A note on the compactness of the resolvent of the age-diffusion operator

Abstract

The generator of the semigroup associated with linear age-structured population models including spatial diffusion is shown to have compact resolvent.

1 Introduction and main results

The dynamics of a population consisting of individuals structured by age and spatial position is governed by equations of the form

$$\begin{aligned} \partial _t u+ \partial _au \,&= A(a)u, \qquad t>0\, ,\quad a\in (0,a_m)\, , \end{aligned}$$

(1.1a)

$$\begin{aligned} u(t,0)&=\int _0^{a_m}b(a)\, u(t,a)\, \textrm{d}a, \qquad t>0\, , \end{aligned}$$

(1.1b)

$$\begin{aligned} u(0,a)&= u_0(a), \qquad a\in (0,a_m), \end{aligned}$$

(1.1c)

where $u=u(t,a)$ denotes the population density $u:{\mathbb {R}}^+\times J\rightarrow E_0$ with values in a Banach space $E_0$ representing the spatial heterogeneity. The age of individuals is denoted by $a\in J:=[0,a_m]$ with maximal age $a_m\in (0,\infty )$. Spatial movement of individuals is described by the age-dependent (usually: differential) operator A, for which we assume that there is $\rho >0$ with

$$\begin{aligned} A\in C^\rho \big (J,{\mathcal {H}}(E_1,E_0)\big ). \end{aligned}$$

(1.2a)

${\mathcal {H}}(E_1,E_0)$ stands for the space of generators of analytic semigroups on $E_0$ with domain $E_1$ equipped with the operator norm in ${\mathcal {L}}(E_1,E_0)$. Death processes of individuals are incorporated into the operator A and spatial boundary conditions into the domain of definition $E_1$. We impose that

(1.2b)

that is, $E_1$ is a densely and compactly embedded subspace of the Banach space $E_0$. Equation (1.1b) is the birth law with birth rate b, for which we shall assume that

$$\begin{aligned} b\in L_{\infty }\big (J,{\mathcal {L}}(E_\theta )\big ), \quad \theta \in \{0,\vartheta \}, \end{aligned}$$

(1.2c)

for some fixed $\vartheta \in (0,1)$. Here, the space $E_\vartheta := (E_0,E_1)_\vartheta$ is an interpolation space with admissible interpolation functor $(\cdot ,\cdot )_\vartheta$; that is, $E_1$ is dense in $E_\vartheta$ (see [1, I.Section 2.11]). In fact, given any admissible interpolation functor $(\cdot ,\cdot )_\theta$ with $\theta \in (0,1)$ we use the notion $E_\theta := (E_0,E_1)_\theta$ and note from (1.2b) and [1, I.Theorem 2.11.1] that

(1.3)

We set

$$\begin{aligned} {\mathbb {E}}_\theta :=L_1(J,E_\theta ),\quad \theta \in [0,1]. \end{aligned}$$

In (1.1c), $u_0$ represents the initial population. The main features of (1.1) are that the evolution equation (1.1a) has a hyperbolic character if spatial movement is neglected (i.e. $A=0$) or a parabolic character if aging is neglected and that condition (1.1b) is nonlocal with respect to age.

Age-structured population models (with and without spatial diffusion) have been the subject of intensive research in the past; we refer e.g. to [13] and the references therein for more details and concrete examples.

Realistic models for the evolution of age- and spatially structured populations are nonlinear and involve density-dependent vital rates b or nonlinear operators A incorporating nonlinear death processes or nonlinear diffusion. The study of linear equations like (1.1) is important for the understanding of such more complex models. For instance, they arise as (part of the) linearization when investigating stability properties of equilibria [9]. It is well-known that the solutions to the linear problem (1.1) may be represented by a semigroup on the phase space ${\mathbb {E}}_0=L_1(J,E_0)$. We refer to [4, 5, 7, 10, 12, 13] and the references therein for the investigation of this semigroup though this list is far from being complete.

