# Some Compactness and Interpolation Results for Linear Boltzmann Equation

Authors: Nadjeh Redjel; Abdelkader Dehici
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We discuss some compactness results inLp(1≤p<∞) spaces related to the spectral theory of neutron transport equations for general classes of collision operators and Radon measures having velocity spaces as supports covering most physical models. We show in particular that the asymptotic spectrum of the transport operator is independent of p.


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## Body

## 1. Introduction and Notations

The Boltzmann equation (1872) is an integrodifferential equation of the kinetic theory which is devoted to the study of evolutionary behavior of the gas in the one particle phase space of position and velocity. The time evolution of the state of a gas which is contained in a vesselD bounded by solid walls is determined on one hand by the behavior of the gas molecules at collisions with each other and on the other hand by the influence of the walls as well as by external forces; in the case where there are no external forces, this state is described by a scalar function f(x,v,t) which models the density function of gas particles having position x∈D and velocity v∈R3 at time t∈R. The integral of this function ∫‍∫D×R3‍fx,v,tdxdv gives the expectation value (statistical average) of the total mass of gas contained in the phase space D×R3. Under some assumptions, function f must satisfy the Boltzmann equation⋆∂f∂tx,v,t=-v·∇xfx,v,t+Jfx,·,tvcompleted by boundary and initial conditions. The first term in ⋆ is called streaming operator which is responsible for the motion of the particles between collisions, while the second one J(f(x,·,t)), which is bilinear, describes the mechanism of collisions. A solution to the initial boundary value problem for ⋆ and a proof of H-theorem are given by treating it under its abstract form (for more details, see [1]).This equation is applied also to the transport of photons involved in studies of nuclear reactors, including calculations on the protection against radiation and calculations of warm-up of materials. The quantum behavior of neutrons occurs in collisions with nuclei, but for physicists these events of collisions can be considered as one-time events and instantaneous, which only the consequences are interested in. According to the energy of the incident neutron and the nucleus with which it interacts, different types of reactions can occur. The neutron can be absorbed or broadcasted or it causes the fission of the nucleus. Each reaction is characterized by the microscopic cross section. Between collisions, neutrons behave as classical particles, described by their position and speed. Uncharged (neutral particles), they move in a straight line at least for short distances for which we neglect the effect of the gravitation. The neutronic equations are naturally linear. Indeed, the neutron-neutron interactions can be neglected vis-a-vis neutron-matter interactions. The relationship between the neutron density and the density of the propagation medium (water, uranium oxyde,…) is of the order 10-15, which justifies this approximation. This assumption leads to simplifying the nonlinear version of the Boltzmann equation used in the kinetic theory of gases.Without delayed neutrons, these equations can be written under the form(1)∂ψ∂tx,v,t+v·∇xψx,v,t-σvψx,v,t+∫Vκx,v,v′ψx,v′,tdμv′=0with initial data ψ(x,v,0)=ψ0(x,v), where (x,v)∈D×V. D is a smooth open subset of Rn and μ(·) is a positive Radon measure on Rn such that μ({0})=0 and V (admissible velocity space) denotes the support of μ. The function ψ(x;v;t) describes the distribution of the neutrons in a nuclear reactor occupying the region D. The functions σ(·) and κ(·,·,·) are called, respectively, the collision frequency and the scattering kernel.Here, the boundary conditions which represent the interaction between the particles and ambient medium are given by a boundary bounded operatorH satisfying (2)ψ-=Hψ+,where ψ- (resp., ψ+) is the restriction of ψ to Γ- (resp., Γ+) with Γ- (resp., Γ+) being the incoming (resp., outcoming) part of the phase space boundary and H is a linear bounded operator from a suitable function space on Γ+ to a similar one on Γ-. The classical boundary conditions (vacuum boundary, specular reflections, diffuse reflections, and periodic and mixed type boundary conditions) are special examples of our framework.Let(x,v)∈D¯×V. We define the positive real numbers t±(x;v) by(3)t±x,v=sup⁡t>0;x±sv∈D,∀0<s<t.Physically, t±(x,v) is the time taken by a neutron initially in x∈D with animated speed ±v to achieve (for the first time) the boundary of D.We denote byΓ± the set(4)Γ±=x,v∈∂D×V;±v·nx≥0,where nx is the outer unit normal vector at x∈∂D.Let1≤p<∞; we introduce the functional spaces (5)Wp=ψ∈Xpsuch  thatv·∇xψ∈Xp,where(6)Xp≔LpD×V;dxdμv.The spaces of traces are Lp±≔Lp(Γ±;|v·nx|dγ(x)dμ(v)). Here dγ(·) is the Lebesgue measure on ∂D.Recall that, for everyψ∈Wp, we can define the traces ψ± on Γ±; unfortunately, these traces do not belong to Lp±. The traces lie only in Lp,loc⁡± or precisely in a certain weighted Lp space (see [2–4], for details).Define(7)Wp~=ψ∈Wp;ψ±∈Lp±.In this case H∈L(Lp+,Lp-)(1≤p<∞) and the associated advection operator TH is given as follows: (8)TH:DTH⊆Xp⟶Xp,φ⟶THφ=-v·∇xφx,v-σvφx,v,with domain (9)DTH=ψ∈Wp~such thatψ-=Hψ+,where the collision frequency σ(·)∈L+∞(V) (in other words, a positive bounded function).Letλ∈C; consider the boundary value problem(10)λψx,v+v·∇xψx,v+σvψx,v=φx,v,ψ-=Hψ+,where φ∈Xp and the unknown ψ must belong to D(TH). Let(11)λ⋆≔μ-essinfv∈V⁡σv.For Reλ+λ⋆>0, (10) can be solved formally by (12)ψx,v=ψx-t-x,vv,ve-λ+σvt-x,v+∫0t-x,ve-λ+σvsφx-sv,vds.Moreover, if (x,v)∈Γ+, (10) becomes (13)ψ+x,v=ψ-e-λ+σvτx,v+∫0τx,ve-λ+σvsφx-sv,vds,where τ(x,v)=t+(x,v)+t-(x,v). On the other hand, for every (x,v)∈D¯×V, we have (x-t-(x,v)v,v)∈Γ- (for more details on the time numbers t+, t-, and τ, see [1]).For the abstract formulation of (12) and (13), we define the following operators depending on the parameter λ: (14)Mλ:Lp-⟶Lp+,u⟶Mλu≔ue-λ+σvτx,v;Bλ:Lp-⟶Xp,u⟶Bλu≔ue-λ+σvt-x,v;Gλ:Xp⟶Lp+,φ⟶∫0τx,ve-λ+σvsφx-sv,vds;Cλ:Xp⟶Xp,φ⟶∫0t-x,ve-λ+σvsφx-sv,vds.Straightforward calculations using Hölder’s inequality show that all these operators are bounded on their respective spaces. More precisely, we have, for Reλ>-λ⋆,(15)Mλ≤1,Bλ≤pReλ+λ⋆-1/p,Gλ≤qReλ+λ⋆-1/q,Cλ≤1Reλ+λ⋆1p+1q=1.


