Abstract
These notes resume a lecture given in the Cimpa School “Evolutionary equations with applications in natural sciences” held in South Africa (Muizenberg, July 22–August 2, 2013). However, the oral style of the lecture has been changed and the bibliography augmented. This version benefited also from helpful remarks and suggestions of a referee whom I would like to thank. The notes deal with various functional analytic tools and results around spectral analysis of neutron transport-like operators. A first section gives a detailed introduction (mostly without proofs) to fundamental concepts and results on spectral theory of (non-selfadjoint) operators in Banach spaces; in particular, we provide an introduction to spectral analysis of semigroups in Banach spaces and its consequences on their time asymptotic behaviour as time goes to infinity.
In memory of Seiji Ukaï
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1 Introduction
These notes resume a lecture given in the Cimpa School “Evolutionary equations with applications in natural sciences” held in South Africa (Muizenberg, July 22–August 2, 2013). However, the oral style of the lecture has been changed and the bibliography augmented. This version benefited also from helpful remarks and suggestions of a referee whom I would like to thank. The notes deal with various functional analytic tools and results around spectral analysis of neutron transport-like operators. A first section gives a detailed introduction (mostly without proofs) to fundamental concepts and results on spectral theory of (non-selfadjoint) operators in Banach spaces; in particular, we provide an introduction to spectral analysis of semigroups in Banach spaces and its consequences on their time asymptotic behaviour as time goes to infinity. A special attention is paid to positive semigroups in ordered spaces (i.e. semigroups leaving invariant the cone of positive elements) because of their fundamental interest in neutron transport theory. We focus on the analysis of essential spectra and isolated eigenvalues with finite multiplicities. A second section deals with spectral analysis of weighted shift (or collisionless transport) semigroups. A third section is devoted to spectral analysis of perturbed semigroups in Banach spaces, in particular to stability of essential type for perturbed semigroups. A last section deals with a thorough analysis of compactness problems for general models of neutron transport; the results are very different depending on whether we work in L p spaces (1 < p < ∞) or in (the physical) L 1 space; this issue is the very core of spectral analysis of neutron transport operators and allow the abstract theory to cover them.
Transport theory provides a statistical description of large populations of “particles” moving in a host medium (see e.g. [17]) and is of interest in various fields such as radiative transfer theory, nuclear reactor theory, gas dynamics, plasma physics, structured population models in mathematical biology etc. Among the most classical kinetic equations, we mention the one governing the transport of neutrons through the uranium fuel elements of a nuclear reactor. The aim of this lecture is to present various functional analytic tools and results motivated by this class of equations. In a nuclear reactor, the proportion of neutrons with respect to the atoms of the host medium, is infinitesimal (about 10−11), so the possible collisions between neutrons are negligible in comparison with the collisions of neutrons with the atoms of the host material. Thus (in absence of feedback temperature) neutron transport equations as well as radiative transfer equations for photons are genuinely linear. The population of particles is described by a density function f(t, x, v) of particles at time t > 0, at position x and with velocity v. In particular
is the expected number of particles at time t > 0. One sees immediately that L 1 spaces are natural settings in transport theory! Various models are used in nuclear reactor theory:
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Inelastic model for neutron transport
$$\displaystyle{ \frac{\mathit{\partial f }} {\mathit{\partial t}} + v.\frac{\mathit{\partial f }} {\mathit{\partial x}} +\sigma (x,v)f(t,x,v) =\int _{V }k(x,v,v^{{\prime}})f(t,x,v^{{\prime}})\mathit{dv}^{{\prime}} }$$where \((x,v) \in \varOmega \times V,\ \varOmega \subset \mathbb{R}^{3},V = \left \{v \in \mathbb{R}^{3};\ c_{0} \leq \left \vert v\right \vert \leq c_{1}\right \}\) (\(0 \leq \ c_{0} < c_{1} < \infty )\) and dv is Lebesgue measure, with initial condition f(0, x, v) = f 0(x, v) and boundary condition
$$\displaystyle{ f(t,x,v)_{\mid \varGamma _{-}} = 0\ }$$where
$$\displaystyle{ \varGamma _{-}:= \left \{(x,v) \in \partial \varOmega \times V;\ v.n(x) < 0\right \} }$$and n(x) is the unit exterior normal at \(x \in \partial \varOmega\). The collision frequency σ(. , . ) and the scattering kernel k(. , . , . ) are nonnegative.
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Multiple scattering: This physical model differs from the previous “reactor model” by the fact that \(\varOmega =\mathbb{R} ^{3}\) (no boundary condition) but σ(x, v) and \(k(x,v,v^{{\prime}})\) are compactly supported in space.
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The presence of delayed neutrons
Besides the prompt neutrons (appearing instantaneously in a fission process), some neutrons may appear after a time delay as a decay product of radioactive fission fragments and induce a suitable source term in the usual equation
$$\displaystyle{ \frac{\mathit{\partial f }} {\mathit{\partial t}} + v.\frac{\mathit{\partial f }} {\mathit{\partial x}} +\sigma (x,v)f(t,x,v) =\int _{\mathbb{R}^{3}}k(x,v,v^{{\prime}})f(t,x,v^{{\prime}})\mathit{dv}^{{\prime}} +\sum _{ i=1}^{m}\lambda _{ i}g_{i} }$$which is thus coupled to m differential equations
$$\displaystyle{ \frac{\mathit{dg}_{i}} {\mathit{dt}} = -\lambda _{i}g_{i} +\int _{\mathbb{R}^{3}}k_{i}(x,v,v^{{\prime}})f(t,x,v^{{\prime}})\mathit{dv}^{{\prime}}\ \ \ (1 \leq i \leq m) }$$where \(\lambda _{i} > 0\,(1 \leq i \leq m)\) are the radioactive decay constants; see [48, Chapter 4] and references therein.
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Multigroup models (motivated by numerical calculations)
$$\displaystyle{ \frac{\partial f_{i}} {\mathit{\partial t}} + v.\frac{\partial f_{i}} {\mathit{\partial x}} +\sigma _{i}(x,v)f_{i}(t,x,v) =\sum _{ j=1}^{m}\int _{ V _{j}}k_{i_{,j}}(x,v,v^{{\prime}})f_{ j}(t,x,v^{{\prime}})\mu _{ j}(\mathit{dv}^{{\prime}}), }$$\((1 \leq i \leq m)\ \) where the spheres
$$\displaystyle{ V _{j}:= \left \{v \in \mathbb{R}^{3},\ \left \vert v\right \vert = c_{ j}\right \},\ \ 1 \leq j \leq m,\ \ (c_{j} > 0) }$$are endowed with surface Lebesgue measures μ j and f i (t, x, v) is the density of neutrons (at time t > 0 located at x ∈ Ω) with velocity v ∈ V i .
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Partly inelastic models
$$\displaystyle{ \frac{\mathit{\partial f }} {\mathit{\partial t}} + v.\frac{\mathit{\partial f }} {\mathit{\partial x}} +\sigma (x,v)f(t,x,v) = K_{e}f + K_{i}f }$$in L p(Ω × V ) where, e.g. \(V = \left \{v \in \mathbb{R}^{3};\ c_{0} \leq \left \vert v\right \vert \leq c_{1}\right \}\). The inelastic scattering operator is just
$$\displaystyle{ K_{i}f =\int _{V }k(x,v,v^{{\prime}})f(x,v^{{\prime}})\mathit{dv}^{{\prime}}\ \ }$$while the elastic scattering operator is given by
$$\displaystyle{ K_{e}f =\int _{S^{2}}k(x,\rho,\omega,\omega ^{{\prime}})f(x,\rho \omega ^{{\prime}})dS(\omega ^{{\prime}}) }$$where v = ρ ω. The presence of an elastic scattering operator acting only on the angles ω ∈ S 2 of velocities changes strongly the spectral structure of neutron transport operators [35, 68].
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Diffusive models
$$\displaystyle{ \frac{\mathit{\partial f }} {\mathit{\partial t}} -\bigtriangleup _{x}f +\sigma (x,v)f =\int _{ 0}^{+\infty }k(x,v,v^{{\prime}})f(t,x,v^{{\prime}})\mathit{dv}^{{\prime}} }$$(motivated also by numerical calculations) where the transport operator \(\frac{\mathit{\partial f }} {\mathit{\partial t}} + v.\frac{\mathit{\partial f }} {\mathit{\partial x}}\) is replaced by the parabolic operator \(\frac{\mathit{\partial f }} {\mathit{\partial t}} -\bigtriangleup _{x}f\) where △ x denotes the Laplacian in space variable x ∈ Ω with Dirichlet boundary condition, (here v > 0 denotes a “ kinetic energy” instead of a velocity); see e.g. [14, p. 133]. In the same spirit, we mention that diffusion (i.e. heat) equations with Dirichlet boundary condition turn out to be asymptotic approximations (as \(\varepsilon \rightarrow 0\)) of usual neutron transport equations appropriately rescaled by means of a small parameter \(\varepsilon\) (typically the mean free path); see e.g. [6] and references therein. We find in [49] an approach of the diffusion approximation of neutron transport (on the torus) via spectral theory.
In this lecture, we ignore the presence of delayed neutrons but deal with an abstract velocity measure μ(dv) (with support V ) covering a priori different models, e.g. Lebesgue measure on \(\mathbb{R}^{n}\) or on spheres or even combinations of the two.
In absence of scattering event (i.e. \(k(x,v,v^{{\prime}}) = 0\)) the density of neutral particles (e.g. neutrons) is governed by
with initial condition f 0 and is solved explicitly by the method of characteristics
where
is the first exit time function. This defines a positive C 0-semigroup \(\left (U(t)\right )_{t\geqslant 0}\) on \(L^{p}(\varOmega \times \mathbb{R}^{3};\mathit{dx} \otimes d\mu )\)
called the advection semigroup. Its generator T is given (at least for smooth domains Ω) by
(see e.g. [12, 13] for a trace theory in neutron transport theory). Then the treatment of the full equation follows naturally by perturbation theory. For instance, if the scattering operator
is bounded on \(L^{p}(\varOmega \times \mathbb{R}^{3})\) then, by standard perturbation theory,
generates a positive C 0-semigroup \(\left (V (t)\right )_{t\geqslant 0}\) which solves the full neutron transport equation.
There are two basic eigenvalue problems in nuclear reactor theory:
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Criticality eigenvalue problem
This problem consists in looking for (γ, g) where γ > 0 and g is a nontrivial nonnegative solution to
$$\displaystyle\begin{array}{rcl} 0& =& -v.\frac{\partial g} {\mathit{\partial x}} -\sigma (x,v)g(x,v) +\int _{V }k_{s}(x,v,v^{{\prime}})g(x,v^{{\prime}})\mu (\mathit{dv}^{{\prime}}) {}\\ & & +\frac{1} {\gamma } \int _{V }k_{f}(x,v,v^{{\prime}})g(x,v^{{\prime}})\mu (\mathit{dv}^{{\prime}}),\ \ \ g_{\mid \varGamma _{ -}} = 0; {}\\ \end{array}$$here \(k_{s}(x,v,v^{{\prime}})\) and \(k_{f}(x,v,v^{{\prime}})\) are the scattering kernel and the fission kernel, see e.g. [41, 66].
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The “time eigenelements”
This problem consists in looking for (λ, g) with nontrivial g such that
and in relating them to time asymptotic behaviour (\(t \rightarrow +\infty \)) of the semigroup \(\left (V (t)\right )_{t\geqslant 0}.\)
In this lecture, we focus on the second class of problems. There exists a considerable literature on the subject; we refer to [48] and references therein for the state of the art up to 1997. In these lecture, we present mostly new developments on this topic.
