A Note on the Compactness of the Resolvent of the Age-Diffusion Operator

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


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 where u = u(t, a) denotes the population density u : R + × J → E 0 with values in a Banach space E 0 representing the spatial heterogeneity.The age of individuals is denoted by a ∈ J := [0, a m ] with maximal age a m ∈ (0, ∞).Spatial movement of individuals is described by the age-dependent (usually: differential) operator A, for which we assume that there is ρ > 0 with A ∈ C ρ J, H(E 1 , E 0 ) . (1.2a) 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 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 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 for some fixed ϑ ∈ (0, 1).Here, the space E ϑ := (E 0 , E 1 ) ϑ is an interpolation space with admissible interpolation functor (•, •) ϑ (see [1]).In fact, given any admissible interpolation functor (•, •) θ with θ ∈ (0, 1) we use the notion E θ := (E 0 , E 1 ) θ and set 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 [12].It is well-known that the solutions to the linear problem (1.1) may be represented by a semigroup on the phase space E 0 = L 1 (J, E 0 ).We refer to [13,7,4,5,10,9] and the references therein for the investigation of this semigoup though this list is far from being complete.
It follows from [10,Theorem 2.3 (c)] that the generator A : dom(A) ⊂ E 0 → E 0 of this semigroup is given by where Π(a, σ) ∈ L(E 0 ) ; a ∈ J , 0 ≤ σ ≤ a denotes the parabolic evolution operator on E 0 with regularity subspace E 1 in the sense of [ to (1.3), where D(A) means the domain dom(A) equipped with the graph norm.Precise information on D(A) -which is defined implicitly initially -is thus pertinent, also with regard to evolution problems involving time-dependent operators A = A(t), see [11].For instance, one can show [11, Lemma 2.1] that is a core for A; that is, a dense subspace of D(A).Owing to (1.2b) and [6,Corollary 4], this space is also compactly embedded into E 0 .Moreover, from [10, Theorem 2.3 (c)] it follows that D(A) ֒→ E θ for each θ ∈ [0, 1), where the spaces E θ 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 A has compact resolvent.In fact, for every θ ∈ [0, 1), the embedding D(A) ֒→ E θ is compact.
Compact resolvent means that (λ − A) −1 is a compact operator on E 0 for every λ > 0 large enough.In order to prove Theorem 1.1, the definition of D(A) entails to derive compactness of the mapping To this end we shall follow the lines of [2].
Besides compactness of the embedding D(A) ֒→ E 0 , Theorem 1.1 also yields information on the spectrum of A, e.g. that it consists of eigenvalues only.This information was derived previously in [10] by showing that the semigroup (e tA ) t≥0 is eventually compact (under slightly stronger assumptions).However, for perturbations of the form A + B (arising, for instance, when linearizing equations incorporating nonlinear vital rates), the eventual compactness of the corresponding semigroup has recently been obtained in [12], but only for particular perturbations B. Nonetheless, Theorem 1.1 ensures for general perturbations B ∈ L(E α , E 0 ) with α ∈ [0, 1) the compactness of the resolvent of A + B and thus provides spectral properties for perturbed operators A + B: Corollary 1.2.Suppose (1.2).Let B ∈ L(E α , E 0 ) for some α ∈ [0, 1).Then A + B with domain dom(A) generates a strongly continuous semigroup (e t(A+B) ) t≥0 on E 0 and has compact resolvent.In particular, the spectrum σ(A + B) is a pure point spectrum without finite accumulation point.
If E 0 is an ordered Banach space and then the semigroup (e tA ) t≥0 generated by A is positive on E 0 , see [10]; that is, A is resolvent positive.In case that E 0 is a Banach lattice, more information on the spectral bound is available, see [3]: Corollary 1.3.Let E 0 be a Banach lattice and suppose (1.2) and (1.4).
for some α ∈ [0, 1), then A + B generates a positive semigroup (e t(A+B) ) t≥0 on E 0 with e tA ≤ e t(A+B) for t ≥ 0.Moreover, s(A) ≤ s(A + B) and ) is an eigenvalue of A + B possessing an eigenvector ψ ∈ dom(A) with ψ ≥ 0. E.g. this is the case if b(a)Π(a, 0) ∈ L + (E 0 ) is strongly positive 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 is the evolution operator associated with −λ+A.Moreover, according to [1, II.Lemma 5.1.3]there are ̟ ∈ R and M ϑ ≥ 1 with We may choose λ > 0 in the resolvent set of A so large such that for the operator ) and (2.1)).Consider a bounded sequence (φ j ) j∈N in E 0 .Then, for j ∈ N, , and the definition of A implies Π λ (a, σ) φ j (σ) dσ da We adopt the proof of [2] in order to prove that the set {u j ; j ∈ N}, given by is relatively compact in E 0 .We split this into two steps: (i) Let µ > 0 be fixed and define where we used the evolution property 1 Analogously to the derivation of (2.4), the sequence Note that the space E ϑ embeds compactly into E 0 due to (1.2b).
For the equi-integrability recall from [1, Section II.2.1] that the evolution operator has the continuity property Π ∈ C(∆ * J , L(E 0 )), where ∆ * J := {(a, σ) ∈ J ×J ; σ < a}.Uniform continuity on compact subsets of ∆ * J implies that, given ε > 0, there is η We use this and (2.1) to derive, for h ∈ (0, η) with 0 and therefore with tilde referring to trivial extension.Since (φ j ) j∈N is bounded in E 0 and ε > 0 was arbitrary, we deduce That is, {v µ j ; j ∈ N} is equi-integrable.Therefore, taking into account (2.5) we are in a position to apply [6, Theorem 3] and derive that (ii) We consider the limit µ → 0. Given arbitrary ε ∈ (0, a m ), we argue as in part (i) The uniqueness assertion of [1, II.Corollary 4.4.2]ensures that the corresponding evolution operator then also extends Π.

whenever (a
. Thus, for 0 < µ < η(ε) and a ∈ J with a ≥ ε, we obtain from (2.1) that Therefore, Together with (2.6) we conclude that {u j ; j ∈ N} is relatively compact in E 0 .

Proof of Corollary 1.3
Let E 0 be a Banach lattice and assume, in addition to (1.2), also (1.4).Corollary 1.3 follows from [3,Proposition 12.11] (except that therein the perturbation B ∈ L + (E 0 ) is a bounded operator on E 0 ), the proof is the same.Indeed, it was shown for B ∈ L + (E α , E 0 ) in [10, Theorem 1.2] that the semigroup (e t(A+B) ) t≥0 generated by A + B is positive and satisfies e t(A+B) φ = e tA φ + t 0 e (t−s)A) B e s(A+B) φ ds , t ≥ 0 , φ ∈ E 0 .
Therefore, e tA ≤ e t(A+B) for t ≥ 0. We then argue as in [3,Proposition 12.11]  Finally, that s(A + B) > −∞ is an eigenvalue with positive eigenvector follows from the Krein-Rutman Theorem [3,Theorem 12.15] since A + B is resolvent positive and has a compact resolvent according to Corollary 1.2.In particular, if b(a)Π(a, 0) ∈ L + (E 0 ) is strongly positive for a in a subset of J of positive measure, then s(A) ∈ R according to [10,Proposition 4.2].This proves Corollary 1.3.
Key words and phrases.Age structure, diffusion, semigroups.