### 1.1. Collision Operators

The collision operatorK given as a perturbation of the advection transport operator TH is defined on Xp by (16)K:Xp⟶Xpψ⟶∫Vκx,v,v′ψx,v′dμv′.Note that the operator K is local in x; it describes the physics scattering and production of particles (fission), so it can be viewed as mapping: (17)K·:x∈D⟶Kx∈LLpV.We assume that K(·) is strongly measurable,(18)x∈D⟶Kxφ∈LpVis measurable for anyφ∈LpV,and bounded, (19)esssupx∈D⁡KxLLpV<∞.It follows that K defines a bounded operator on the space Lp(D×V) according to the formula(20)φ∈LpD×V(Lp(D×V)≃Lp(D;Lp(V))) and(21)KxLLpD×V≤esssupx∈D⁡KxLLpV.The final assumption on K is (22)Kx∈KLpValmost everywhere,where K(Lp(V)) denotes the set of compact linear operators on the space Lp(V).We give now the concept of regular collision operators introduced by Mokhtar-Kharroubi [5].Definition 1.
A collision operator,(23)K·:x∈D⟶Kx∈LLpV,is said to be regular if K(x) is compact on Lp(V) almost everywhere on D and (24)K·:x∈D⟶LLpVis a “Bochner measurable function”.The interest of the class of regular collision operators lies in the following lemma.Lemma 2 (see [5, Proposition 4.1]).
A regular collision operatorK can be approximated, in the uniform topology, by a sequence {Kn} of collision operators with kernels of the form (25)∑i∈Ifixgiξhiξ′,where fi∈L∞(D), gi∈Lp(V) and hi∈Lq(V)(1/p+1/q=1) (I is finite).It is easy to observe that (1) can be written under the following abstract Cauchy problem:(26)∂ψ∂t=TH+Kψt,t>0,ψ0=ψ0.Spectral theory of transport operators has known a major development since the pioneering papers of Lehner and Wing and Jörgens in the late 1950s [6–8]. A considerable literature has been devoted to the spectral analysis of the transport operator. This one is studied by means of the nature of the parameters of the equation (nature of boundary conditions, nature of the domain of positions or velocity space, and nature of the collision operator). Let us quote, for example, [1, 4–6, 9–48].In general, the time asymptotic behavior of solutions of (1) is analyzed under two angles: resolvent approach and the semigroup approach.(1) Resolvent Approach. For 1<p<∞, this approach is based essentially on the compactness (or the compactness of one iterate) of the bounded linear operator (λ-TH)-1K. Indeed, Vidav [44] observed that if this condition is satisfied, it leads via an analytic Fredholm alternative to the fact that the set σ(TH+K)∩{λ∈C:Reλ>sH} (σ is the spectrum, while sH is the spectral bound of the operator TH) composed (at most) a set of isolated eigenvalues with finite algebraic multiplicities {λi}i∈J, where {λi,Reλi≥α} is a finite set for each α>sH. If p=1, it suffices to treat the weak compactness by taking into account the fact that the square of weakly compact operator on this space is compact [49, Corollary 13, p. 510]. Recall that, among relevant results in this direction, we can cite the works of Mokhtar-Kharroubi [5, 36], Latrach [24–27], and Song [43].Thus, ifTH generates a c0-semigroup (U(t);t≥0), by Dyson-Phillips theorem of perturbation, TH+K generates a c0-semigroup (V(t);t≥0) given by the following formula (see [50, Corollary 7.5, p. 29]):(27)Vtψ0=12iπlimγ→∞⁡∫v-iγv+iγeλtλ-T-K-1ψ0dλ,t>0,where v is sufficiently large by deforming the contour of integration in Dunford’s formula. Recover the residues corresponding to the poles (eigenvalues of TH+K); we can obtain a good comprehension of the asymptotic behavior of solution when the initial data ψ0 belongs to D(TH+K)2 (unfortunately this regular condition is not natural).(2) Semigroup Approach. Even if σ(TH+K)∩{λ∈C:Reλ>sH} is reduced to isolated eigenvalues of finite algebraic multiplicities, the set σ(V(t))∩{η∈C:|η|>esHt} can contain the continuous spectrum due to the absence of a spectral mapping theorem for the mapping λ→etλ. Vidav [45] has shown that the time asymptotic behavior of V(t)t≥0 is connected to the analysis of its spectrum and the compactness of remainder terms of the Dyson-Phillips expansion Rn(t)=∑j=n∞‍Uj(t) (where U0(t)=U(t) and Un(t)=∫0t‍U(t-s)KUn-1(s)ds for all n≥1) is an appropriate tool to exclude the eventual presence of the continuous spectrum and to restore the following spectral mapping theorem:(28)σVt∩η∈C:η>esHt=etσTH+K∩etλ:λ>sH.This technique has the advantage of not imposing any condition on the initial data; it has been used by [36, 44, 45, 51] and other authors to study the time asymptotic behavior of solutions of transport equations for absorbing boundary conditions (H=0) or ψ|Γ-=0; in other words, it has been used in the case where each neutron which arrives at a point of ∂D and coming from the interior of D disappears, and no neutron arrives from outside and where D is bounded. Many contributions have been made in this direction, showing in particular the compactness of the second-order remainder of the Dyson-Phillips expansion, sometimes through heavy calculations in the case of non absorbing boundary conditions. Recently