We note that this conventional neutron transport theory deals with the expected (or mean) behaviour of neutrons. In order to describe the fluctuations from the mean value of neutron populations, probabilistic formulations of neutron chain fissions were proposed very early, in particular in [7]. This leads to nonlinear problems governing divergent neutron chain fissions. Such problems are strongly related to spectral theory of usual (linear) neutron transport operators, see [30, 46, 57, 64].
We end this introduction by some historical notes. The beginning of spectral theory of neutron transport dates back to the beautiful and seminal paper by J. Lehner and M. Wing [36] devoted to a simplified model (constant cross sections) in slab geometry. The time asymptotic behaviour of neutron transport semigroups in bounded geometries is well-understood for a long time in the case when the velocities are bounded away from zero; this is a classical result by K. Jorgens:
Theorem 1 ([31])
Let Ω be bounded and convex, let
and let the scattering kernel k(.,.,.) be bounded. If c 0 > 0 then V (t) is compact on L 2 (Ω × V ) for t large enough. In particular, for any \(\alpha \in \mathbb{R}\)
consists at most of finitely many eigenvalues with finite algebraic multiplicities \(\left \{\lambda _{1},\ldots.\lambda _{m}\right \}\) with spectral projections \(\left \{P_{1},\ldots.P_{m}\right \}\) and there exists β < α such that
where \(D_{j}:= (T -\lambda _{j})P_{j}.\)
The picture gets more complicated when arbitrarily small velocities must be taken into account. In this case, the (essential) spectrum of the generator T (of the advection semigroup \(\left \{U(t);t\geqslant 0\right \}\)) on L 2(Ω × V ) consists of a half-plane
where “typically” \(\lambda ^{{\ast}} =\inf \sigma (x,v),\) see S. Albertoni and B. Montagnini [2]. Moreover, important compactness results were obtained very early, (see e.g. Demeru-Montagnini [16], Borysiewicz-Mika [8] and S. Ukai [74]) implying, for most physical scattering kernels, that the scattering operator K is T-compact on L 2(Ω × V ) i.e.
is compact where D(T) is endowed with the graph norm. It follows that the spectrum of A = T + K consists of a left half-plane \(\left \{\lambda \in \mathbb{C};\ \mathit{Re}\lambda \leq -\lambda ^{{\ast}}\right \}\) and at most of isolated eigenvalues with finite algebraic multiplicities located in the right half-plane
(Note that it may happen that this set of isolated eigenvalues is empty for small bodies [2].) Then the time asymptotic behaviour of the solution is traditionally dealt with by means of inverse Laplace transform (Dunford calculus)
(with ρ large enough). If for some \(\varepsilon > 0\)
(with spectral projections \(\left \{P_{1},\ldots.P_{m}\right \}\)) is finite and non-empty then, by shifting the path of integration and picking up the residues, we get an asymptotic expansion
for smooth initial data f; see, e.g. M. Borysiewicz and J. Mika [8] (see also M. Mokhtar-Kharroubi [45]). The drawbackof the approach is that we need very regular initial data (say f ∈ D(A 2)) to estimate the transcient part of the solution. To remedy this situation, a more relevant approach, initiated by I. Vidav [76], consists in studying the spectrum of the semigroup \(\left (V (t)\right )_{t\geqslant 0}\) itself instead of the spectrum of its generator because of the lack (in general) of a spectral mapping theorem relating spectra of semigroups and spectra of their generators. The perturbed semigroup \(\left (V (t)\right )_{t\geqslant 0}\) is expanded into a Dyson–Phillips series
where U 0(t) = U(t) is the advection semigroup and
Theorem 2 ([76])
If some remainder term \(R_{n}(t):=\sum _{ j=n}^{\infty }U_{j}(t)\) is compact for large t then \(\sigma (V (t)) \cap \left \{\mu;\left \vert \mu \right \vert > e^{-\lambda ^{{\ast}}t }\right \}\) consists at most of isolated eigenvalues with finite multiplicities. In particular, \(\forall \varepsilon > 0,\ \)
is finite and
in operator norm where \(\beta < -\lambda ^{{\ast}} +\varepsilon.\)
Vidav’s result had relevant applications to realistic models of kinetic theory much later; see Y. Shizuta [71], G. Greiner [25], J. Voigt [77, 79], P. Takak [72], M. Mokhtar-Kharroubi [42, 43] and L. Weiss [82]. The role of positivity in peripheral spectral theory of neutron transport was emphasized by I. Vidav [75], T. Hiraoka-S. Ukaï [29], Angelescu-Protopopescu [4] and more recently, in others directions, e.g. by G. Greiner [26], J. Voigt [78] and M. Mokhtar-Kharroubi [43–45, 47].
2 Fundamentals of Spectral Theory
This section is a crash course (mostly without proofs) on the fundamental concepts and results on spectral theory of closed linear operators on complex Banach spaces with a special emphasis on generators of strongly continuous semigroups. Because of their importance in transport theory, the basic spectral properties of positive operators (i.e. leaving invariant the positive cone of a Banach lattice) are also given. Finally, we show the role of peripheral spectral theory of positive semigroups in their time asymptotic behaviour as \(t \rightarrow +\infty.\) Apart from Subsection 2.10, the material of this section is widely covered by the general references [15, 20–22, 32, 61, 73] and will be used in the sequel without explicit mention. Subsection 2.10 presents a class of positive semigroups whose real spectra can be described completely; this class covers weighted shift (i.e. advection) semigroups we deal with in Sect. 3.
2.1 Basic Definitions and Results
We start with some basic definitions and results. Let X be a complex Banach space and let
be a linear operator defined on a subspace D(T). We say that T is a closed operator if its graph
is closed in X × X. We define the resolvent set of T by
the spectrum of T by
and the resolvent operator by
In particular, if there exists \(x \in D(T) -\left \{0\right \}\) and \(\lambda \in \mathbb{C}\) such that Tx = λ x then λ ∈ σ(T). In this case, λ is an eigenvalue of T and
is the corresponding eigenspace. In contrast to finite dimensional spaces, in general, σ(T) is not reduced to eigenvalues! For instance, one can show that the spectrum of the multiplication operator on \(C(\left [0,1\right ])\) (endowed with the sup norm)
where Tf(x) = xf(x) is equal to \(\left [0,1\right ]\) and that T has no eigenvalue. For unbounded operators, the spectrum may be empty or equal to \(\mathbb{C}!\) For example, let \(X = C(\left [0,1\right ];\ \mathbb{C})\) endowed with the sup-norm and
Then \(\forall \lambda \in \mathbb{C}\), \(x \in \left [0,1\right ] \rightarrow e^{\lambda x} \in \mathbb{C}\) is an eigenfunction of T so \(\sigma (T) =\mathbb{C}.\) If we replace (T, D(T)) by
then \(\forall \lambda \in \mathbb{C}\) and \(\forall g \in X,\ \) the equation
is uniquely solvable; thus \(\rho (\hat{T}) =\mathbb{C}\) and \(\sigma (\hat{T}) = \varnothing.\)
It is useful to decompose the spectrum of T as follows: The point spectrum
The approximate point spectrum
this terminology is motivated by the fact that λ ∈ σ ap (T) if and only if there exists a sequence \((x_{n})_{n} \subset D(T)\) such that
The residual spectrum
We note that
is a non-disjoint union. Among the first results, we note:
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\((\lambda -T)^{-1}: X \rightarrow X\) is a bounded operator for λ ∈ ρ(T), i.e. \((\lambda -T)^{-1} \in \mathcal{L}(X)\), (by the closed graph theorem).
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ρ(T) is an open subset of \(\mathbb{C}\) (so σ(T) is closed) and
$$\displaystyle{ \lambda \in \rho (T) \rightarrow (\lambda -T)^{-1} \in \mathcal{L}(X) }$$is holomorphic.
More precisely, if μ ∈ ρ(T) then λ ∈ ρ(T) if \(\left \vert \lambda -\mu \right \vert < \left \Vert (\mu -T)^{-1}\right \Vert ^{-1}\) and then
It follows that \(\left \vert \lambda -\mu \right \vert \geqslant \left \Vert (\mu -T)^{-1}\right \Vert ^{-1}\) for any λ ∈ σ(T) and then
In particular \(\left \Vert (\mu -T)^{-1}\right \Vert \rightarrow \infty \ \ \) as \(\mathit{dist}(\mu,\sigma (T)) \rightarrow 0.\)
Bounded operators \(T \in \mathcal{L}(X)\) enjoy specific properties:
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σ(T) is bounded and non-empty.
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The spectral radius of \(T \in \mathcal{L}(X)\), defined by
$$\displaystyle{ r_{\sigma }(T):=\sup \left \{\left \vert \lambda \right \vert;\ \lambda \in \sigma (T)\right \}, }$$is equal to \(\lim _{n\rightarrow \infty }\left \Vert T^{n}\right \Vert ^{ \frac{1} {n} } =\inf _{n}\left \Vert T^{n}\right \Vert ^{ \frac{1} {n} }.\)
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In particular \(r_{\sigma }(T) \leq \left \Vert T\right \Vert\) and (λ − T)−1 is given by a Laurent’s series
$$\displaystyle{ (\lambda -T)^{-1} =\sum _{ 1}^{\infty }\lambda ^{-n}T^{n-1}\ \ \ (\left \vert \lambda \right \vert > r_{\sigma }(T)) }$$with \(T^{m} = \frac{1} {\mathit{2i}\pi }\int _{C}\lambda ^{m}(\lambda -T)^{-1}d\lambda\) where C is any circle (positively oriented) centered at the origin with radius > r σ (T).
If \(T: D(T) \subset X \rightarrow X\) is densely defined linear operator, we can define its dual operator
by
with domain
We note that T ′ is closed but not necessarily densely defined. But if X is reflexive then T ′ is densely defined,\(\ \left (T^{{\prime}}\right )^{{\prime}} = T\), \(\sigma (T^{{\prime}}) =\sigma (T)\) and \((\lambda -T^{{\prime}})^{-1} = \left ((\lambda -T)^{-1}\right )^{{\prime}}\). In particular if \(T \in \mathcal{L}(X)\) then \(r_{\sigma }(T^{{\prime}}) = r_{\sigma }(T).\)
We end this section with a spectral mapping theorem for bounded operators. Let \(T \in \mathcal{L}(X)\) and let \(\varOmega \ni \lambda \rightarrow f(\lambda ) \in \mathbb{C}\) be holomorphic on some open neighborhood Ω of σ(T). Then there exists an open set ω such that \(\sigma (T) \subset \omega \subset \overline{\omega }\subset \varOmega\) and ∂ ω consists of finitely many simple closed curves that do not intersect. One defines a Dunford integral
where ∂ ω is properly oriented (the definition does not depend on the choice of ω). In particular if f(λ) is a polynomial then f(T) coincides with the usual meaning of f(T). Then we have a spectral mapping theorem
2.2 Spectral Decomposition and Riesz Projection
Let X be a complex Banach space such that
(direct sum) where X i (i = 1, 2) are closed subspaces. Let \(P: x \in X \rightarrow Px\) be the (continuous) projection on X 1 along X 2. Let
be a closed linear operator such that P(D(T)) ⊂ D(T) and X i (i = 1, 2) are invariant under T. The parts T i (i = 1, 2) of T on X i (i = 1, 2) are defined by
We say that T is reduced by X i (i = 1, 2). Then
(not necessarily a disjoint union),
Similar results hold for any finite direct sum: \(X = X_{1} \oplus \ldots \oplus X_{n};\ \) see e.g. [73, Theorem 5.4, p. 289].