It follows from [12, Theorem 2.3 (c)] that the generator ${\mathbb {A}}:\textrm{dom}({\mathbb {A}})\subset {\mathbb {E}}_0\rightarrow {\mathbb {E}}_0$ of this semigroup is given by

$$\begin{aligned} \textrm{dom}({\mathbb {A}})&:=\bigg \{\psi \in C(J,E_0)\,;\, \exists \, \zeta _\psi \in {\mathbb {E}}_0\, \text {such that}~\psi ~\text {is the mild solution to}\\&\qquad \qquad \qquad \partial _a\psi = A(a) \psi -\zeta _\psi (a),\ a\in J,\ \ \psi (0)=\int _0^{a_m} b(a) \psi (a)\,\textrm{d}a \bigg \} \end{aligned}$$

and

$$\begin{aligned} {\mathbb {A}}\psi := \zeta _\psi . \end{aligned}$$

Mild solution here means that

$$\begin{aligned} \psi (a)=\Pi (a,0)\psi (0)-\int _0^a\Pi (a,\sigma )\,\zeta _\psi (\sigma )\,\textrm{d}\sigma ,\quad a\in J, \end{aligned}$$

where

$$\begin{aligned} \big \{\Pi (a,\sigma )\in {\mathcal {L}}(E_0)\,;\, a\in J,\, 0\le \sigma \le a\big \} \end{aligned}$$

denotes the parabolic evolution operator on $E_0$ with regularity subspace $E_1$ in the sense of [1, Section II.2.1] (existence and uniqueness of $\Pi$ is ensured by (1.2a) and [1, II. Corollary 4.4.2]). With this notation, the linear problem (1.1) can be reformulated as the Cauchy problem

$$\begin{aligned} u'(t)={\mathbb {A}}u(t),\quad t\ge 0,\qquad u(0)=u_0 \end{aligned}$$

(1.4)

in ${\mathbb {E}}_0$. Since ${\mathbb {A}}$ generates a strongly continuous semigroup $(e^{t{\mathbb {A}}})_{t\ge 0}$ on ${\mathbb {E}}_0$, there is for each initial value $u_0\in \textrm{dom}({\mathbb {A}})$ a unique solution

$$\begin{aligned} u\in C^1({\mathbb {R}}^+,{\mathbb {E}}_0)\cap C({\mathbb {R}}^+,D({\mathbb {A}})) \end{aligned}$$

to (1.4), where $D({\mathbb {A}})$ means the domain $\textrm{dom}({\mathbb {A}})$ equipped with the graph norm. Precise information on $D({\mathbb {A}})$ – which is defined implicitly initially – is thus pertinent, also with regard to evolution problems involving time-dependent operators ${\mathbb {A}}={\mathbb {A}}(t)$, see [11]. For instance, one can show [11, Lemma 2.1] that

$$\begin{aligned} \left\{ \psi \in {\mathbb {E}}_1\cap W_1^1(J,E_0)\,;\,\psi (0)=\int _0^{a_m}b(a)\, \psi (a)\, \textrm{d}a\right\} \end{aligned}$$

is a core for ${\mathbb {A}}$; that is, a dense subspace of $D({\mathbb {A}})$. Owing to (1.2b) and [6, Corollary 4], this space is also compactly embedded into ${\mathbb {E}}_0$. Moreover, from [12, Theorem 2.3 (c)] it follows that $D({\mathbb {A}})\hookrightarrow {\mathbb {E}}_\theta$ for each $\theta \in [0,1)$, where the spaces ${\mathbb {E}}_\theta$ play an important role in the study of nonlinear problems as domains of definitions for the nonlinearities. Herein we prove that these embeddings are compact:

Theorem 1.1

Suppose (1.2). Then ${\mathbb {A}}$ has compact resolvent. In fact, for every $\theta \in [0,1)$, the embedding $D({\mathbb {A}})\hookrightarrow {\mathbb {E}}_\theta$ is compact.

Compact resolvent means that $(\lambda -{\mathbb {A}})^{-1}$ is a compact operator on ${\mathbb {E}}_0$ for every $\lambda >0$ large enough. In order to prove Theorem 1.1, the definition of $D({\mathbb {A}})$ entails to derive compactness of the mapping

$$\begin{aligned} \zeta \mapsto \int _0^a\Pi (a,\sigma )\,\zeta (\sigma )\,\textrm{d}\sigma \end{aligned}$$

in ${\mathbb {E}}_0=L_1(J,E_0)$. To this end we shall follow the lines of [2].