and always for absorbing boundary conditions, dealing with regular collision operators by assuming that the domain of positions has a finite volume (not necessarily bounded), Mokhtar-Kharroubi [40] has established the compactness of the first remainder term of the Dyson-Phillips expansion on Lp(D×V)(1<p<∞). This analysis simplifies considerably the spectral analysis of transport equations and extends all known results made in the framework of the study of the compactness of the second-order remainder term; this is due to the fact that if Rn(t) is compact, thus Rn+1(t) is also compact, and it implies that (U(t))t≥0 and (V(t))t≥0 have the same essential spectra and consequently the same essential types. Unfortunately, this argument cannot be applied to the case where p=1 since its proof was obtained in the framework of L2(D×V) (and extended to Lp(D×V) space (1<p<∞) via some interpolation techniques) using some properties of Fourier transform and Hilbert-Schmidt operators. Better than that, Mokhtar-Kharroubi conjectured that the first remainder term of the Dyson-Phillips expansion R1(t),t>0 is not compact on L1(D×V); additionally, its weakly compactness is an open problem (see [5, Problem 7, p. 94]).In this work, we study the impact of compactness results onp-independence of the asymptotic spectrum of the transport operator AH. These results are established by means of some geometrical properties of the space of positions D and the Radon measure μ having the velocity space V as a support and the natures of the collision operator K and the boundary linear operator H.


## 1.1. Collision Operators

The collision operatorK given as a perturbation of the advection transport operator TH is defined on Xp by (16)K:Xp⟶Xpψ⟶∫Vκx,v,v′ψx,v′dμv′.Note that the operator K is local in x; it describes the physics scattering and production of particles (fission), so it can be viewed as mapping: (17)K·:x∈D⟶Kx∈LLpV.We assume that K(·) is strongly measurable,(18)x∈D⟶Kxφ∈LpVis measurable for anyφ∈LpV,and bounded, (19)esssupx∈D⁡KxLLpV<∞.It follows that K defines a bounded operator on the space Lp(D×V) according to the formula(20)φ∈LpD×V(Lp(D×V)≃Lp(D;Lp(V))) and(21)KxLLpD×V≤esssupx∈D⁡KxLLpV.The final assumption on K is (22)Kx∈KLpValmost everywhere,where K(Lp(V)) denotes the set of compact linear operators on the space Lp(V).We give now the concept of regular collision operators introduced by Mokhtar-Kharroubi [5].Definition 1.
A collision operator,(23)K·:x∈D⟶Kx∈LLpV,is said to be regular if K(x) is compact on Lp(V) almost everywhere on D and (24)K·:x∈D⟶LLpVis a “Bochner measurable function”.The interest of the class of regular collision operators lies in the following lemma.Lemma 2 (see [5, Proposition 4.1]).
A regular collision operatorK can be approximated, in the uniform topology, by a sequence {Kn} of collision operators with kernels of the form (25)∑i∈Ifixgiξhiξ′,where fi∈L∞(D), gi∈Lp(V) and hi∈Lq(V)(1/p+1/q=1) (I is finite).It is easy to observe that (1) can be written under the following abstract Cauchy problem:(26)∂ψ∂t=TH+Kψt,t>0,ψ0=ψ0.Spectral theory of transport operators has known a major development since the pioneering papers of Lehner and Wing and Jörgens in the late 1950s [6–8]. A considerable literature has been devoted to the spectral analysis of the transport operator. This one is studied by means of the nature of the parameters of the equation (nature of boundary conditions, nature of the domain of positions or velocity space, and nature of the collision operator). Let us quote, for example, [1, 4–6, 9–48].In general, the time asymptotic behavior of solutions of (1) is analyzed under two angles: resolvent approach and the semigroup approach.(1) Resolvent Approach. For 1<p<∞, this approach is based essentially on the compactness (or the compactness of one iterate) of the bounded linear operator (λ-TH)-1K. Indeed, Vidav [44] observed that if this condition is satisfied, it leads via an analytic Fredholm alternative to the fact that the set σ(TH+K)∩{λ∈C:Reλ>sH} (σ is the spectrum, while sH is the spectral bound of the operator TH) composed (at most) a set of isolated eigenvalues with finite algebraic multiplicities {λi}i∈J, where {λi,Reλi≥α} is a finite set for each α>sH. If p=1, it suffices to treat the weak compactness by taking into account the fact that the square of weakly compact operator on this space is compact [49, Corollary 13, p. 510]. Recall that, among relevant results in this direction, we can cite the works of Mokhtar-Kharroubi [5, 36], Latrach [24–27], and Song [43].Thus, ifTH generates a c0-semigroup (U(t);t≥0), by Dyson-Phillips theorem of perturbation, TH+K generates a c0-semigroup (V(t);t≥0) given by the following formula (see [50, Corollary 7.5, p. 29]):(27)Vtψ0=12iπlimγ→∞⁡∫v-iγv+iγeλtλ-T-K-1ψ0dλ,t>0,where v is sufficiently large by deforming the contour of integration in Dunford’s formula. Recover the residues corresponding to the poles (eigenvalues of TH+K); we can obtain a good comprehension of the asymptotic behavior of solution when the initial data ψ0 belongs to D(TH+K)2 (unfortunately this regular condition is not natural).