Let now \(T: D(T) \subset X \rightarrow X\) be a closed linear operator such that σ(T) is a disjoint union of two non-empty closed subsets σ 1 and σ 2 and let σ 1 be compact. Then there exists Γ, a finite number of rectifiable simple closed curves properly oriented enclosing an open set O which contains σ 1 and such that σ 2 is included in the exterior of O. Then
and \(X = X_{1} \oplus X_{2}\) (X 1 = PX and X 2 = (I − P)X = Ker P) reduces T (i.e. X i are T invariant), \(\sigma (T_{i}) =\sigma _{i}\) where \(T_{i}:= T_{\mid X_{i}}\) and T 1 is bounded. P is the spectral projection associated with σ 1. If σ 1 consists of finitely many points \((\lambda _{1},\ldots,\lambda _{n})\) then
(where Γ j is e.g. a small circle enclosing λ j ). P j is the spectral projection associated with λ j . We study now the structure of the resolvent around an isolated singularity. Let μ ∈ σ(T) be an isolated point of σ(T). There exists a Laurent’s series around μ
where
where C is a small circle positively oriented centered at μ. In particular, the residues
is the spectral projection P. In addition
We have
so μ is a pole of the resolvent (i.e. there exists k > 0 such that \(U_{-k}\neq 0\) and \(U_{-n} = 0\ \forall n > k\)) if and only if there exists k > 0 such that \(U_{-k}\neq 0\) and U −(k+1) = 0. Then k is the order of the pole. In this case, μ is an eigenvalue of T and PX = Ker(μ − T)k. The algebraic multiplicity \(m_{a} \leq +\infty \) of μ is the dimension of PX. Conversely, if m a < +∞, i.e. P is of finite rank, then \((\mu -T)^{m_{a}}P = 0\) and then μ is a pole of the resolvent of order ≤ m a . Actually, the order k of the pole is the smallest \(j \in \mathbb{N}\) such that (μ − T)j P = 0. The subspace Ker(μ − T)k contains the generalized eigenvectors; it coincides with the eigenspace if and only if PX = Ker(μ − T), i.e. k = 1 (simple pole); μ is said to be a semi-simple eigenvalue. We say that μ is algebraically simple if m a = 1.
2.3 Application to Riesz-Schauder Theory
As a first illustration of the interest of Riesz projections, we show why the non zero eigenvalues of compact operators have finite algebraic multiplicities. Let \(T: X \rightarrow X\) be a compact operator (i.e. maps bounded sets into relatively compact ones). Then \(\sigma (T)/\left \{0\right \}\) consists at most of isolated eigenvalues. Let α ∈ σ(T) with \(\alpha \neq 0.\) Define T λ (in the neighborhood of α) by
Then \((\lambda -T)(\lambda ^{-1} + T_{\lambda }) = I\) implies that \(T_{\lambda } = T(\lambda ^{-1}T_{\lambda } +\lambda ^{-2}I)\) is compact. So (C being a small circle around α positively oriented) the spectral projection
is compact too. Since U −1 has a closed range then the open mapping theorem and Riesz theorem imply that U −1 has finite-dimensional range. Hence α has a finite algebraic multiplicity.
This result extends to power compact operators. Indeed, let \(T \in \mathcal{L}(X)\) and \(n \in \mathbb{N}\) (\(n\geqslant 2)\) such that T n is compact. The spectral mapping theorem
implies that \(\sigma (T)/\left \{0\right \}\) consists at most of isolated points. Let α ∈ σ(T) with \(\alpha \neq 0.\) Then, for λ close to \(\alpha,(\lambda ^{n} - T^{n}) = (\lambda ^{n-1}I +\lambda ^{n-2}T +\ldots +T^{n-1})(\lambda -T)\) implies
(where C λ is compact) so the spectral projection
is compact and we argue as previously.
2.4 Spectral Mapping Theorem for a Resolvent
Let \(T: D(T) \subset X \rightarrow X\) be closed linear operator and λ 0 ∈ ρ(T). The spectral links between T and its resolvent \((\lambda _{0} - T)^{-1}\) are completely described by:
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\(\sigma \left [(\lambda _{0} - T)^{-1}\right ]\setminus \!\left \{0\right \} = (\lambda _{0}-\sigma (T))^{-1}\) (so \(r_{\sigma }\!\left [(\lambda _{0} - T)^{-1}\right ] = \left [\mathit{dist}(\lambda _{0},\sigma (T))\right ]^{-1}\))
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\(\sigma _{p}\left [(\lambda _{0} - T)^{-1}\right ]\setminus \left \{0\right \} = (\lambda _{0} -\sigma _{p}(T))^{-1}\)
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\(\sigma _{\mathit{ap}}\left [(\lambda _{0} - T)^{-1}\right ]\setminus \left \{0\right \} = (\lambda _{0} -\sigma _{\mathit{ap}}(T))^{-1}\)
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\(\sigma _{\mathit{res}}\left [(\lambda _{0} - T)^{-1}\right ]\setminus \left \{0\right \} = (\lambda _{0} -\sigma _{\mathit{res}}(T))^{-1}\)
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μ is an isolated point of σ(T) if and only if (λ 0 −μ)−1 is an isolated point of \(\sigma \left [(\lambda _{0} - T)^{-1}\right ].\) In this case, the residues and the orders of the pole of (λ − T)−1 at μ and of \(\left [\lambda -(\lambda _{0} - T)^{-1}\right ]^{-1}\) at \((\lambda _{0}-\mu )^{-1}\) coincide.
See [20, Chapter IV]. These properties are of interest e.g. when we deal with Riesz-Schauder theory of operators with compact resolvent.
2.5 Fredholm Operators
A closed operator \(T: D(T) \subset X \rightarrow X\) is said to be a Fredholm operator if \(\dim \mathit{Ker}(T) < \infty \) and the range R(T) of T is closed with finite codimension (i.e. \(\dim \frac{X} {R(T)} < \infty \)). Let \(T: D(T) \subset X \rightarrow X\) be closed linear operator; its Fredholm domain is defined by
Then ρ F (T) is open and ρ(T) ⊂ ρ F (T). If λ 0 is an isolated eigenvalue of T with finite algebraic multiplicity then \(\ \lambda _{0} \in \rho _{F}(T)\), (see [32, Chapter IV]).
We recall that \(T \in \mathcal{L}(X)\) is Fredholm if and only if there exists \(S \in \mathcal{L}(X)\) such that I − ST and I − TS are finite rank operators (see [21] p. 190). The essential spectrum of T is defined by
Let \(\mathcal{K}(X) \subset \mathcal{L}(X)\) be the closed ideal of compact operators. The Calkin algebra
is endowed with the quotient norm (for \(\hat{T}:= T + \mathcal{K}(X)\))
Then
The essential norm of \(T \in \mathcal{L}(X)\) is defined by
In particular, \(\left \Vert T\right \Vert _{\mathit{ess}} \leq \left \Vert T\right \Vert \ \) and the essential norm \(\left \Vert.\right \Vert _{\mathit{ess}}\) is submultiplicative, i.e.
The essential radius of \(T \in \mathcal{L}(X)\) is defined by
Then
In addition
The unbounded component of ρ F (T) consists of resolvent set and at most of isolated eigenvalues with finite algebraic multiplicities, (see [21, p. 204]). Then the essential radius of \(T \in \mathcal{L}(X)\) is also given by
where σ discr (T) refers to the isolated eigenvalues of T with finite algebraic multiplicities. Note that for any \(\varepsilon > 0,\ \sigma (T) \cap \left \{\left \vert \lambda \right \vert \geqslant r_{\mathit{ess}}(T)+\varepsilon \right \}\) consists at most of finitely many eigenvalues with finite algebraic multiplicities. We point out that there exist several non equivalent definitions of essential spectrum for bounded operators but the corresponding essential radius is the same for all them, see [19, Corollary 4.11, p. 44].
2.6 Semigroups and Generators
Let X be a complex Banach space. By a C 0-semigroup on X we mean a family \(\left (S(t)\right )_{t\geqslant 0}\) of bounded linear operators on X indexed by \(t\geqslant 0\) such that S(0) = I, S(t)S(s) = S(t + s) and such that the strong continuity condition holds:
is continuous for all x ∈ X. By the uniform boundedness theorem, \(\left (S(t)\right )_{t\geqslant 0}\) is locally bounded in \(\mathcal{L}(X)\). The infinitesimal generator of \(\left (S(t)\right )_{t\geqslant 0}\) is the unbounded linear operator defined by
with domain
Then T is closed and densely defined. In addition, D(T) is invariant under S(t) and \(S(t)Tx = TS(t)x\ \ \forall x \in D(T).\) Finally, \(\forall x \in D(T),\)
and
see e.g. [15].
If
is subadditive (i.e. p(t + s) ≤ p(t) + p(s)) and locally bounded from above then
see e.g. [15]. Since
is subadditive and locally bounded from above then
In particular \(\left (S(t)\right )_{t\geqslant 0}\) is exponentially bounded, i.e.
ω is called the type or growth bound of \(\left (S(t)\right )_{t\geqslant 0}\). In addition, for any t > 0
We recall that \(\left \{\mathit{Re}\lambda >\omega \right \} \subset \rho (T)\) and
where the integral converges in operator norm. Thus \(\sigma (T) \subset \left \{\mathit{Re}\lambda \leq \omega \right \}\) and the spectral bound of T
We end this section with the famous Hille–Yosida–Phillips–Miyadera–Feller theorem (commonly called Hille–Yosida theorem) which provides a general framework for a huge amount of linear evolution equations of mathematical physics and probability theory [22].
Theorem 3
Let \(T: D(T) \subset X \rightarrow X\) be a closed densely defined linear operator. Then T is the generator of a C 0 -semigroup \(\left (S(t)\right )_{t\geqslant 0}\) satisfying the estimate \(\left \Vert S(t)\right \Vert \leq Me^{\alpha t}\ \forall t \geqq 0\) if and only if \(\sigma (T) \subset \left \{\mathit{Re}\lambda \leq \alpha \right \}\) and
We note that if X is a reflexive complex Banach space and if \(\left (S(t)\right )_{t\geqslant 0}\) is a C 0-semigroup with generator T then the dual semigroup \(\left (S^{{\prime}}(t)\right )_{t\geqslant 0}\) is strongly continuous and its generator is given by T ′. In particular \(\left (S(t)\right )_{t\geqslant 0}\) and \(\left (S^{{\prime}}(t)\right )_{t\geqslant 0}\) have the same type while T and T ′ have the same spectral bound.
2.7 Partial Spectral Mapping Theorems for Semigroups
In general, there exist partial spectral links between a C 0-semigroup and its generator, see [20, Chapter IV].