Besides compactness of the embedding $D({\mathbb {A}})\hookrightarrow {\mathbb {E}}_0$, Theorem 1.1 also yields information on the spectrum of ${\mathbb {A}}$, e.g. that it consists of eigenvalues only. This information was derived previously in [12] by showing that the semigroup $(e^{t{\mathbb {A}}})_{t\ge 0}$ is eventually compact (under slightly stronger assumptions). However, for perturbations of the form ${\mathbb {A}}+{\mathbb {B}}$ (arising, for instance, when linearizing equations incorporating nonlinear vital rates), the eventual compactness of the corresponding semigroup has recently been obtained in [9], but only for particular perturbations ${\mathbb {B}}$. Nonetheless, Theorem 1.1 ensures for general perturbations ${\mathbb {B}}\in {\mathcal {L}}({\mathbb {E}}_\alpha ,{\mathbb {E}}_0)$ with $\alpha \in [0,1)$ the compactness of the resolvent of ${\mathbb {A}}+{\mathbb {B}}$ and thus provides spectral properties for perturbed operators ${\mathbb {A}}+{\mathbb {B}}$:

Corollary 1.2

Suppose (1.2). Let ${\mathbb {B}}\in {\mathcal {L}}({\mathbb {E}}_\alpha ,{\mathbb {E}}_0)$ for some $\alpha \in [0,1)$. Then ${\mathbb {A}}+{\mathbb {B}}$ with domain $\textrm{dom}({\mathbb {A}})$ generates a strongly continuous semigroup $(e^{t({\mathbb {A}}+{\mathbb {B}})})_{t\ge 0}$ on ${\mathbb {E}}_0$ and has compact resolvent. In particular, the spectrum $\sigma ({\mathbb {A}}+{\mathbb {B}})$ is a pure point spectrum without finite accumulation point.

If $E_0$ is an ordered Banach space and

$$\begin{aligned} A(a)~\text {is resolvent positive for each}~a\in J \end{aligned}$$

(1.5a)

and

$$\begin{aligned} b\in L_\infty (J,{\mathcal {L}}_+(E_0)), \end{aligned}$$

(1.5b)

then the semigroup $(e^{t{\mathbb {A}}})_{t\ge 0}$ generated by ${\mathbb {A}}$ is positive on ${\mathbb {E}}_0$, see [12, Theorem 1.2]; that is, ${\mathbb {A}}$ is resolvent positive. In case that $E_0$ is a Banach lattice, more information on the spectral bound

$$\begin{aligned} s({\mathbb {A}}):=\sup \{\textrm{Re}\,\lambda \,;\,\lambda \in \sigma ({\mathbb {A}})\} \end{aligned}$$

is available, see [3]:

Corollary 1.3

Let $E_0$ be a Banach lattice and suppose (1.2) and (1.5). If ${\mathbb {B}}\in {\mathcal {L}}_+ ({\mathbb {E}}_\alpha ,{\mathbb {E}}_0)$ for some $\alpha \in [0,1)$, then ${\mathbb {A}}+{\mathbb {B}}$ generates a positive semigroup $(e^{t({\mathbb {A}}+{\mathbb {B}})})_{t\ge 0}$ on ${\mathbb {E}}_0$ with $e^{t{\mathbb {A}}} \le e^{t({\mathbb {A}}+{\mathbb {B}})}$ for $t\ge 0$. Moreover, $s({\mathbb {A}})\le s({\mathbb {A}}+{\mathbb {B}})$ and

$$\begin{aligned} (\lambda -{\mathbb {A}})^{-1}\le (\lambda -{\mathbb {A}}-{\mathbb {B}})^{-1},\quad \lambda >s({\mathbb {A}}+{\mathbb {B}}). \end{aligned}$$

If $s({\mathbb {A}}+{\mathbb {B}})>-\infty$, then $s({\mathbb {A}}+{\mathbb {B}})$ is an eigenvalue of ${\mathbb {A}}+{\mathbb {B}}$ possessing an eigenvector $\psi \in \textrm{dom}({\mathbb {A}})$ with $\psi \ge 0$. E.g. this is the case if $b(a)\Pi (a,0)\in {\mathcal {L}}_+(E_0)$ is strongly positive¹ for a in a subset of J of positive measure.

We prove the above statements in the next sections. To this end, we assume throughout the following (1.2).

