Y_+\{0\}$, so that

$$ \int_0^T \langle S(t)g, \phi \rangle dt = \left\langle \int_0^T S(t)dtg, \phi \right\rangle > 0, $$

by assumption. We get (4.11) by observing that the function $t \mapsto \langle S(t)g, \phi \rangle$ must be positive somewhere on $[0, T]$. For the reverse implication we refer to [50, 168], where is studied an example of growth-fragmentation operator associated to mitosis satisfying the irreducibility condition (4.11) but not the integral strong positivity condition (4.14) nor the irreducibility and aperiodicity condition (4.12), see also Section 9.

We prove (3). We finally assume (4.11). From the above continuity argument, for any $g \in X_+\{0\}$, $\phi \in Y_+\{0\}$ there exist $\tau > \varepsilon > 0$ such that $\langle S(t)g, \phi \rangle > 0$ for any $t \in [\tau - \varepsilon, \tau + \varepsilon]$. As a
4.3. **The geometry of the first eigenvalue problem.** We come back on and state a result about the geometry of the first eigenvalue.

We consider an operator $\mathcal{L}$ on $X$ which satisfies the conclusion (C1) about the existence of a solution $(\lambda_1, f_1, \phi_1)$ to the first eigentriplet problem. We next assume

(H1') the weak maximum principle

$$ (4.16) \qquad \lambda > \lambda_1, f \in D(\mathcal{L}), (\lambda - \mathcal{L})f \ge 0 \quad \text{imply} \quad f \ge 0 $$

and its Kato's inequalities counterpart

$$ (4.17) \qquad (\operatorname{sign} f) \mathcal{L} f \le \mathcal{L} |f|, \quad (\operatorname{sign}_+ f) \mathcal{L} f \le \mathcal{L} f_+ $$

as well as

(H4) the strong maximum principle

$$ (4.18) \qquad \lambda \in \mathbb{R}, f \in X_+ \cap D(\mathcal{L}), (\lambda - \mathcal{L})f \ge 0 \quad \text{imply} \quad f \gg 0 \text{ or } f = 0. $$

We also assume the same properties for the adjoint operator $\mathcal{L}^*$.

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 of a solution $(\lambda_1, f_1, \phi_1)$ to the first eigentriplet 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 maximum principle (H4).

Then the following hold

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

$$ (4.19) \qquad \|\phi_1\| = 1, \quad \langle \phi_1, f_1 \rangle = 1. $$

ii) $\lambda_1$ is algebraically simple:

$$ (4.20) \qquad N((\mathcal{L} - \lambda_1)^k) = \operatorname{Span}(f_1), \quad \forall k \ge 1, $$

$$ (4.21) \qquad N((\mathcal{L}^* - \lambda_1)^k) = \operatorname{Span}(\phi_1), \quad \forall k \ge 1. $$

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

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

**Remark 4.14.** (1) It is worth emphasizing again that (4.16) and (4.17) 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 (4.18) is true when $S_{\mathcal{L}}$ enjoys additional strong positivity (or irreducibility) condition as formulated in (4.12), (4.13), (4.14) or (4.15). 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 ([238]), 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 [185]) and to the more recent contributions [278, Thm. 5.3], [41, Thm. 14.15] and [231, 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 [278, Thm. 4.3] and [231, Thm. 5.1]. We learnt this kind of technique in the (less abstract and general) proof of the uniqueness part of [313, Lem. 2.1].

(4) From ii), we deduce that $\mathcal{L}$ decomposes according to $X = X_0 \oplus X_1$ with $X_1 := \operatorname{Span} f_1$ and $X_0 := (\operatorname{Span} \phi_1)^\perp = \{f \in X; \langle f, \phi_1 \rangle = 0\}$ in the sense of [229, § 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 $\mathcal{L} : X_1 \to X_1$.