Theorem 4
Let X be a complex Banach space and \(\left (S(t)\right )_{t\geqslant 0}\ \) be a C 0 -semigroup on X with generator T.Then:
-
(i)
\(e^{t\sigma _{\mathit{ap}}(T)} \subset \sigma _{\mathit{ap}}(S(t))\setminus \left \{0\right \}.\)
-
(ii)
\(e^{t\sigma _{p}(T)} =\sigma _{p}(S(t))\setminus \left \{0\right \}.\)
-
(iii)
\(e^{t\sigma _{\mathit{res}}(T)} =\sigma _{\mathit{res}}(S(t))\setminus \left \{0\right \}.\)
-
(iv)
\(m_{g}(\lambda,T) \leq m_{g}(e^{\lambda t},S(t))\)
-
(v)
\(m_{a}(\lambda,T) \leq m_{a}(e^{\lambda t},S(t))\)
-
(vi)
\(k(\lambda,T) \leq k(e^{\lambda t},S(t)).\)
Here m g (resp. m a , resp. k) refers to geometric multiplicity (resp. algebraic multiplicity, resp. multiplicity of a pole). We note that the possible failure of the spectral mapping theorem stems from the approximate point spectrum. The link between the eigenvalues of \(\left (S(t)\right )_{t\geqslant 0}\ \) and those of its generator T is clarified further by:
Theorem 5
Let X be a complex Banach space and \(\left (S(t)\right )_{t\geqslant 0}\ \) be a C 0 -semigroup on X with generator T. Then:
-
(i)
\(\mathit{Ker}(\mu -T) = \cap _{t\geqslant 0}\mathit{Ker}(e^{\mu t} - S(t)).\)
-
(ii)
\(\mathit{Ker}(e^{\mu t} - S(t)) = \overline{\mathit{lin}}_{n\in \mathbb{Z}}\mathit{Ker}(\mu +\frac{\mathit{2i}\pi n} {t} - T)\ \forall t > 0.\)
Theorem 6 ([24] Proposition 1.10 or [20] p. 283)
Let X be a complex Banach space and \(\left (S(t)\right )_{t\geqslant 0}\ \) be a C 0 -semigroup on X with generator T and let t > 0 be fixed. Let e μt be a pole of S(t) of order k and let Q be the corresponding residue. Then
-
(i)
For every \(n \in \mathbb{Z},\ \ \mu +\frac{\mathit{2i}\pi n} {t}\) is (at most) a pole of (λ − T) −1 of order at most k and residue P n .
-
(ii)
\(QX = \overline{\mathit{lin}}_{n\in \mathbb{Z}}P_{n}X.\)
Corollary 1
Let X be a complex Banach space and \(\left (S(t)\right )_{t\geqslant 0}\ \) be a C 0 -semigroup on X with generator T and let t > 0 be fixed. Let \(\alpha \neq 0\) be an isolated eigenvalue of S(t) with finite algebraic multiplicity and with residue Q. Then \(Q =\sum _{ j=1}^{n}P_{j}\ \) where the P j are the residues of (λ − T) −1 at \(\left \{\lambda _{1},\ldots,\lambda _{n}\right \}\) , the (finite and nonempty) set of eigenvalues of T such that \(e^{\lambda _{i}t} =\alpha.\)
2.8 Essentially Compact Semigroups
The fact that \(\left \Vert.\right \Vert _{\mathit{ess}}\) is submultiplicative implies that
is subadditive. It is also locally bounded from above so
In particular \(\forall \alpha >\omega _{\mathit{ess}}\) \(\exists M_{\alpha }\geqslant 1\) such that
ω ess is called the essential type (or essential growth bound) of \(\left (S(t)\right )_{t\geqslant 0}\). For any t > 0
A C 0-semigroup \(\left (S(t)\right )_{t\geqslant 0}\) on a complex Banach space X is said to be essentially compact if its essential type is less than its type (i.e. ω ess < ω). Such semigroups have a nice finite-dimensional asymptotic structure.
Theorem 7
Let X be a complex Banach space and \(\left (S(t)\right )_{t\geqslant 0}\ \) be an essentially compact C 0 -semigroup on X with generator T. Then:
-
(i)
\(\sigma (T) \cap \left \{\mathit{Re}\lambda >\omega _{\mathit{ess}}\right \}\) consists of a nonempty set of isolated eigenvalues with finite algebraic multiplicities.
-
(ii)
For any ω ′ such that \(\omega _{\mathit{ess}} <\omega ^{{\prime}} <\omega,\ \sigma (T) \cap \left \{\mathit{Re}\lambda \geqslant \omega ^{{\prime}}\right \}\) consists of a finite set (depending on ω ′ ) \(\left \{\lambda _{1},\ldots,\lambda _{m}\right \}\) of eigenvalues of T.
-
(iii)
Let P j be the residues of (λ − T) −1 at λ j and let \(P:=\sum _{ j=1}^{m}P_{j}\) . Then the projector P reduces \(\left (S(t)\right )_{t\geqslant 0}\) and
$$\displaystyle{ S(t) =\sum _{ j=1}^{m}e^{\lambda _{j}t}e^{tD_{j} }P_{j} + O(e^{(\omega ^{{\prime}}-\varepsilon )t }) }$$(for some \(\varepsilon > 0\) ) where \(D_{j}:= (T -\lambda _{j})P_{j}\) are nilpotent bounded operators ( \(D_{j}^{k_{j}} = 0\) where k j is the order of the pole λ j ).
Proof
Let ω ′ be such that ω ess < ω ′ < ω. Let t > 0 be fixed. Then \(e^{\omega _{\mathit{ess}}t} < e^{\omega ^{{\prime}}t } < e^{\omega t}\) and
consists of a finite (and nonempty) set of eigenvalues with finite algebraic multiplicities \(\left \{\mu _{1},\ldots,\mu _{n}\right \}\) while
for some \(\varepsilon > 0\). For each j (1 ≤ j ≤ n) let \(\left \{\lambda _{j}^{1},\ldots,\lambda _{j}^{l_{j}}\right \}\) be the (finite and nonempty) set of eigenvalues λ of T such that \(e^{\lambda t} =\mu _{j}.\) Then the residue of the pole μ j of the resolvent of S(t) is given by
where \(P_{j}^{k}\) is the residue of the \(\lambda _{j}^{k}\) of the resolvent of T. Let \(Q =\sum _{ j=1}^{n}Q_{j}\) be the spectral projection corresponding to the eigenvalues \(\left \{\mu _{1},\ldots,\mu _{n}\right \}\) of S(t) in \(\left \{\mu;\ \left \vert \mu \right \vert \geqslant e^{\omega ^{{\prime}}t }\right \}\). One sees that \(Q =\sum _{ j=1}^{n}\sum _{k=1}^{l_{j}}P_{j}^{k}\) is nothing but the spectral projection corresponding to the eigenvalues of T in \(\left \{\mathit{Re}\lambda \geqslant \omega ^{{\prime}}\right \}\). We decompose S(t) as S(t)Q + S(t)(I − Q). We know that \(\sigma (S(t)_{\mid \mathit{Im}Q}) = \left \{\mu _{1},\ldots,\mu _{n}\right \}\) while \(\sigma (S(t)_{\mid \mathit{Ker}Q}) \subset \left \{\mu;\ \left \vert \mu \right \vert < e^{(\omega ^{{\prime}}-\varepsilon )t }\right \}\) so the type of S(t)∣Ker Q is \(\leq \omega ^{{\prime}}-\varepsilon.\) Finally, \(S(t)_{\mid \mathit{Im}Q}\) is generated by the bounded operator
so \(S(t)_{\mid \mathit{Im}Q} =\sum _{ j=1}^{m}e^{\lambda _{j}t}e^{tD_{j}}P_{j}.\)
2.9 Peripheral Spectral Theory and Applications
In ordered Banach spaces, positive semigroups (i.e. leaving invariant the positive cone) enjoy nice spectral properties. For the sake of simplicity, we restrict ourselves to Lebesgue spaces
where \(\left (\varOmega,\mathcal{A},\mu \right )\) is a measure space (i.e. Ω is a set, \(\mathcal{A}\) is a σ-algebra of subsets of Ω and μ is a σ-finite measure on \(\mathcal{A}\)) although most of the results hold in general Banach lattices. For short, we will write L p(μ) (or just L p) instead of \(L^{p}\left (\varOmega,\mathcal{A},\mu \right )\). Let \(L_{+}^{p}\left (\mu \right )\) be the cone of nonnegative a.e. functions. Then
More precisely
where
In particular
where \(\left \vert f\right \vert (x):= \left \vert f(x)\right \vert\). An operator G ∈ L(X) is said to be positive if \(Gf \in L_{+}^{p}\left (\mu \right )\ \forall f \in L_{+}^{p}\left (\mu \right )\). We write \(G\geqslant 0.\) In this case
and consequently
It follows that if \(0 \leq G_{1} \leq G_{2}\) with \(G_{i} \in \mathcal{L}(L^{p})\)(i = 1, 2) then \(\left \Vert G_{1}\right \Vert \leq \left \Vert G_{2}\right \Vert\). This last property applied to the iterates shows that \(r_{\sigma }(G_{1}) \leq r_{\sigma }(G_{2})\). It is easy to see that \(G \in \mathcal{L}(L^{p})\) is positive if and only if its dual operator \(G^{{\prime}}\in \mathcal{L}(L^{p^{{\prime}} })\) is positive.
A C 0-semigroup \(\left (S(t)\right )_{t\geqslant 0}\) on X is said to be positive if \(\forall t > 0\), S(t) is a positive operator. A C 0-semigroup \(\left (S(t)\right )_{t\geqslant 0}\) with type ω and generator T is positive if and only if the resolvent (λ − T)−1 is positive for λ > ω; this follows from
and the exponential formula
A fundamental result for positive C 0 -semigroups \(\left (S(t)\right )_{t\geqslant 0}\) on Lebesgue spaces \(L^{p}\left (\mu \right )\) is that the type of \(\left (S(t)\right )_{t\geqslant 0}\) coincides with the spectral bound s(T) of its generator T (see e.g. [20]). Another fundamental spectral property of positive operators \(G \in \mathcal{L}(X)\) (in general Banach lattices) is that the spectral radius belongs to the spectrum
Let us show an analogous property for a generator T of a positive semigroup \(\left (S(t)\right )_{t\geqslant 0}\):
Indeed, note first that
so \(\left \Vert (\lambda -T)^{-1}\right \Vert \leq \left \Vert (\mathit{Re}\lambda - T)^{-1}\right \Vert \ \ \left (\forall \mathit{Re}\lambda > s(T)\right ).\) By assumption there exists a sequence β n n ⊂ σ(T) such that \(\mathit{Re}\beta _{n} \rightarrow s(T)\). We build a sequence \(\left (\lambda _{n}\right )_{n}\) with \(\mathit{Re}\lambda _{n} > s(T)\) (so \(\left (\lambda _{n}\right )_{n} \subset \rho (T)\)), \(\mathit{Im}\lambda _{n} = \mathit{Im}\beta _{n}\) and \(\mathit{Re}\lambda _{n} \rightarrow s(T)\). Then \(\left \vert \lambda _{n} -\beta _{n}\right \vert \rightarrow 0\) and \(\left \Vert (\lambda _{n} - T)^{-1}\right \Vert \rightarrow +\infty.\) Thus \(\left \Vert (\mathit{Re}\lambda _{n} - T)^{-1}\right \Vert \rightarrow +\infty \) and consequently s(T) ∈ σ(T).
Let \(G \in \mathcal{L}(L^{p})\) be positive. We say that G is irreducible if \(\forall f \in L_{+}^{p}(\mu ),\ f\neq 0\) and \(\forall g \in L_{+}^{p^{{\prime}} }(\mu ),\ g\neq 0\) there exists \(n \in \mathbb{N}\) (depending a priori on f and g) such that
For p < +∞, this is equivalent to saying that there is no closed subspace \(L^{p}(\varOmega ^{{\prime}},\mu )\) (with μ(Ω ′) > 0 and μ(Ω∕Ω ′) > 0) invariant by G. For instance, if Gf > 0 a.e. \(\forall f \in L_{+}^{p}(\varOmega ),\ f\neq 0\) (we say that G is positivity-improving) then G is irreducible. A positive C 0 -semigroup \(\left (S(t)\right )_{t\geqslant 0}\) is said to be irreducible if \(\forall f \in L_{+}^{p}(\mu ),\ f\neq 0\) and \(\forall g \in L_{+}^{p^{{\prime}} }(\mu ),\ g\neq 0\) there exists t > 0 (depending a priori on f and g) such that
For p < +∞, this is equivalent to saying that there is no closed subspace \(L^{p}(\varOmega ^{{\prime}},\mu )\) (with μ(Ω ′) > 0 and μ(Ω∕Ω ′) > 0) invariant by all S(t). A positive C 0 -semigroup \(\left (S(t)\right )_{t\geqslant 0}\) with generator T is irreducible if and only if (λ − T)−1 is positivity-improving for some λ > s(T). This follows easily from
We recall a useful result combining compactness and irreducibility:
Theorem 8 ([63])
If \(G \in \mathcal{L}(X)\) is compact and irreducible then r σ (G) > 0.