2 Proof of Theorem 1.1

I. Compact resolvent

First note that, for $\lambda \in {\mathbb {R}}$, the family

$$\begin{aligned} \Pi _{\lambda }(a,\sigma ):=e^{-\lambda (a-\sigma )}\Pi (a,\sigma ),\qquad a\in J,\quad 0\le \sigma \le a, \end{aligned}$$

is the evolution operator associated with $-\lambda +A$. Moreover, according to [1, II. Lemma 5.1.3] there are $\varpi \in {\mathbb {R}}$ and $M_\vartheta \ge 1$ with

$$\begin{aligned} \Vert \Pi (a,\sigma )\Vert _{{\mathcal {L}}(E_\vartheta )}+(a-\sigma )^\vartheta \,\Vert \Pi (a,\sigma )\Vert _{{\mathcal {L}}(E_0,E_\vartheta )}\le M_\vartheta e^{\varpi (a-\sigma )},\quad 0\le \sigma \le a\le a_m, \end{aligned}$$

(2.1)

where $\vartheta \in (0,1)$ stems from (1.2c). We may choose $\lambda >0$ in the resolvent set of ${\mathbb {A}}$ so large such that for the operator

$$\begin{aligned} Q_\lambda :=\int _0^{a_m} b(a)\Pi _{\lambda }(a,0)\,\textrm{d}a\in {\mathcal {L}}(E_\vartheta ) \end{aligned}$$

we have $\Vert Q_\lambda \Vert _{{\mathcal {L}}(E_\vartheta )}<1$ (see (1.2c) and (2.1)). Consider a bounded sequence $(\phi _j)_{j\in {\mathbb {N}}}$ in ${\mathbb {E}}_0$. Then, for $j\in {\mathbb {N}}$,

$$\begin{aligned} \psi _j:=(\lambda -{\mathbb {A}})^{-1}\phi _j\in \textrm{dom}({\mathbb {A}}), \end{aligned}$$

and the definition of ${\mathbb {A}}$ implies

$$\begin{aligned} \psi _j(a) =\Pi _{\lambda }(a,0)\psi _j(0)+\int _0^a\Pi _{\lambda }(a,\sigma ) \phi _j(\sigma )\,\textrm{d}\sigma ,\quad a\in J, \end{aligned}$$

(2.2)

and

$$\begin{aligned} (1-Q_\lambda )\psi _j(0)= \int _0^{a_m} b(a)\int _0^a \Pi _{\lambda }(a,\sigma )\, \phi _j(\sigma ) \,\textrm{d}\sigma \,\textrm{d}a. \end{aligned}$$

(2.3)

In particular, since by (1.2c) and (2.1) we have

$$\begin{aligned} \Bigg \Vert&\int _0^{a_m} b(a)\int _0^a \Pi _{\lambda }(a,\sigma )\, \phi _j(\sigma ) \,\textrm{d}\sigma \,\textrm{d}a \Bigg \Vert _{E_\vartheta } \\&\le M_\vartheta \,\Vert b\Vert _{L_\infty (J,{\mathcal {L}}(E_\vartheta ))}\, \int _0^{a_m} \int _0^a e^{(-\lambda +\varpi ) (a-\sigma )}(a-\sigma )^{-\vartheta }\, \Vert \phi _j(\sigma ) \Vert _{E_0}\,\textrm{d}\sigma \,\textrm{d}a\\&= M_\vartheta \,\Vert b\Vert _{L_\infty (J,{\mathcal {L}}(E_\vartheta ))}\,\int _0^{a_m} \Vert \phi _j(\sigma ) \Vert _{E_0} \int _\sigma ^{a_m} e^{(-\lambda +\varpi ) (a-\sigma )} (a-\sigma )^{-\vartheta } \,\textrm{d}a\,\textrm{d}\sigma \le c \Vert \phi _j\Vert _{{\mathbb {E}}_0}, \end{aligned}$$

the sequence

$$\begin{aligned} \left( \int _0^{a_m} b(a)\int _0^a \Pi _{\lambda }(a,\sigma )\, \phi _j(\sigma ) \,\textrm{d}\sigma \,\textrm{d}a\right) _{j\in {\mathbb {N}}} \end{aligned}$$

is bounded in $E_\vartheta$. Thus, (2.3) and $\Vert Q_\lambda \Vert _{{\mathcal {L}}(E_\vartheta )}<1$ imply that

$$\begin{aligned} (\psi _j(0))_{j\in {\mathbb {N}}}\ \text { is bounded in}\ E_\vartheta . \end{aligned}$$