(2) Semigroup Approach. Even if σ(TH+K)∩{λ∈C:Reλ>sH} is reduced to isolated eigenvalues of finite algebraic multiplicities, the set σ(V(t))∩{η∈C:|η|>esHt} can contain the continuous spectrum due to the absence of a spectral mapping theorem for the mapping λ→etλ. Vidav [45] has shown that the time asymptotic behavior of V(t)t≥0 is connected to the analysis of its spectrum and the compactness of remainder terms of the Dyson-Phillips expansion Rn(t)=∑j=n∞‍Uj(t) (where U0(t)=U(t) and Un(t)=∫0t‍U(t-s)KUn-1(s)ds for all n≥1) is an appropriate tool to exclude the eventual presence of the continuous spectrum and to restore the following spectral mapping theorem:(28)σVt∩η∈C:η>esHt=etσTH+K∩etλ:λ>sH.This technique has the advantage of not imposing any condition on the initial data; it has been used by [36, 44, 45, 51] and other authors to study the time asymptotic behavior of solutions of transport equations for absorbing boundary conditions (H=0) or ψ|Γ-=0; in other words, it has been used in the case where each neutron which arrives at a point of ∂D and coming from the interior of D disappears, and no neutron arrives from outside and where D is bounded. Many contributions have been made in this direction, showing in particular the compactness of the second-order remainder of the Dyson-Phillips expansion, sometimes through heavy calculations in the case of non absorbing boundary conditions. Recently and always for absorbing boundary conditions, dealing with regular collision operators by assuming that the domain of positions has a finite volume (not necessarily bounded), Mokhtar-Kharroubi [40] has established the compactness of the first remainder term of the Dyson-Phillips expansion on Lp(D×V)(1<p<∞). This analysis simplifies considerably the spectral analysis of transport equations and extends all known results made in the framework of the study of the compactness of the second-order remainder term; this is due to the fact that if Rn(t) is compact, thus Rn+1(t) is also compact, and it implies that (U(t))t≥0 and (V(t))t≥0 have the same essential spectra and consequently the same essential types. Unfortunately, this argument cannot be applied to the case where p=1 since its proof was obtained in the framework of L2(D×V) (and extended to Lp(D×V) space (1<p<∞) via some interpolation techniques) using some properties of Fourier transform and Hilbert-Schmidt operators. Better than that, Mokhtar-Kharroubi conjectured that the first remainder term of the Dyson-Phillips expansion R1(t),t>0 is not compact on L1(D×V); additionally, its weakly compactness is an open problem (see [5, Problem 7, p. 94]).In this work, we study the impact of compactness results onp-independence of the asymptotic spectrum of the transport operator AH. These results are established by means of some geometrical properties of the space of positions D and the Radon measure μ having the velocity space V as a support and the natures of the collision operator K and the boundary linear operator H.


## 2. Main Results

### 2.1. Compactness Results andp-Independence of σs(AH)

We assume that the measureμ satisfies the following specific geometrical property:(29)∫c1≤∥x∥≤c2dμx∫0c3χAtxdtasA⟶0,for every 0<c1<c2<∞ and c3<∞, where |A| is the Lebesgue measure of the set A and χA is the indicator function of A.Remark 3.
As indicated in [5, Remark 4.3], the above condition is satisfied by the Lebesgue measure on Rn or on spheres (multigroup model).We start our analysis by the following fundamental compactness result which will be used in the rest of this section.Theorem 4.
Ifp=1, let H∈L(L1+,L1-),H<1 be a weakly compact boundary linear operator and let μ be a Radon measure satisfying the condition (29). Assume that the collision operator K is regular. Thus, (i)
ifD is bounded, then K(λ-TH)-1K is weakly compact on L1(D×V,dxdμ);(ii)
ifD is a bounded convex set in Rn and H is let to be compact, then K(λ-TH)-1K is compact on L1(D×V,dxdμ).Proof.
IfH<1, TH generates a c0-semigroup (UH(t);t≥0) on X1, and then its resolvent exists as bounded linear operator satisfying (30)λ-TH-1=ΓλH+Cλ,where ΓλH=∑n≥0‍BλH(MλH)nGλ, and (31)λ-TH-1≤1Reλ+λ⋆,Reλ>-λ⋆.Thus,(32)Kλ-TH-1K=KΓλHK+KCλK.Obviously, if H is weakly compact, then ΓλH is weakly compact. On the other hand, it is easy to observe that Cλ is nothing but the resolvent of the streaming operator with vacuum boundary conditions T0. Now, under condition (29) and by applying [5, Theorem 4.4(ı)], we obtain that K(λ-T0)-1K is weakly compact on X1 if D is bounded. Moreover, if D is convex, then, by applying [5, Theorem 4.4(ıı)], we get the compactness of the operator K(λ-T0)-1K. Since the compactness and weak compactness property concerning bounded linear operators is stable under summation, we obtain the desired result.Question 1.