The fact that \(r_{\sigma }(G^{n}) = r_{\sigma }(G)^{n}\) implies easily:
Corollary 2
If some power of G ∈ L(X) is compact and positivity-improving then r σ (G) > 0.
The following result can be found in [61, Chapter CIII].
Theorem 9
Let \(\left (S(t)\right )_{t\geqslant 0}\) be a positive C 0 -semigroup on \(L^{p}\left (\mu \right )\) with generator T. If s(T) is a pole of the (λ − T) −1 then the boundary spectrum
consists of poles of the resolvent and is cyclic in the sense that there exists \(\alpha \geqslant 0\) such that
Corollary 3
Let \(\left (S(t)\right )_{t\geqslant 0}\) be a positive C 0 -semigroup on \(L^{p}\left (\varOmega,\mathcal{A},\mu \right )\) with generator T. We assume that \(\left (S(t)\right )_{t\geqslant 0}\) is essentially compact (i.e. ω ess < ω). Then
i.e. s(T) is the leading eigenvalue and is strictly dominant (i.e. \(\exists \varepsilon > 0;\ \mathit{Re}\lambda \leq s(T) -\varepsilon \ \forall \lambda \in \sigma (T),\ \lambda \neq s(T)\) ).
Proof
According to the theorem above, σ b (T) is either unbounded or reduces to \(\left \{s(T)\right \}\). The fact that ω ess < ω implies that σ b (T) is finite. ■
By combining essential compactness and positivity arguments we get a fundamental functional analytic result:
Theorem 10 ([61] Prop 3.5, p. 310)
Let \(\left (S(t)\right )_{t\geqslant 0}\) be an irreducible C 0 -semigroup on \(L^{p}\left (\varOmega,\mathcal{A},\mu \right )\) with generator T. We assume that \(\left (S(t)\right )_{t\geqslant 0}\) is essentially compact (i.e. ω ess < ω). Then s(T) is the leading eigenvalue, is strictly dominant and is algebraically simple. In particular there exists \(\varepsilon > 0\) such that
where u is the (strictly positive almost everywhere) eigenfunction of T associated to s(T) and v is the (strictly positive almost everywhere) eigenfunction of T ′ associated to s(T ′ ) = s(T) with the normalization ∫u(x)v(x)μ(dx) = 1.
2.10 Semigroups with Dense Local Quasinilpotence Subspace
This subsection deals with a class of positive semigroups whose real spectra can be described completely. This class is well-suited to weighted shift semigroups we consider in the next section. We resume here some abstract results from [56]. For the sake of simplicity, we restrict ourselves to complex Lebesgue spaces X = L p(μ) (1 ≤ p ≤ ∞). Let \((S(t))_{t\geqslant 0}\) be a positive semigroup on L p(μ). We define its local quasinilpotence subset by
where \(\left \vert f\right \vert\) is the absolute value of f ∈ L p(μ).
Lemma 1
Y is a subspace of L p (μ) invariant under \((S(t))_{t\geqslant 0}.\)
Proof
-
(i)
Linearity: Clearly λ f ∈ Y if f ∈ Y. Let \(\varepsilon > 0,\ f,g \in Y\) be given. There exists \(\overline{t} > 0\) depending on them such that
$$\displaystyle{ \left \Vert S(t)\left \vert f\right \vert \right \Vert \leq \varepsilon ^{t}\text{ and }\left \Vert S(t)\left \vert g\right \vert \right \Vert \leq \varepsilon ^{t}\ \ \forall t\geqslant \overline{t}. }$$So \(\left \Vert S(t)\left \vert f + g\right \vert \right \Vert \leq \left \Vert S(t)\left (\left \vert f\right \vert + \left \vert g\right \vert \right )\right \Vert \leq 2\varepsilon ^{t}\ \ \forall t\geqslant \overline{t}\) and
$$\displaystyle{ \left \Vert S(t)\left \vert f + g\right \vert \right \Vert ^{\frac{1} {t} } \leq 2^{\frac{1} {t} }\varepsilon \leq 2\varepsilon \ \ \forall t\geqslant \max (\overline{t},1). }$$ -
(ii)
Invariance: Let τ > 0, f ∈ Y.
$$\displaystyle\begin{array}{rcl} \left \Vert S(t)\left \vert S(\tau )f\right \vert \right \Vert ^{\frac{1} {t} }& \leq &\left \Vert S(t)\left (S(\tau )\left \vert f\right \vert \right )\right \Vert ^{\frac{1} {t} } {}\\ & =& \left \Vert S(t+\tau )\left (\left \vert f\right \vert \right )\right \Vert ^{\frac{1} {t} } = \left (\left \Vert S(t+\tau )\left (\left \vert f\right \vert \right )\right \Vert ^{ \frac{1} {t+\tau } }\right )^{\frac{t+\tau } {t} } \rightarrow 0 {}\\ \end{array}$$as \(t \rightarrow +\infty;\) i.e. S(τ)f ∈ Y.
Theorem 11
Let \((S(t))_{t\geqslant 0}\) be a positive semigroup on L p (μ) with type ω. If its local quasinilpotence subspace is dense in L p (μ) then \(\left [0,e^{\omega t}\right ] \subset \sigma _{\mathit{ap}}(S(t)).\)
Proof
Let t > 0 be fixed. Let 0 < μ < e ω t and y ∈ Y. The equation
can be solved by
provided that this series converges. This is the case since
In particular \(x\geqslant 0\) for \(y\geqslant 0\) and
There exists \(z_{k} \in L_{+}^{p}(\mu )\) such that \(\left \Vert z_{k}\right \Vert = 1\) and
By the denseness of Y, ∃ y k ∈ Y such that \(\left \Vert y_{k}\right \Vert = 1\)
We may assume that \(y_{k}\geqslant 0\) since \(\left \Vert S(t)^{k}\left \vert y_{k}\right \vert \right \Vert \geqslant \left \Vert S(t)^{k}y_{k}\right \Vert \ \) and \(\left \vert y_{k}\right \vert \in Y.\) The solution \(\hat{x}_{k}\) of
satisfies
So
and then \(\lim _{k\rightarrow +\infty }\left \Vert \hat{x}_{k}\right \Vert = \infty.\) Finally \(x_{k}:= \frac{\hat{x}_{k}} {\left \Vert \hat{x}_{k}\right \Vert }\) is such that
i.e. μ ∈ σ ap (S(t)). The closedness of σ ap (S(t)) ends the proof. ■
Lemma 2
Let \((S(t))_{t\geqslant 0}\) be a positive semigroup on L p (μ) with generator T. Let Y be the local quasinilpotence subspace of \((S(t))_{t\geqslant 0}.\) Then, for any λ > ω,
Proof
For any y ∈ Y and any \(\varepsilon > 0\) there exists \(t_{y,\varepsilon } > 0\) such that
i.e. (write \(\varepsilon = e^{-A}\))
so \(\exists M_{y,A}\geqslant 0\) such that
Hence
and
which ends the proof since A > 0 is arbitrary. ■
Theorem 12
Let \((S(t))_{t\geqslant 0}\) be a positive semigroup on L p (μ) with generator T. Let s(T) be the spectral bound of T. If the local quasinilpotence subspace of \((S(t))_{t\geqslant 0}\) is dense in L p (μ) then
Proof
Let λ < s(T) < μ be fixed. Consider
Arguing as for the semigroup, we show the existence of \((x_{k})_{k}\) with \(\left \Vert x_{k}\right \Vert = 1\) and
i.e. \(\frac{1} {\mu -\lambda } \in \sigma _{\mathit{ap}}((\mu -T)^{-1})\) or equivalently λ ∈ σ ap (T). The closedness of σ ap (T) ends the proof. ■
Corollary 4
Let \((S(t))_{t\geqslant 0}\) be a positive semigroup on L p (μ) with type ω and generator T. We assume that the local quasinilpotence subspace of \((S(t))_{t\geqslant 0}\) is dense in L p (μ).
-
(i)
If σ(T) is invariant under translations along the imaginary axis then
$$\displaystyle{ \sigma (T) = \left \{\lambda \in \mathbb{C};\ \mathit{Re}\lambda \leq \omega \right \}. }$$ -
(ii)
If σ(S(t)) is invariant under rotations then
$$\displaystyle{ \sigma (S(t)) = \left \{\mu \in \mathbb{C};\ \left \vert \mu \right \vert \leq e^{\omega t}\right \}. }$$
3 Spectral Analysis of Advection Semigroups
Neutron transport theory is mainly a perturbation theory (by scattering operators) of suitable weighted shift semigroups called advection semigroups
where
is the (first) exit time function from the spatial domain Ω. We describe here the spectra of such semigroups. This section resumes essentially [56]; (an alternative approach is given in [78]).
3.1 On Advection Semigroups
Let \(\varOmega \subset \mathbb{R}^{n}\) be an open subset and let μ be a positive Borel measure on \(\mathbb{R}^{n}\) with support V. Let
be measurable and such that
Let
be the so-called exit time function. Then
defines a positive semigroup on \(L^{p}(\varOmega \times V;\mathit{dx} \otimes \mu (\mathit{dv}))\) (for any \(1 \leq p \leq +\infty \)), strongly continuous when p < +∞, see e.g. [78]. The dual streaming semigroups in \(L^{p^{{\prime}} }(\varOmega \times V )\ \) are given by
3.2 Invariance Property of Transport Operators
Let \(\mu \left \{0\right \} = 0\) and
For any η > 0
is an isometric isomorphism.
Theorem 13 ([78])
\(M_{\eta }^{-1}S(t)M_{\eta } = e^{i\eta t}S(t).\) In particular σ(S(t)) is invariant by rotations.
Proof
We have
so
so \(M_{\eta }^{-1}S(t)M_{\eta } = e^{i\eta t}S(t)\). Hence \(\sigma (M_{\eta }^{-1}S(t)M_{\eta }) =\sigma (e^{i\eta t}S(t)).\) On the other hand, by similarity,
and
so we are done. ■
As previously we have:
Theorem 14
Let T be the generator of a streaming semigroup \((S(t))_{t\geqslant 0}\) . Then \(M_{\eta }^{-1}TM_{\eta } = T + i\eta I.\) In particular σ(T) is invariant by translation along the imaginary axis.
Proof
Let f ∈ D(T). Then
so M η f ∈ D(T) and \(TM_{\eta }f = M_{\eta }Tf + i\eta M_{\eta }f\ \) or \(M_{\eta }^{-1}TM_{\eta } = T + i\eta I.\) By similarity, \(\sigma (T) =\sigma (M_{\eta }^{-1}TM_{\eta }) =\sigma (T) + i\eta \forall \eta \in \mathbb{R}\). ■
3.3 Decomposition of the Phase Space
We consider the partition of the phase space Ω × V according to
This induces a direct sum
where, we identify \(L^{p}(E_{i})\) to the closed subspace of functions f ∈ L p(Ω × V ) vanishing almost everywhere on Ω × V ∖ E i . If some set E i has zero measure then we drop out L p(E i ) from the direct sum above.