(2.4)

We adopt the proof of [2] in order to show that the set $\{u_j\,;\, j\in {\mathbb {N}}\}$, given by

$$\begin{aligned} u_j(a):=\int _0^a\Pi _{\lambda }(a,\sigma ) \phi _j(\sigma )\,\textrm{d}\sigma ,\quad a\in J,\quad j\in {\mathbb {N}}, \end{aligned}$$

is relatively compact in ${\mathbb {E}}_0$. We split this into two steps:

(i) Let $\mu >0$ be fixed and define

$$\begin{aligned} v_j^\mu (a):=\Pi _{\lambda }(\mu +a,a)u_j(a)=\int _0^a\Pi _{\lambda }(\mu +a,\sigma ) \phi _j(\sigma )\,\textrm{d}\sigma ,\quad a\in J,\quad j\in {\mathbb {N}}, \end{aligned}$$

where we used the evolution property²

$$\begin{aligned} \Pi _{\lambda }(\mu +a,a)\Pi _{\lambda }(a,\sigma )=\Pi _{\lambda }(\mu +a,\sigma ),\quad \sigma \le a\le \mu +a. \end{aligned}$$

Analogously to the derivation of (2.4), the sequence

$$\begin{aligned} (v_j^\mu )_{j\in {\mathbb {N}}}~\text {is bounded in}~{\mathbb {E}}_\vartheta . \end{aligned}$$

(2.5)

Recall from (1.3) that the space $E_\vartheta$ embeds compactly into $E_0$.

For the equi-integrability recall from [1, II. Section 2.1] that the evolution operator has the continuity property $\Pi \in C(\Delta _J^*,{\mathcal {L}}(E_0))$, where $\Delta _J^*:=\{(a,\sigma )\in J\times J\,;\, \sigma < a\}$. Uniform continuity on compact subsets of $\Delta _J^*$ implies that, given $\varepsilon >0$, there is $\eta :=\eta (\varepsilon ,\mu )>0$ such that

$$\begin{aligned} \Vert \Pi _{\lambda }(a_1,\sigma _1)-\Pi _{\lambda }(a_2,\sigma _2)\Vert _{{\mathcal {L}}(E_0)}\le \varepsilon \end{aligned}$$

whenever $(a_i,\sigma _i)\in \Delta _J^*$, $\mu \le a_i-\sigma _i$ for $i=1,2$ and $\vert (a_1,\sigma _1)-(a_2,\sigma _2)\vert \le \eta$. We use this and (2.1) to derive, for $h\in (0,\eta )$ with $0<a\le a+h\le a_m$,

$$\begin{aligned} \Vert v_j^\mu (a+h)-v_j^\mu (a)\Vert _{E_0}&\le \int _a^{a+h}\Vert \Pi _{\lambda }(\mu +a+h,\sigma )\Vert _{{\mathcal {L}}(E_0)}\,\Vert \phi _j(\sigma )\Vert _{E_0}\,\textrm{d}\sigma \\&\quad + \int _0^{a}\Vert \Pi _{\lambda }(\mu +a+h,\sigma )-\Pi _{\lambda }(\mu +a,\sigma )\Vert _{{\mathcal {L}}(E_0)}\,\Vert \phi _j(\sigma )\Vert _{E_0}\,\textrm{d}\sigma \\&\le c(a_m)\int _a^{a+h} \Vert \phi _j(\sigma )\Vert _{E_0}\,\textrm{d}\sigma +\varepsilon \int _0^{a}\Vert \phi _j(\sigma )\Vert _{E_0}\,\textrm{d}\sigma \end{aligned}$$

and therefore

$$\begin{aligned} \int _0^{a_m}\Vert {\tilde{v}}_j^\mu (a+h)-{\tilde{v}}_j^\mu (a)\Vert _{E_0}\,\textrm{d}a&\le h c(a_m)\Vert \phi _j \Vert _{{\mathbb {E}}_0}+\varepsilon a_m \Vert \phi _j \Vert _{{\mathbb {E}}_0}, \end{aligned}$$

with tilde referring to trivial extension. Since $(\phi _j)_{j\in {\mathbb {N}}}$ is bounded in ${\mathbb {E}}_0$ and $\varepsilon >0$ was arbitrary, we deduce

$$\begin{aligned} \lim _{h\rightarrow 0}\, \sup _{j\in {\mathbb {N}}}\int _0^{a_m}\Vert \tilde{v}_j^\mu (a+h)-{\tilde{v}}_j^\mu (a)\Vert _{E_0}\,\textrm{d}a =0. \end{aligned}$$