It is known that for1<p<∞ and if the boundary linear operator H is compact, then TH generates a c0-semigroup (UH(t);t≥0) on Xp (see [31, Theorem 6.8]); is the result still true for p=1 under the condition that H is weakly compact multiplicative boundary operator (H≥1)?Let(Di,μi),i=0,1, be measure spaces with σ-finite positive measures μi.Theorem 5 (Riesz-Thorin theorem).
Assume that1≤pi, qi≤∞, for i=0,1, and let T be a linear operator which maps Lpi(D0,μ0) continuously into Lqi(D1,μ1) with norm Mi. If 0<θ<1 and 1/p=1-θ/p0+θ/p1, 1/q=1-θ/q0+θ/q1, then T maps Lp(D0,μ0) continuously into Lq(D1,μ1) with norm M≤M01-θM1θ.This theorem shows that the boundedness of linear operators can be interpolated betweenLp-spaces. In 1960, Krasnoselskii [52] showed that compactness can be also interpolated. Thus, we can announce the following result.Theorem 6.
Assume that1≤pi, qi≤∞, for i=0,1, and let T:Lpi(D0,μ0)→Lqi(D1,μ1) be compact. If 0<θ<1 and 1/p=1-θ/p0+θ/p1,1/q=1-θ/q0+θ/q1, then T:Lp(D0,μ0)→Lq(D1,μ1) is also compact.By combining Theorem6 and [53, Corollary 1.6.2], the following lemma can be derived.Lemma 7.
Let1≤p0, p1<∞. Assume that a linear operator T:Lp0(D)∩Lp1(D)→Lp0(D)∩Lp1(D) can be extended to bounded linear operators on Lpi(D),(i=0,1) such that at least one of them is power compact. Then(i)
T can be extended to a power compact operator on Ls(D) for each s∈(p0,p1);(ii)
denote the extension ofT to Ls(D) by Ts. If Tpi is power compact, then σ(Ts)=σ(Tpi) for all s∈(p0,p1) and the spectral projections corresponding to nonzero eigenvalues are independent of p.LetTHp (resp., AHp) be the closed densely defined operator TH (resp., AH) on Xp, (1≤p<∞). We denote by Kp and (UHp(t);t≥0) the bounded linear operators K and (UH(t);t≥0) defined on Xp. Let σsp(AH)=σ(AHp)∩{λ∈C/Reλ>-λ⋆} (the asymptotic spectrum of the operator AHp).Now, we establish the fundamental result of this work which describesp-independence of σsp(AH).Theorem 8.
Under assumptions of Theorem4, we have(i)
σsp(AH)=σs1(AH) for all p>1;(ii)
ifλ∈σsp(AH)=σs1(AH), we have N((λ-AHp)m)=N((λ-AH1)m) for every positive integer m and every p>1, where N(T) designates the null space of the linear operator T. As a consequence, both the geometrical multiplicity and algebraic multiplicity of λ are p-independent.Proof.
Letp>1; we have(33)KpLp∩L1=K1Lp∩L1,UHptLp∩L1=UH1tLp∩L1,t≥0.The resolvent of THp can be written as the Laplace transform of UHp(t) as follows:(34)λ-THp-1φ=∫0∞e-λtUHptφdt.Thus for λ∈C such that Reλ>-λ⋆, we get(35)λ-THpLp∩L1-1=λ-TH1Lp∩L1-1.By applying Theorem 4, we obtain the compactness of ((λ-TH1)-1K1)2 on L1(D×V). On the other hand, Lemma 7 implies the compactness of ((λ-THp)-1Kp)2 on Lp(D×V) and consequently σ(((λ-THp)-1Kp)2)=σ(((λ-TH1)-1K1)2) for every p>1. Hence, Gohberg-Schmul’yan theorem [22, Theorem 11.4] shows that σsp(AH) consists of discrete eigenvalues with finite algebraic multiplicity. Moreover, it is easy to observe that 1∈σ(((λ-THp)-1Kp)2) if and only if λ∈σsp(AH); therefore, taking into account assertion (ii) in Lemma 7, we get that σsp(AH)=σs1(AH) for all p≥1.
Next, following estimation (31), we obtain that limReλ→+∞(λ-THp)-1Kp=0. This gives that, for Reλ sufficiently large, we have(36)I-λ-THp-1Kp-1=∑j=0∞λ-THp-1Kpj.Thus, for Reλ sufficiently large, it follows that(37)I-λ-THp-1KpLp∩L1-1=I-λ-TH1-1K1Lp∩L1-1.Using the analyticity of the operator valuation function λ→(I-(λ-THp)-1Kp)-1 on the set {λ∈C/Reλ>-λ⋆}∖σsp(AH) for each p≥1, we get that for(38)λ∈μ∈C/Reμ>-λ⋆∖σspAH=μ∈C/Reμ>-λ⋆∖σs1AHwe have(39)I-λ-THp-1KpLp∩L1-1=I-λ-TH1-1K1Lp∩L1-1.Using the formula (λ-AHp)-1=(I-(λ-THp)-1Kp)-1(λ-THp)-1 for each p≥1 and λ∈{μ∈C/Reμ>-λ⋆}∖σsp(AH), one sees that(40)λ-AHpLp∩L1-1=λ-AH1Lp∩L1-1for λ∈{μ∈C/Reμ>-λ⋆}∖σsp(AH).