Theorem 15
The subspaces \(L^{p}(E_{i})\ (i = 1,2,3)\) are invariant under \((S(t))_{t\geqslant 0.}.\) For each i = 1,2,3, we denote by \((S_{i}(t))_{t\geqslant 0}\) the part of \((S(t))_{t\geqslant 0.}\) on L p (E i ) and by T i its generator. Then
We have also similar results where σ(.) is replaced by σ p (.) or σ ap (.). In addition, if σ(.,.) is bounded then \((S_{3}(t))_{t\geqslant 0}\) extends to a positive group.
Proof
We check that the direct sum \(L^{p}(\varOmega \times V ) = L^{p}(E_{1}) \oplus L^{p}(E_{2}) \oplus L^{p}(E_{3})\) reduces \((S(t))_{t\geqslant 0.}\ \) We restrict ourselves to \(L^{p}(E_{1}).\) Let \(f \in L^{p}(E_{1})\), i.e. f vanishes almost everywhere on \(E_{2} \cup E_{3}\). We have to show that \(S(t)f \in L^{p}(E_{1})\) i.e. S(t)f vanishes almost everywhere on \(E_{2} \cup E_{3}\). Since
is zero for t > s(x, v), we assume from the start that t ≤ s(x, v). One notes that \((x,v) \in E_{2} \cup E_{3} \Leftrightarrow s(x,-v) = +\infty \) and
so that \((x - tv,-v) \in E_{2} \cup E_{3}\) and f(x − tv, v) = 0. Since the projection P i on \(L^{p}(E_{i})\) along \(L^{p}(\varOmega \times V \ \setminus \ E_{i})\) commutes with \((S(t))_{t\geqslant 0.}\) then the direct sum above reduces also the generator T. Finally, on E 3 (if σ(. , . ) is bounded) \((S_{3}(t))_{t\geqslant 0}\) extends to a positive group where
3.4 Spectra of the First Reduced Advection Semigroup
Lemma 3
Let t > 0 be fixed. For any \(f \in L^{p}(E_{1})\)
Proof
We have to compute the norm of S 1(t)f on the set
so \(\left \Vert S_{1}(t)f\right \Vert ^{p}\) is equal to
Since s(x − tv, −v) = t + s(x, −v) is finite if and only if s(x, −v) is finite then the change of variable
gives s(y, −v) > t and
The type of \((S_{1}(t))_{t\geqslant 0}\) is equal to \(-\lambda _{1}^{{\ast}}\) where
because
so
and
We have
Indeed, it suffices to show that the local quasinilpotence subspace of \((S_{1}(t))_{t\geqslant 0}\) is dense in \(L^{p}(E_{1}).\) Let
We note that \(\cup _{m}L^{p}(O_{m})\) is dense in \(L^{p}(E_{1})\) because of
Finally
shows that, for \(f \in L^{p}(O_{m}),\left \Vert S_{1}(t)f\right \Vert = 0\) for t > m so \(\cup _{m}L^{p}(O_{m})\) is included in the local quasinilpotence subspace of \((S_{1}(t))_{t\geqslant 0}\). ■
3.5 Spectra of the Second Reduced Advection Semigroup
We deal now with \((S_{2}(t))_{t\geqslant 0}\) on \(L^{p}(E_{2})\) where
We consider first the case 1 < p < +∞. Indeed, by duality \(\sigma (S_{2}(t)) =\sigma (S_{2}^{{\prime}}(t))\) where
Thus \(\left \Vert S_{2}(t)f\right \Vert ^{p^{{\prime}} }\) is equal to
Introducing the sets
one sees that \(\cup _{m}L^{p^{{\prime}} }(O_{m}^{{\prime}})\) is dense in \(L^{p^{{\prime}} }(E_{2})\) because of
Since in \(L^{p^{{\prime}} }(O_{m}^{{\prime}}),\left \Vert S_{2}(t)f\right \Vert = 0\) for t > m then the local quasinilpotence subspace of \((S_{2}^{{\prime}}(t))_{t\geqslant 0}\) is dense. This ends the proof because \(\sigma (S_{2}(t)) =\sigma (S_{2}^{{\prime}}(t))\) and \(\sigma (T_{2}) =\sigma (T_{2}^{{\prime}})\). ■
3.6 Spectra of the Third Reduced Advection (Semi)group
Theorem 16
Let \(\mathcal{S}:=\sigma (T_{3})\cap \mathbb{R}\) be the real spectrum of T 3 . Then
Moreover, \(sup\mathcal{S} = -\lambda _{3}^{{\ast}}\) and \(\inf \mathcal{S} = -\lambda _{3}^{{\ast}{\ast}}\) where
Proof
The fact that σ(T 3) is invariant by translation along the imaginary axis and that \(e^{t\sigma (T_{3})}\) is invariant under the rotations is a general feature of streaming semigroups in arbitrary geometry. The spectral mapping property for the real spectrum is due to the fact that \((S_{3}(t))_{t\in \mathbb{R}}\) is a positive C 0-group (see [27]). The type \(-\lambda _{3}^{{\ast}}\) of \((S_{3}(t))_{t\geqslant 0}\) is obtained as for \((S_{1}(t))_{t\geqslant 0}\) or \((S_{2}(t))_{t\geqslant 0}.\) Finally, \(\lambda _{3}^{{\ast}}\) is the spectral bound of the generator of \((S_{3}(-t))_{t\geqslant 0}\), (i.e. − T 3) and is obtained similarly. ■
Theorem 17
If \(\sigma:\varOmega \times V \rightarrow \mathbb{R}_{+}\) is space-homogeneous then \(\mathcal{S}:=\sigma (T_{3})\cap \mathbb{R}\) is nothing but the essential range of −σ(.).
See the details in [56]; in particular, \(S:=\sigma (T_{3})\cap \mathbb{R}\) need not be connected. The description of \(\sigma (T_{3})\cap \mathbb{R}\) for general collision frequency\(\ \sigma:\varOmega \times V \rightarrow \mathbb{R}_{+}\) seems to be open. When \(\varOmega =\mathbb{R} ^{n}\), the situation is well understood for bounded and compactly supported (in space) collision frequencies; see [28].
3.7 Reminders on Sun-Dual Theory
To study σ(S 2(t)) in L 1 spaces, we need to recall some material. Let X be a complex Banach space and let \((S(t))_{t\geqslant 0}\) be a C 0-semigroup on X with generator T. Let \((S^{{\prime}}(t))_{t\geqslant 0}\) be the dual semigroup on the dual space X ′. If X is not reflexive then a priori \((S^{{\prime}}(t))_{t\geqslant 0}\) is not strongly continuous. Let
the subspace of strong continuity of \((S^{{\prime}}(t))_{t\geqslant 0}.\) Then
-
L ⊙ is a closed subspace of X ′ invariant under \((S^{{\prime}}(t))_{t\geqslant 0}.\)
-
\(L^{\odot } = \overline{D(T^{{\prime}})}\) (the closure in X ′).
We denote by \((S^{\odot }(t))_{t\geqslant 0}\) the restriction of \((S^{{\prime}}(t))_{t\geqslant 0}\) to L ⊙ (sun-dual C 0 -semigroup). Its generator is given by
and we have \(\sigma (T) =\sigma (T^{{\prime}}) =\sigma (T^{\odot })\) and \(\sigma (S(t)) =\sigma (S^{{\prime}}(t)) =\sigma (S^{\odot }(t));\ \) see e.g. [20, Chapter IV].
3.8 Sun-Dual Theory for Advection Semigroups
We consider \((S_{2}(t))_{t\geqslant 0}\) in \(L^{1}(E_{2})\) where
and assume that
Since
it suffices to identify \(\sigma (S_{2}^{\odot }(t))\). Because of the boundedness of σ,
Actually, we are going to work in the smaller closed subspace
Lemma 4
\((S^{{\prime}}(t))_{t\geqslant 0}\) leaves invariant \(L_{0}^{\odot }.\)
Proof
and \(s(y + tv,v) = s(y,v) + t \rightarrow \infty \) if and only if \(s(y,v) \rightarrow \infty \) so that \(S^{{\prime}}(t)f \in L_{0}^{\odot }\) if \(f \in L_{0}^{\odot }\). ■
Let \(\left (S_{0}^{\odot }(t)\right )_{t\geqslant 0}\) be the restriction of \(\left (S_{2}^{\odot }(t)\right )_{t\geqslant 0}\) to \(L_{0}^{\odot }\) and let \(T_{0}^{\odot }\) be its generator. Then \(\sigma _{\mathit{ap}}(S_{0}^{\odot }(t)) \subset \sigma _{\mathit{ap}}(S_{2}^{\odot }(t))\) and \(\sigma _{\mathit{ap}}(T_{0}^{\odot }) \subset \sigma _{\mathit{ap}}(T_{2}^{\odot })\). In particular, \(\sigma _{\mathit{ap}}(S_{0}^{\odot }(t)) \subset \sigma (S_{2}(t))\) and \(\sigma _{\mathit{ap}}(T_{0}^{\odot }) \subset \sigma (T_{2})\). Let
Theorem 18
\(L_{00}^{\odot }\) is dense in \(L_{0}^{\odot }.\)
Corollary 5
The local quasinilpotence subspace of \(\left (S_{0}^{\odot }(t)\right )_{t\geqslant 0}\) is dense in \(L_{0}^{\odot }.\)
Proof of Corollary 5:
The local quasinilpotence subspace of \(\left (S_{0}^{\odot }(t)\right )_{t\geqslant 0}\) contains \(L_{00}^{\odot }\). ■
Before proving Theorem 18, we need:
Lemma 5
L ⊙ is an algebra.
Proof
For each \(m \in \mathbb{N}\), let \(\gamma _{m}: \left [0,+\infty \right [ \rightarrow \left [0,1\right ]\) be smooth (say C 1) and such that
Lemma 6
\(\forall m \in \mathbb{N},\ (x,v) \rightarrow \gamma _{m}(s(x,v))\) belongs to \(L_{00}^{\odot }.\)
Proof
We have just to show that \((x,v) \rightarrow \gamma _{m}(s(x,v))\) belongs to L ⊙. Since γ m is Lipschitz then
■
Proof of Theorem 18:
Let \(f \in L_{0}^{\odot }\) then \(\forall m \in \mathbb{N},(x,v) \rightarrow \gamma _{m}(s(x,v))f(x,v)\) belongs to \(L_{00}^{\odot }\).
since \(f \in L_{0}^{\odot }\). ■
By the general theory,
where \(\omega _{0}^{\odot }\) is the type of \(\left (S_{0}^{\odot }(t)\right )_{t\geqslant 0}\). We have to identify \(\omega _{0}^{\odot }!\ \) The fact that \(\left \Vert S_{0}^{\odot }(t)\right \Vert \leq \left \Vert S_{2}^{{\prime}}(t)\right \Vert = \left \Vert S_{2}(t)\right \Vert\) implies
On the other hand,
belongs to \(L_{0}^{\odot }\) and \(\left \Vert g_{m}\right \Vert \leq 1\) so that
But γ m (s(y, v) + t) = 1 if s(y, v) ≤ m − t so
Finally
whence \(\omega _{2} \leq \omega _{0}^{\odot }\) and we are done. ■
A similar theory can be built for general vector fields. Indeed, consider\(\mathcal{F}:\mathbb{R} ^{n} \rightarrow \mathbb{R}^{n}\) a Lipschitz vector field and denote by Φ(x, t) the unique global solution to
Let \(\varOmega \subset \mathbb{R}^{n}\) be an open set and let
be the exit times from Ω (with the convention that \(\inf \varnothing = +\infty )\). We define a weighted shift semigroup
where
We introduce the sets
and
Then \(L^{p}(\varOmega _{i})\) (i = 1, 2, 3) are invariant under \((U(t))_{t\geqslant 0}\) and we can extend the previous spectral theory of advection semigroups, see [39].