That is, $\{v_j^\mu \,;\, j\in {\mathbb {N}}\}$ is equi-integrable. Therefore, taking into account (2.5) we are in a position to apply [6, Theorem 3] and derive that

$$\begin{aligned} \{v_j^\mu \,;\, j\in {\mathbb {N}}\}\ \text { is relatively compact in } {\mathbb {E}}_0 \text { for } \mu >0. \end{aligned}$$

(2.6)

(ii) We consider the limit $\mu \rightarrow 0$. Given arbitrary $\varepsilon \in (0,a_m)$, we argue as in part (i) to find $\eta (\varepsilon )>0$ such that

$$\begin{aligned} \Vert \Pi _{\lambda }(a_1,\sigma _1)-\Pi _{\lambda }(a_2,\sigma _2)\Vert _{{\mathcal {L}}(E_0)}\le \varepsilon \end{aligned}$$

whenever $(a_i,\sigma _i)\in \Delta _J^*$, $\varepsilon \le a_i-\sigma _i$ for $i=1,2$ and $\vert (a_1,\sigma _1)-(a_2,\sigma _2)\vert \le \eta (\varepsilon )$. Thus, for $0<\mu <\eta (\varepsilon )$ and $a\in J$ with $a\ge \varepsilon$, we obtain from (2.1) that

$$\begin{aligned} \Vert v_j^\mu (a)-u_j(a)\Vert _{E_0}&\le \int _0^{a-\varepsilon }\Vert \Pi _{\lambda }(\mu +a,\sigma )-\Pi _{\lambda }(a,\sigma )\Vert _{{\mathcal {L}}(E_0)}\,\Vert \phi _j(\sigma )\Vert _{E_0}\,\textrm{d}\sigma \\&\quad + \int _{a-\varepsilon }^a \big (\Vert \Pi _{\lambda }(\mu +a,\sigma )\Vert _{{\mathcal {L}}(E_0)}+\Vert \Pi _{\lambda }(a,\sigma )\Vert _{{\mathcal {L}}(E_0)}\big )\,\Vert \phi _j(\sigma )\Vert _{E_0}\,\textrm{d}\sigma \\&\le \varepsilon \int _0^{a-\varepsilon } \Vert \phi _j(\sigma )\Vert _{E_0}\,\textrm{d}\sigma + c(a_m)\int _{a-\varepsilon }^a \Vert \phi _j(\sigma )\Vert _{E_0}\,\textrm{d}\sigma , \end{aligned}$$

while, for $0\le a\le \varepsilon$, we have

$$\begin{aligned} \Vert v_j^\mu (a)-u_j(a)\Vert _{E_0}&\le c(a_m)\int _0^{\varepsilon } \Vert \phi _j(\sigma )\Vert _{E_0}\,\textrm{d}\sigma . \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert v_j^\mu -u_j\Vert _{{\mathbb {E}}_0}&\le c(a_m)\, \varepsilon \, \Vert \phi _j\Vert _{{\mathbb {E}}_0}+\varepsilon \, a_m\, \Vert \phi _j\Vert _{{\mathbb {E}}_0} \end{aligned}$$

so that, since $\varepsilon >0$ was arbitrary,

$$\begin{aligned} \lim _{\mu \rightarrow 0}\, \sup _{j\in {\mathbb {N}}}\, \Vert v_j^\mu -u_j\Vert _{{\mathbb {E}}_0}=0. \end{aligned}$$

Together with (2.6) we conclude that $\{u_j\,;\, j\in {\mathbb {N}}\}$ is relatively compact in ${\mathbb {E}}_0$.