Now, if we denote byPλ(AHp) the spectral projection corresponding to an eigenvalue ζ of AHp, then for β>0 sufficiently small(41)PλAHp=12iπ∫z-ζ=βz-AHp-1dz.According to (40) and (41), it follows that for each λ∈σsp(AH)=σs1(AH)(42)PλAHpLp∩L1=PλAH1Lp∩L1.Since the space of infinitely differentiable functions with compact supports C0∞(D×V) is dense in Xp, then Pλ(AHp)(C0∞(D×V)) is dense in Pλ(AHp)(Xp), but these two vector spaces are finite-dimensional; hence(43)PλAHpC0∞D×V=PλAHpXp,p≥1.According to (42) and (43), we obtain that Pλ(AHp)(Xp)=Pλ(AH1)(X1) for all p≥1 and λ∈σsp(AH). Afterwards, since, for every k≥1, we have N((λ-AHp)k)⊂Pλ(AHp)(Xp)=Pλ(AH1)(X1) and N((λ-AH1)k)⊂Pλ(AH1)(X1)=Pλ(AHp)(Xp), it follows that N((λ-AH1)k)=N((λ-AHp)k)⊂Xp∩X1.
In the spirit of the above theorem, we can prove the following result without weak compactness hypothesis on the boundary linear operatorH and the geometrical property (29) with boundedness of D but based on the weak compactness of one remainder of the Dyson-Phillips expansion.
Letϱsp(AH)=σ(AHp)⋂{λ∈C/Reλ>s(TH)}, where s(TH) is the spectral bound of the operator TH in Xp for all p≥1.Proposition 9.
LetK be a regular collision operator and let H∈L(Lp+,Lp-)(1≤p<∞) such that TH generates a c0-semigroup (UH(t);t≥0) on Xp. If one of the remainder terms of the Dyson-Phillips series RnH(t) is weakly compact on X1, then(i)
ϱsp(AH)=ϱs1(AH) for all p>1;(ii)
ifλ∈ϱsp(AH)=ϱs1(AH), then N((λ-AHp)m)=N((λ-AH1)m) for every positive integer m and every p>1. As a consequence, both the geometrical multiplicity and algebraic multiplicity of λ are p-independent.Proof.
Following the proof of Theorem8, it suffices to show that there exists m≥1 such that (λ-TH-1K)m is compact for λ∈C such that Reλ>s(TH). Indeed, assume that there exists n≥1 such that RnH(t) is weakly compact on X1. Then, according to [5, Theorem 2.6], Un(t)=([UK]n∗U)(t) is weakly compact and consequently by [5, Theorem 2.3] the strong integral(44)∫0Ne-λtUntdtis weakly compact onX1.On the other hand, we have(45)∫0Ne-λtUntdt⟶∫0∞e-λtUntdtinLX1.Hence, ∫0∞‍e-λtUn(t)dt is weakly compact on X1. Since the Laplace transform of ([UK]n∗U)(t) is nothing but ((λ-TH)-1K)n(λ-TH)-1, this gives the weak compactness of ((λ-TH)-1K)n(λ-TH)-1 for λ∈C such that Reλ>ω, where ω is the type of (UH(t);t≥0); by analytic arguments, we obtain that ((λ-TH)-1K)n(λ-TH)-1 is weakly compact for λ∈C such that Reλ>s(TH) and implies compactness of ((λ-TH)-1K)2n+2 which gives the needed result.Now, we focus our study on the case of slab geometry.Theorem 10.
IfD=]-a,a[,V=[-1,1],μ=v (the Lebesgue measure on R) and H is a bounded boundary linear operator from Lp+ to Lp-, then(i)
σsp(AH)=σs1(AH) for all p>1;(ii)
ifλ∈σsp(AH)=σs1(AH), then N((λ-AHp)m)=N((λ-AH1)m) for every positive integer m and every p>1. As a consequence, both the geometrical multiplicity and algebraic multiplicity of λ are p-independent.Proof.
Here, the time of sojourn of particles inD is bounded from below by 2a; indeed, in this case we have inf{τ(x,v);(x,v)∈Γ+}=2a>0. As an immediate consequence, TH generates a c0-semigroup for any boundary linear operator H [29, Remark 6]. Moreover, we have(46)λ-TH-1≤αReλ+λ⋆,∀λ∈Λ0,where α is a positive constant depending on H and Λ0={λ∈C/Reλ>λ0} with(47)λ0=-λ⋆,ifH≤1,-λ⋆+12aln⁡H,ifH>1.On the other hand, we have ((λ-TH)-1K)2 is compact for all 1≤p<∞ [25, Theorem 2.1]. By adopting the same techniques given in the proof of Theorem 8, we obtain the desired result.Remark 11.
We note thatK(λ-TH)-1 is not weakly compact in general on L1(D×V) (see [18]).Question 2.
In the case of slab geometry, is(λ-TH)-1K weakly compact on L1(]-a,a[×[-1,1]) under the assumption that μ is a diffuse (nonatomic) measure on R?


## 2.1. Compactness Results andp-Independence of σs(AH)

We assume that the measureμ satisfies the following specific geometrical property:(29)∫c1≤∥x∥≤c2dμx∫0c3χAtxdtasA⟶0,for every 0<c1<c2<∞ and c3<∞, where |A| is the Lebesgue measure of the set A and χA is the indicator function of A.Remark 3.
As indicated in [5, Remark 4.3], the above condition is satisfied by the Lebesgue measure on Rn or on spheres (multigroup model).We start our analysis by the following fundamental compactness result which will be used in the rest of this section.Theorem 4.
Ifp=1, let H∈L(L1+,L1-),H<1 be a weakly compact boundary linear operator and let μ be a Radon measure satisfying the condition (29). Assume that the collision operator K is regular. Thus, (i)
ifD is bounded, then K(λ-TH)-1K is weakly compact on L1(D×V,dxdμ);(ii)
ifD is a bounded convex set in Rn and H is let to be compact, then K(λ-TH)-1K is compact on L1(D×V,dxdμ).Proof.