4 Spectra of Perturbed Operators
This section deals with functional analytic results on stability of essential spectra for perturbed generators or perturbed semigroups on Banach spaces. Let X be a complex Banach space and \(D \subset \mathbb{C}\) be open and connected. A compact operator valued meromorphic mapping
(\(\mathcal{C}(X) \subset \mathcal{L}(X)\) is the closed subspace of compact operators) is called essentially meromorphic on D if A is holomorphic on D except at a discrete set of points z k ∈ D where A has poles with Laurent expansions
where \(A_{n}(z_{k})(n = -1,-2,\ldots,-m_{k})\ \) are finite rank operators. We recall now a fundamental analytic Fredholm alternative:
Theorem 19 ([65] Corollary II)
Let X be a complex Banach space and \(D \subset \mathbb{C}\) be open and connected. Let
be essentially meromorphic. Then
-
(i)
Either λ = 1 is an eigenvalue of all A(z)
-
(ii)
or \(\left [I - A(z)\right ]^{-1}\) exists except for a discrete set of points and \(\left [I - A(z)\right ]^{-1}\) is essentially meromorphic on D.
Let \(T: D(T) \subset X \rightarrow X\) be a closed operator. We define its “essential resolvent set” as
where σ discr (T) refers to the isolated eigenvalues of T with finite algebraic multiplicities. This set is open. We note that if \(T \in \mathcal{L}(X)\) then the unbounded component of ρ e (T) coincides with the unbounded component of the Fredholm domain ρ F (T) (see [21, p. 204]). We give first a result from [77] and some of its consequences.
Theorem 20
Let \(T: D(T) \subset X \rightarrow X\) be a closed operator and let Ω be a connected component of ρ e (T). Let \(B: D(T) \rightarrow X\) be T-bounded such that there exists \(n \in \mathbb{N}\) and
We assume that there exists some \(\lambda \in \varOmega \cap \rho (T)\) such that \(I -\left [B(\lambda -T)^{-1}\right ]^{n}\) is invertible (i.e. 1 is not an eigenvalue of \(\left [B(\lambda -T)^{-1}\right ]^{n}\) ). Then Ω ⊂ρ e (T + B) and
Proof
We note that (λ − T)−1 is essentially meromorphic on Ω. Then \(B_{\lambda }:= B(\lambda -T)^{-1}\) and \(B_{\lambda }^{n} = \left [B(\lambda -T)^{-1}\right ]^{n}\) are also essentially meromorphic on Ω. Since \(B_{\lambda }^{n}\) is operator compact valued then, by the analytic Fredholm alternative \((I - B_{\lambda }^{n})^{-1}\) is also essentially meromorphic on Ω. On the other hand
shows that
is also essentially meromorphic on Ω and then so is
i.e. Ω ⊂ ρ e (T + B). Finally
is also compact on \(\varOmega \cap \rho (T + B)\). ■
We give now a more precise version of the theorem above under an additional assumption.
Corollary 6
Let \(T: D(T) \subset X \rightarrow X\) be a closed operator and let Ω be a connected component of ρ e (T). Let \(B: D(T) \rightarrow X\) be T-bounded such that there exists \(n \in \mathbb{N}\) and
We assume that there exists a sequence \(\left (\lambda _{j}\right )_{j} \subset \varOmega \cap \rho (T)\) such that
Then \(\left (\lambda _{j}\right )_{j} \subset \rho (T + B)\) for j large enough and \(\left \Vert B(\lambda _{j} - T - B)^{-1}\right \Vert \rightarrow 0\) as \(j \rightarrow +\infty \) . Furthermore Ω is a component of ρ e (T + B).
Proof
We note that for j large enough
Let Ω ′ be the component of ρ e (T + B) which contains Ω. We know that
is compact \((\lambda \in \varOmega \cap \rho (T + B))\). By analyticity, this extends to \(\varOmega ^{{\prime}}\cap \rho (T + B).\) By considering T as (T + B) − B and reversing the arguments in the previous theorem one gets \(\varOmega ^{{\prime}} \subset \rho _{e}(T)\) and consequently Ω = Ω ′. ■
Corollary 7
Let \(T,B \in \mathcal{L}(X)\) . We assume that \(\left [B(\lambda -T)^{-1}\right ]^{n}\ \ \) is compact on the unbounded component of ρ(T), i.e.
where Ω is the unbounded connected component of ρ e (T). The unbounded components of ρ e (T) and ρ e (T + B) coincide and then
Proof
Since \(\left \Vert (\lambda -T)^{-1}\right \Vert \rightarrow 0\) as \(\left \vert \lambda \right \vert \rightarrow \infty \) we apply the corollary above. ■
Corollary 8
Let \(T,B \in \mathcal{L}(X).\)
-
(i)
If B is compact then \(r_{e}(T) = r_{e}(T + B).\)
-
(ii)
If X = L 1 (μ) and B is weakly compact then \(r_{e}(T) = r_{e}(T + B).\)
Proof
The case (i) is clear with n = 1. In the case (ii) B(λ − T)−1 is also weakly compact on L 1(μ) and consequently its square is compact. ■
4.1 Strong Integral of Operator Valued Mappings
Let (Ω, μ) be a finite measure space and let X, Y be two Banach spaces. Let
be bounded and strongly measurable in the sense that for each x ∈ X
is (Bochner) measurable. We define the strong integral of G on Ω as the bounded operator
We note that strongly continuous mappings appear everywhere in semigroup theory!
Theorem 21 ([33, 80, 82])
Under the conditions above, assume in addition that \(\forall \omega \in \varOmega,G(\omega ) \in \mathcal{C}(X,Y )\) (i.e. G(ω) is a compact operator). Then
In the statement above, we can replace “compact” by “weakly compact” [69]. Direct proofs in Lebesgue spaces relying on Kolmogorov’s compactness criterion and the Dunford-Pettis criterion of weak compactness are given in [50].
4.2 Spectra of Perturbed Generators
Theorem 22
Let X be a complex Banach space. Let \(\left (U(t)\right )_{t\geqslant 0}\) be a C 0 -semigroup with generator T and let \(\ K \in \mathcal{L}(X)\) . We denote by \(\left (V (t)\right )_{t\geqslant 0}\) the C 0 -semigroup generated by T + K. We assume that K is T-power compact i.e. there exists \(n \in \mathbb{N}\) such that (for λ in some right half-plane included in ρ(T + K))
Then the components of ρ e (T) and ρ e (T + K) containing a right half-plane coincide. In particular
consists at most of isolated eigenvalues with finite algebraic multiplicities.
The proof follows from Corollary 6 since \(\left \Vert (\lambda -T)^{-1}\right \Vert \rightarrow 0\) as \(\mathit{Re}\lambda\) goes to + ∞. This theorem is a refinement of a first version due to I. Vidav [75].
4.3 Dyson–Phillips Expansions
The perturbed semigroup \(\left (V (t)\right )_{t\geqslant 0}\) is related to the unperturbed one \(\left (U(t)\right )_{t\geqslant 0}\) by an integral equation (Duhamel equation)
The integrals are interpreted in a strong sense i.e.
The Duhamel equation is solved by standard iterations
and
Finally, \(\left (V (t)\right )_{t\geqslant 0}\) is given by a Dyson–Phillips series
where
By remainder terms of Dyson–Phillips expansions we mean
For any strongly continuous mappings \(f,g:\mathbb{R} _{+} \rightarrow \mathcal{L}(X)\), we define their convolution by (the strong integral)
and note \(\left [f\right ]^{n}\) the n-fold convolution of f with the convention \(\left [f\right ]^{1} = f\). Then we can express U j (. ) for \(j\geqslant 1\) as
Theorem 23 ([48] Chapter 2)
Let \(n \in \mathbb{N}_{{\ast}}\) be given. Then U n (t) is compact for all \(t\geqslant 0\) if and only if \(R_{n}(t):=\sum _{ j=n}^{+\infty }U_{j}(t)\) is compact for all \(t\geqslant 0\) . If X = L 1 (μ) then we can replace “compact” by “weakly compact”.
Proof
If U n (t) is compact for all \(t\geqslant 0\) then
is compact for all \(t\geqslant 0\) as a strong integral of “compact operator” valued mappings. By induction U j (t) is compact (for all \(t\geqslant 0\)) for all j ⩾ n and then R n (t) is compact for all t ⩾0 since the series converges in operator norm. Conversely, let R n (t) be compact for all t ⩾0. Then
shows that R n+1(t) is also compact for all t ⩾0 as a strong integral of “compact operator” valued mappings. Finally \(U_{n}(t) = R_{n}(t) - R_{n+1}(t)\) is also compact for all t ⩾0. ■
The following result is given in [81] for unbounded perturbations. We give here a slightly different (and simpler) presentation of the proof thanks to the boundedness of K.
Theorem 24
Let X be a complex Banach space. Let \(\left (U(t)\right )_{t\geqslant 0}\) be a C 0 -semigroup with generator T and let \(\ K \in \mathcal{L}(X)\) . We denote by \(\left (V (t)\right )_{t\geqslant 0}\) the C 0 -semigroup generated by T + K. If some remainder term
is compact for t large enough then \(\omega _{e}(V ) \leq \omega _{e}(U)\) where ω e (V ) (resp. ω e (U)) is the essential type of \(\left (V (t)\right )_{t\geqslant 0}\) (resp. \(\left (U(t)\right )_{t\geqslant 0}\) ). If X = L 1 (μ) then we can replace “compact” by “weakly compact”.
Proof
Let β > ω e (U) then there exists a finite range projection P β commuting with \(\left (U(t)\right )_{t\geqslant 0}\) such that for any β ′ > β
(with P β = 0 if β > ω(U) the type of \(\left (U(t)\right )_{t\geqslant 0}\)). On the other hand, by stability of essential radius by compact (or weakly compact if X = L 1(μ)) perturbation
for t large enough. We note that
where C j (t) is a sum of convolutions where each convolution involves at least one term of the form \(U(I - P_{\beta })KP_{\beta },\ UP_{\beta }KP_{\beta }\) or \(UP_{\beta }K(I - P_{\beta })\). Such terms are compact (of finite rank) because of P β so that the convolutions are compact for all time as strong integrals of “compact operator” valued mappings. Thus C j (t) is compact for all t ⩾0. Once again, the stability of essential radius by compact perturbation gives for t large enough
Observe that
where \(\left (\tilde{U}(t)\right )_{t\geqslant 0}\) is the semigroup \(\left (U(t)(I - P_{\beta })\right )_{t\geqslant 0}\). More generally
By using the estimate
an elementary calculation shows that \(\left \Vert \left [\tilde{U}K\right ]^{j} {\ast}\tilde{ U}\right \Vert \leq c_{j}t^{j}e^{\beta ^{{\prime}}t }\) so
where p n (t) is a polynomial of degree n. To end the proof, let \(\beta ^{{\prime\prime}} >\beta ^{{\prime}}\). Then there exists a constant \(M_{\beta ^{{\prime\prime}}}\) such that
for t large enough. Let ω e (V ) be the essential type of \(\left (V (t)\right )_{t\geqslant 0}\). The fact that
implies that \(\omega _{e}(V ) \leq \beta ^{{\prime\prime}}\). Hence \(\omega _{e}(V ) \leq \omega _{e}(U)\) since \(\beta ^{{\prime}} >\omega _{e}(U)\) and \(\beta ^{{\prime\prime}} >\beta ^{{\prime}}\) are chosen arbitrarily. ■
Remark 1
A classical weaker estimate ω e (V ) ≤ ω(U) (where ω(U) is the type of \(\left (U(t)\right )_{t\geqslant 0}\)) is due to I. Vidav [76]. The estimate \(\omega _{e}(V ) \leq \omega _{e}(U)\) is also derived in [70, 82] by using the properties of measure of noncompactness of strong integrals.