(iii) Finally, since

$$\begin{aligned} \Vert \Pi _{\lambda } (a+h,0)-\Pi _{\lambda } (a,0)\Vert _{{\mathcal {L}}(E_\vartheta ,E_0)}\le c h^{\vartheta },\quad 0\le a\le a+h\le a_m, \end{aligned}$$

according to [1, II.Equation(5.3.8)], we infer from (2.4) and the Arzelà–Ascoli Theorem that $(\Pi _{\lambda } (\cdot ,0)\psi _j(0))_{j\in {\mathbb {N}}}$ is relatively compact in $C(J,E_0)$. Consequently, the previous step (ii) and (2.2) entail that $(\psi _j)_{j\in {\mathbb {N}}}$ is relatively compact in ${\mathbb {E}}_0$. Therefore, ${\mathbb {A}}$ has compact resolvent.

II. Compact embedding

Since ${\mathbb {A}}$ has compact resolvent, the embedding $D({\mathbb {A}})\hookrightarrow {\mathbb {E}}_0$ is compact. The compact embedding into ${\mathbb {E}}_\theta$ for $\theta \in (0,1)$ fixed follows by an interpolation argument (note, however, that ${\mathbb {E}}_\theta$ need not be obtained exactly by an interpolation functor). Indeed, choosing $\varepsilon >0$ with $\theta <1-3\varepsilon$ we set $\theta _k:=\theta (1-k\varepsilon )^{-1}$ for $k=1,2,3$ so that $0<\theta _1<\theta _2<\theta _3<1$. Then [1, I. Equation (2.5.2)] yields for the continuous interpolation functor $(\cdot ,\cdot )_{\theta _3,\infty }^0$ that

$$\begin{aligned} \big ({\mathbb {E}}_0,{\mathbb {E}}_{1-\varepsilon }\big )_{\theta _3,\infty }^0\hookrightarrow \big ({\mathbb {E}}_0,{\mathbb {E}}_{1-\varepsilon }\big )_{\theta _2,1}=\big (L_1(J,E_0),L_1(J,E_{1-\varepsilon })\big )_{\theta _2,1} \end{aligned}$$

while [8, Theorem 1.18.4] ensures for the real interpolation functor $(\cdot ,\cdot )_{\theta _2,1}$ that

$$\begin{aligned} \big (L_1(J,E_0),L_1(J,E_{1-\varepsilon })\big )_{\theta _2,1}\doteq L_1\big (J,(E_0,E_{1-\varepsilon })_{\theta _2,1}\big ). \end{aligned}$$

Finally, [1, I. Remark 2.11.2 (a)] implies

$$\begin{aligned} L_1\big (J,(E_0,E_{1-\varepsilon })_{\theta _2,1}\big )\hookrightarrow L_1\big (J,E_{(1-\varepsilon )\theta _1}\big )={\mathbb {E}}_\theta . \end{aligned}$$

Gathering these findings we obtain the continuous embedding

$$\begin{aligned} \big ({\mathbb {E}}_0,{\mathbb {E}}_{1-\varepsilon }\big )_{\theta _3,\infty }^0\hookrightarrow {\mathbb {E}}_\theta \end{aligned}$$

so that there is $c>0$ with

$$\begin{aligned} \Vert \phi \Vert _{{\mathbb {E}}_\theta }\le c \Vert \phi \Vert _{{\mathbb {E}}_0}^{1-\theta _3}\, \Vert \phi \Vert _{{\mathbb {E}}_{1-\varepsilon }}^{\theta _3},\quad \phi \in {\mathbb {E}}_{1-\varepsilon }. \end{aligned}$$

(2.7)

Now, if $(\phi _j)_{j\in {\mathbb {N}}}$ is a bounded sequence in $D({\mathbb {A}})$, then it is also bounded in ${\mathbb {E}}_{1-\varepsilon }$ due to the continuous embedding $D({\mathbb {A}})\hookrightarrow {\mathbb {E}}_{1-\varepsilon }$. Moreover, since the embedding $D({\mathbb {A}})\hookrightarrow {\mathbb {E}}_0$ is compact, we may assume without loss of generality that it is a Cauchy sequence in ${\mathbb {E}}_0$. According to (2.7) it is also a Cauchy sequence in ${\mathbb {E}}_\theta$ and thus converges. Consequently, the embedding $D({\mathbb {A}})\hookrightarrow {\mathbb {E}}_\theta$ is compact. This proves Theorem 1.1. $\square$