IfH<1, TH generates a c0-semigroup (UH(t);t≥0) on X1, and then its resolvent exists as bounded linear operator satisfying (30)λ-TH-1=ΓλH+Cλ,where ΓλH=∑n≥0‍BλH(MλH)nGλ, and (31)λ-TH-1≤1Reλ+λ⋆,Reλ>-λ⋆.Thus,(32)Kλ-TH-1K=KΓλHK+KCλK.Obviously, if H is weakly compact, then ΓλH is weakly compact. On the other hand, it is easy to observe that Cλ is nothing but the resolvent of the streaming operator with vacuum boundary conditions T0. Now, under condition (29) and by applying [5, Theorem 4.4(ı)], we obtain that K(λ-T0)-1K is weakly compact on X1 if D is bounded. Moreover, if D is convex, then, by applying [5, Theorem 4.4(ıı)], we get the compactness of the operator K(λ-T0)-1K. Since the compactness and weak compactness property concerning bounded linear operators is stable under summation, we obtain the desired result.Question 1.
It is known that for1<p<∞ and if the boundary linear operator H is compact, then TH generates a c0-semigroup (UH(t);t≥0) on Xp (see [31, Theorem 6.8]); is the result still true for p=1 under the condition that H is weakly compact multiplicative boundary operator (H≥1)?Let(Di,μi),i=0,1, be measure spaces with σ-finite positive measures μi.Theorem 5 (Riesz-Thorin theorem).
Assume that1≤pi, qi≤∞, for i=0,1, and let T be a linear operator which maps Lpi(D0,μ0) continuously into Lqi(D1,μ1) with norm Mi. If 0<θ<1 and 1/p=1-θ/p0+θ/p1, 1/q=1-θ/q0+θ/q1, then T maps Lp(D0,μ0) continuously into Lq(D1,μ1) with norm M≤M01-θM1θ.This theorem shows that the boundedness of linear operators can be interpolated betweenLp-spaces. In 1960, Krasnoselskii [52] showed that compactness can be also interpolated. Thus, we can announce the following result.Theorem 6.
Assume that1≤pi, qi≤∞, for i=0,1, and let T:Lpi(D0,μ0)→Lqi(D1,μ1) be compact. If 0<θ<1 and 1/p=1-θ/p0+θ/p1,1/q=1-θ/q0+θ/q1, then T:Lp(D0,μ0)→Lq(D1,μ1) is also compact.By combining Theorem6 and [53, Corollary 1.6.2], the following lemma can be derived.Lemma 7.
Let1≤p0, p1<∞. Assume that a linear operator T:Lp0(D)∩Lp1(D)→Lp0(D)∩Lp1(D) can be extended to bounded linear operators on Lpi(D),(i=0,1) such that at least one of them is power compact. Then(i)
T can be extended to a power compact operator on Ls(D) for each s∈(p0,p1);(ii)
denote the extension ofT to Ls(D) by Ts. If Tpi is power compact, then σ(Ts)=σ(Tpi) for all s∈(p0,p1) and the spectral projections corresponding to nonzero eigenvalues are independent of p.LetTHp (resp., AHp) be the closed densely defined operator TH (resp., AH) on Xp, (1≤p<∞). We denote by Kp and (UHp(t);t≥0) the bounded linear operators K and (UH(t);t≥0) defined on Xp. Let σsp(AH)=σ(AHp)∩{λ∈C/Reλ>-λ⋆} (the asymptotic spectrum of the operator AHp).Now, we establish the fundamental result of this work which describesp-independence of σsp(AH).Theorem 8.
Under assumptions of Theorem4, we have(i)
σsp(AH)=σs1(AH) for all p>1;(ii)
ifλ∈σsp(AH)=σs1(AH), we have N((λ-AHp)m)=N((λ-AH1)m) for every positive integer m and every p>1, where N(T) designates the null space of the linear operator T. As a consequence, both the geometrical multiplicity and algebraic multiplicity of λ are p-independent.Proof.
Letp>1; we have(33)KpLp∩L1=K1Lp∩L1,UHptLp∩L1=UH1tLp∩L1,t≥0.The resolvent of THp can be written as the Laplace transform of UHp(t) as follows:(34)λ-THp-1φ=∫0∞e-λtUHptφdt.Thus for λ∈C such that Reλ>-λ⋆, we get(35)λ-THpLp∩L1-1=λ-TH1Lp∩L1-1.By applying Theorem 4, we obtain the compactness of ((λ-TH1)-1K1)2 on L1(D×V). On the other hand, Lemma 7 implies the compactness of ((λ-THp)-1Kp)2 on Lp(D×V) and consequently σ(((λ-THp)-1Kp)2)=σ(((λ-TH1)-1K1)2) for every p>1. Hence, Gohberg-Schmul’yan theorem [22, Theorem 11.4] shows that σsp(AH) consists of discrete eigenvalues with finite algebraic multiplicity. Moreover, it is easy to observe that 1∈σ(((λ-THp)-1Kp)2) if and only if λ∈σsp(AH); therefore, taking into account assertion (ii) in Lemma 7, we get that σsp(AH)=σs1(AH) for all p≥1.
Next, following estimation (31), we obtain that limReλ→+∞(λ-THp)-1Kp=0. This gives that, for Reλ sufficiently large, we have(36)I-λ-THp-1Kp-1=∑j=0∞λ-THp-1Kpj.Thus, for Reλ sufficiently large, it follows that(37)I-λ-THp-1KpLp∩L1-1=I-λ-TH1-1K1Lp∩L1-1.Using the analyticity of the operator valuation function λ→(I-(λ-THp)-1Kp)-1 on the set {λ∈C/Reλ>-λ⋆}∖σsp(AH) for each p≥1, we get that for(38)λ∈μ∈C/Reμ>-λ⋆∖σspAH=μ∈C/Reμ>-λ⋆∖σs1AHwe have(39)I-λ-THp-1KpLp∩L1-1=I-λ-TH1-1K1Lp∩L1-1.Using the formula (λ-AHp)-1=(I-(λ-THp)-1Kp)-1(λ-THp)-1 for each p≥1 and λ∈{μ∈C/Reμ>-λ⋆}∖σsp(AH), one sees that(40)λ-AHpLp∩L1-1=λ-AH1Lp∩L1-1for λ∈{μ∈C/Reμ>-λ⋆}∖σsp(AH).
Now, if we denote byPλ(AHp) the spectral projection corresponding to an eigenvalue ζ of AHp, then for β>0 sufficiently small(41)PλAHp=12iπ∫z-ζ=βz-AHp-1dz.According to (40) and (41), it follows that for each λ∈σsp(AH)=σs1(AH)(42)PλAHpLp∩L1=PλAH1Lp∩L1.Since the space of infinitely differentiable functions with compact supports C0∞(D×V) is dense in Xp, then Pλ(AHp)(C0∞(D×V)) is dense in Pλ(AHp)(Xp), but these two vector spaces are finite-dimensional; hence(43)PλAHpC0∞D×V=PλAHpXp,p≥1.According to (42) and (43), we obtain that Pλ(AHp)(Xp)=Pλ(AH1)(X1) for all p≥1 and λ∈σsp(AH). Afterwards, since, for every k≥1, we have N((λ-AHp)k)⊂Pλ(AHp)(Xp)=Pλ(AH1)(X1) and N((λ-AH1)k)⊂Pλ(AH1)(X1)=Pλ(AHp)(Xp), it follows that N((λ-AH1)k)=N((λ-AHp)k)⊂Xp∩X1.
In the spirit of the above theorem, we can prove the following result without weak compactness hypothesis on the boundary linear operatorH and the geometrical property (29) with boundedness of D but based on the weak compactness of one remainder of the Dyson-Phillips expansion.
Letϱsp(AH)=σ(AHp)⋂{λ∈C/Reλ>s(TH)}, where s(TH) is the spectral bound of the operator TH in Xp for all p≥1.Proposition 9.
LetK be a regular collision operator and let H∈L(Lp+,Lp-)(1≤p<∞) such that TH generates a c0-semigroup (UH(t);t≥0) on Xp. If one of the remainder terms of the Dyson-Phillips series RnH(t) is weakly compact on X1, then(i)
ϱsp(AH)=ϱs1(AH) for all p>1;(ii)
ifλ∈ϱsp(AH)=ϱs1(AH), then N((λ-AHp)m)=N((λ-AH1)m) for every positive integer m and every p>1. As a consequence, both the geometrical multiplicity and algebraic multiplicity of λ are p-independent.Proof.
Following the proof of Theorem8, it suffices to show that there exists m≥1 such that (λ-TH-1K)m is compact for λ∈C such that Reλ>s(TH). Indeed, assume that there exists n≥1 such that RnH(t) is weakly compact on X1. Then, according to [5, Theorem 2.6], Un(t)=([UK]n∗U)(t) is weakly compact and consequently by [5, Theorem 2.3] the strong integral(44)∫0Ne-λtUntdtis weakly compact onX1.On the other hand, we have(45)∫0Ne-λtUntdt⟶∫0∞e-λtUntdtinLX1.Hence, ∫0∞‍e-λtUn(t)dt is weakly compact on X1. Since the Laplace transform of ([UK]n∗U)(t) is nothing but ((λ-TH)-1K)n(λ-TH)-1, this gives the weak compactness of ((λ-TH)-1K)n(λ-TH)-1 for λ∈C such that Reλ>ω, where ω is the type of (UH(t);t≥0); by analytic arguments, we obtain that ((λ-TH)-1K)n(λ-TH)-1 is weakly compact for λ∈C such that Reλ>s(TH) and implies compactness of ((λ-TH)-1K)2n+2 which gives the needed result.Now, we focus our study on the case of slab geometry.Theorem 10.
IfD=]-a,a[,V=[-1,1],μ=v (the Lebesgue measure on R) and H is a bounded boundary linear operator from Lp+ to Lp-, then(i)
σsp(AH)=σs1(AH) for all p>1;(ii)
ifλ∈σsp(AH)=σs1(AH), then N((λ-AHp)m)=N((λ-AH1)m) for every positive integer m and every p>1. As a consequence, both the geometrical multiplicity and algebraic multiplicity of λ are p-independent.Proof.
Here, the time of sojourn of particles inD is bounded from below by 2a; indeed, in this case we have inf{τ(x,v);(x,v)∈Γ+}=2a>0. As an immediate consequence, TH generates a c0-semigroup for any boundary linear operator H [29, Remark 6]. Moreover, we have(46)λ-TH-1≤αReλ+λ⋆,∀λ∈Λ0,where α is a positive constant depending on H and Λ0={λ∈C/Reλ>λ0} with(47)λ0=-λ⋆,ifH≤1,-λ⋆+12aln⁡H,ifH>1.On the other hand, we have ((λ-TH)-1K)2 is compact for all 1≤p<∞ [25, Theorem 2.1]. By adopting the same techniques given in the proof of Theorem 8, we obtain the desired result.Remark 11.
We note thatK(λ-TH)-1 is not weakly compact in general on L1(D×V) (see [18]).Question 2.
In the case of slab geometry, is(λ-TH)-1K weakly compact on L1(]-a,a[×[-1,1]) under the assumption that μ is a diffuse (nonatomic) measure on R?



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