3 Proof of Corollary 1.2

It was shown in [12, Theorem 2.1] that ${\mathbb {A}}+{\mathbb {B}}$ with domain $\textrm{dom}({\mathbb {A}})$ generates a strongly continuous semigroup on ${\mathbb {E}}_0$ when ${\mathbb {B}}\in {\mathcal {L}}({\mathbb {E}}_\alpha ,{\mathbb {E}}_0)$ for some $\alpha \in [0,1)$. Thus, for $\lambda >0$ sufficiently large, both $\lambda -{\mathbb {A}}$ and $\lambda -{\mathbb {A}}-{\mathbb {B}}$ are invertible with

$$\begin{aligned} (\lambda -{\mathbb {A}}-{\mathbb {B}})^{-1}= (\lambda -{\mathbb {A}})^{-1}\big (1-{\mathbb {B}}(\lambda -{\mathbb {A}})^{-1}\big )^{-1}. \end{aligned}$$

Since $(\lambda -{\mathbb {A}})^{-1}\in {\mathcal {L}}({\mathbb {E}}_0)$ is compact according to Theorem 1.1 while

$$\begin{aligned} \big (1-{\mathbb {B}}(\lambda -{\mathbb {A}})^{-1}\big )^{-1}\in {\mathcal {L}}({\mathbb {E}}_0) \end{aligned}$$

since $D({\mathbb {A}}) \hookrightarrow {\mathbb {E}}_\alpha$ (see [12, Theorem 2.3]), it follows that $(\lambda -{\mathbb {A}}-{\mathbb {B}})^{-1}$ is compact. This proves Corollary 1.2. $\square$

4 Proof of Corollary 1.3

Let $E_0$ be a Banach lattice and assume, in addition to (1.2), also (1.5). Corollary 1.3 follows from [3, Proposition 12.11] (except that therein the perturbation ${\mathbb {B}}\in {\mathcal {L}}_+({\mathbb {E}}_0)$ is a bounded operator on ${\mathbb {E}}_0$), the proof is the same. Indeed, it was shown for ${\mathbb {B}}\in {\mathcal {L}}_+({\mathbb {E}}_\alpha ,{\mathbb {E}}_0)$ in [12, Theorem 1.2] that the semigroup $(e^{t({\mathbb {A}}+{\mathbb {B}})})_{t\ge 0}$ generated by ${\mathbb {A}}+{\mathbb {B}}$ is positive and satisfies

$$\begin{aligned} e^{t({\mathbb {A}}+{\mathbb {B}})}\phi =e^{t{\mathbb {A}}}\phi +\int _0^te^{(t-s){\mathbb {A}}}\,{\mathbb {B}}\, e^{s({\mathbb {A}}+{\mathbb {B}})}\,\phi \,\textrm{d}s,\quad t\ge 0,\quad \phi \in {\mathbb {E}}_0. \end{aligned}$$

Therefore, $e^{t{\mathbb {A}}} \le e^{t({\mathbb {A}}+{\mathbb {B}})}$ for $t\ge 0$. We then argue as in [3, Proposition 12.11] that the Laplace transform formula [3, Theorem 12.7] implies

$$\begin{aligned} (\lambda -{\mathbb {A}})^{-1}\le (\lambda -{\mathbb {A}}-{\mathbb {B}})^{-1},\quad \lambda >\max \{s({\mathbb {A}}+{\mathbb {B}}),s({\mathbb {A}})\}, \end{aligned}$$

and then $s({\mathbb {A}}+{\mathbb {B}})\ge s({\mathbb {A}})$ using [3, Corollary 12.9].

Finally, that $s({\mathbb {A}}+{\mathbb {B}})>-\infty$ is an eigenvalue with positive eigenvector follows from the Krein–Rutman Theorem [3, Theorem 12.15] since ${\mathbb {A}}+{\mathbb {B}}$ is resolvent positive and has a compact resolvent according to Corollary 1.2. In particular, if $b(a)\Pi (a,0)\in {\mathcal {L}}_+(E_0)$ is strongly positive for a in a subset of J of positive measure, then $s({\mathbb {A}})\in {\mathbb {R}}$ according to [12, Proposition 4.2]. This proves Corollary 1.3. $\square$