We have seen that if some remainder term \(R_{n}(t):=\sum _{ j=n}^{+\infty }U_{j}(t)\) is compact (or weakly compact when X = L 1(μ)) for large t then \(\omega _{e}(V ) \leq \omega _{e}(U).\) We show now that if some remainder term is compact (or weakly compact when X = L 1(μ)) for all t ⩾0 then ω e (V ) = ω e (U). We need a preliminary result:
Lemma 7 ([48] Chapter 2)
Let X be a complex Banach space. Let \(\left (U(t)\right )_{t\geqslant 0}\) be a C 0 -semigroup with generator T and let \(\ K \in \mathcal{L}(X)\) . We denote by \(\left (V (t)\right )_{t\geqslant 0}\) the C 0 -semigroup generated by T + K. Let \(V (t) =\sum _{ 0}^{+\infty }U_{j}(t)\) be the Dyson–Phillips expansion of \(\left (V (t)\right )_{t\geqslant 0}.\) Let \(U(t) =\sum _{ 0}^{+\infty }V _{j}(t)\) be the Dyson–Phillips expansion of \(\left (U(t)\right )_{t\geqslant 0}\) considered as a perturbation of \(\left (V (t)\right )_{t\geqslant 0}\) (i.e. T = (T + K) + (−K)). For any \(j \in \mathbb{N}_{{\ast}},\ U_{j}(t)\) is compact for all t⩾0 if and only if V j (t) is compact for all t⩾0. If X = L 1 (μ) then we can replace “compact” by “weakly compact”.
By reversing the role of \(\left (U(t)\right )_{t\geqslant 0}\) and \(\left (V (t)\right )_{t\geqslant 0}\) we obtain:
Corollary 9
Let X be a complex Banach space. Let \(\left (U(t)\right )_{t\geqslant 0}\) be a C 0 -semigroup with generator T and let \(\ K \in \mathcal{L}(X)\) . We denote by \(\left (V (t)\right )_{t\geqslant 0}\) the C 0 -semigroup generated by T + K. Let
be the terms of the Dyson–Phillips expansion of \(\left (V (t)\right )_{t\geqslant 0}\) . If for some \(n \in \mathbb{N}_{{\ast}},U_{n}(t)\) is compact (resp. weakly compact if X = L 1 (μ)) for all t⩾0 then
Remark 2
We note that the stability of essential type appears also in [70, 81, 82] but under stronger assumptions.
4.4 Short Digression on Resolvent Approach
The following “resolvent characterization” is due to S. Brendle [9].
Theorem 25
Let \(n \in \mathbb{N}_{{\ast}}\) . Then U n (t) is compact for all t⩾0 if and only if:
-
(i)
\(t\geqslant 0 \rightarrow U_{n}(t) \in \mathcal{L}(X)\) is continuous in operator norm
and
-
(ii)
\(\left [(\alpha +i\beta - T)^{-1}K\right ]^{n}(\alpha +i\beta - T)^{-1}\) is compact for some α > ω(U) and all \(\beta \in \mathbb{R}\) .
This is useful in some applications (for example for kinetic equations involving boundary operators relating the incoming and outgoing fluxes) where the unperturbed semigroup \(\left (U(t)\right )_{t\geqslant 0}\) is not explicit while the resolvent (λ − T)−1 is “tractable”! The cost of the approach is that we need a priori that
is continuous in operator norm. The following result gives sufficient conditions of continuity in operator norm.
Theorem 26 ([9])
Let \(n \in \mathbb{N}.\) If X is a Hilbert space and if
then \(t\geqslant 0 \rightarrow U_{n+2}(t) \in \mathcal{L}(X)\) is continuous in operator norm.
Note that the continuity of \(t\geqslant 0 \rightarrow U_{1}(t) \in \mathcal{L}(X)\) (if we want to show the compactness of V (t) − U(t)) is out of reach of this theorem. We give now Sbihi’s criterion of continuity in operator norm.
Theorem 27 ([67])
Let X be a Hilbert space and let T be dissipative (i.e. \(\mathit{Re}\left (Tx,x\right ) \leq 0\ \forall x \in D(T)\) ). If
then \(t\geqslant 0 \rightarrow U_{1}(t) \in \mathcal{L}(X)\) is continuous in operator norm.
Useful applications of this result are given in [34, 38, 67].
5 Collisional Transport Theory
In this section, we show how the previous functional analytic tools apply to neutron transport theory. We start with an unperturbed (advection) semigroup in \(L^{p}(\varOmega \times V;\ \mathit{dx} \otimes \mu (\mathit{dv}))\)
(with generator T) where
is the first exit time function from the spatial domain Ω. We regard the scattering operator
as a bounded perturbation of T (we refer to [58] and references therein for generation results with unbounded scattering operators) and denote by \(\left (V (t)\right )_{t\geqslant 0}\) the perturbed neutron transport semigroup. We are faced with two main questions:
-
When is \(\left [(\lambda -T)^{-1}K\right ]^{n}\) compact in L p(Ω × V ) for some \(n \in \mathbb{N}_{{\ast}}\ \)?
-
When is some remainder term R m (t) compact in L p(Ω × V )?
We point out that the resolvent (λ − T)−1 of T cannot be compact (e.g. in bounded geometries, σ(T) is a half-plane when 0 ∈ V ! ). The scattering operator
is local with respect to the space-variable x so that K cannot be compact on L p(Ω × V ). The good news is that compactness will emerge from subtle combinations of properties of T and those of K. For information, we recall some classical results: Under quite general assumptions on the scattering kernel k(x, v, v ′), for bounded domains Ω and Lebesgue measure dv on \(\mathbb{R}^{n}\) as velocity measure, the second order remainder term R 2(t) is compact in L p(Ω × V ) (1 < p < ∞) or weakly compact in L 1(Ω × V ); see e.g. [25, 43, 45, 72, 79, 82].
We introduce now a useful class of scattering operators. Let \(\varOmega \subset \mathbb{R}^{n}\ (n\geqslant 1)\) be an open subset and let μ be a positive Radon measure with support V. Let
with 1 ≤ p < +∞. Let
be measurable and such that
is a bounded operator on L p(Ω × V ). Since K is local in space variable, we may interpret it as a family of bounded operators on L p(V ) indexed by the parameter x ∈ Ω i.e. a mapping
Then
5.1 Lp Theory (1 < p < ∞)
In this section, we restrict ourselves to 1 < p < ∞. A scattering operator K is called regular if
-
(i)
\(\left \{K(x);\ x \in \varOmega \right \}\) is a set of collectively compact operators on L p(V ), i.e. the set
$$\displaystyle{ \left \{K(x)\varphi;\ x \in \varOmega,\ \left \Vert \varphi \right \Vert _{L^{p}(V )} \leq 1\right \} }$$is relatively compact in L p(V ).
-
(ii)
For any \(\psi ^{{\prime}}\in L^{p^{{\prime}} }(V ),\) the set \(\left \{K^{{\prime}}(x)\psi ^{{\prime}};\ x \in \varOmega \right \}\) is relatively compact in \(L^{p^{{\prime}} }(V )\) where p ′ is the conjugate exponent and K ′(x) is the dual operator of K(x).
We note that the compactness of K with respect to velocities (at least in L 2(V )) is satisfied by most physical models [8].
Theorem 28 ([51])
The class of regular scattering operators is the closure in the operator norm of \(\mathcal{L}(L^{p}(\varOmega \times V ))\) of the class of scattering operators with kernels
where \(\alpha _{j} \in L^{\infty }(\varOmega ),\ f_{j},g_{j}\ \) continuous with compact supports and J finite.
The first part of the following lemma is given in ([48, Chapter 3, p. 32]) and the second part in [51].
Lemma 8
Let μ be a finite Radon measure on \(\mathbb{R}^{n}\).
-
(i)
If the hyperplanes (through the origin) have zero μ-measure then
$$\displaystyle{ \sup _{e\in S^{n-1}}\mu \left \{v;\ \left \vert v.e\right \vert \leq \varepsilon \right \}\rightarrow 0\ \ \text{as}\ \varepsilon \rightarrow 0. }$$ -
(ii)
If the affine (i.e. translated) hyperplanes have zero μ-measure then
$$\displaystyle{ \sup _{e\in S^{n-1}}\mu \otimes \mu \left \{(v,v^{{\prime}});\ \left \vert (v - v^{{\prime}}).e\right \vert \leq \varepsilon \right \}\rightarrow 0\ \ \text{as}\ \varepsilon \rightarrow 0. }$$
Theorem 29 ([51])
We assume that Ω has finite Lebesgue measure and the scattering operator is regular in \(L^{p}(\varOmega \times V;\mathit{dx} \otimes \mu (\mathit{dv}))\) (1 < p < ∞). If the hyperplanes have zero μ-measure then (λ − T) −1 K and K(λ − T) −1 are compact on L p (Ω × V;dx ⊗μ(dv)).
Remark 3
The compactness of K(λ − T)−1 can be expressed as an averaging lemma in open sets Ω with finite volume, i.e. if г is a bounded subset of D(T) (for the graph norm) then
We note that if
and if Ω is bounded and convex then the compactness can be measured in terms of fractional Sobolev regularity [23], (see also [1]).
Corollary 10
We assume that Ω has finite Lebesgue measure, the scattering operator is regular in \(L^{p}(\varOmega \times V;\mathit{dx} \otimes \mu (\mathit{dv}))\) (1 < p < ∞) and the hyperplanes have zero μ-measure. Then
In particular \(\sigma (T + K) \cap \left \{\mathit{Re}\lambda > s(T)\right \}\) consists at most of isolated eigenvalues with finite algebraic multiplicities where
Theorem 30 ([51])
We assume that Ω has finite Lebesgue measure and the scattering operator is regular in \(L^{p}(\varOmega \times V;\mathit{dx} \otimes \mu (\mathit{dv}))\) (1 < p < ∞). If the hyperplanes have zero μ-measure then
i.e. the first remainder \(R_{1}(t) =\sum _{ j=1}^{+\infty }U_{j}(t)\) is compact on L p (Ω × V ).
Strategy of the proof:
-
\(R_{1}(t) =\sum _{ j=m}^{+\infty }U_{j}(t)\) is compact for all t ⩾0 if and only if \(U_{1}(t) =\int _{ 0}^{t}U(t - s)KU(s)ds\) is. So we deal with U 1(t).
-
U 1(t) depends (linearly and) continuously on the scattering operator K. So, by approximation, we may assume that
$$\displaystyle{ k(x,v,v^{{\prime}}) =\sum _{ j\in J}\alpha _{j}(x)f_{j}(v)g_{j}(v^{{\prime}}) }$$where \(\alpha _{j} \in L^{\infty }(\varOmega ),\ f_{j},g_{j}\ \) continuous with compact supports and J is finite.
-
By linearity, we may even choose
$$\displaystyle{ k(x,v,v^{{\prime}}) =\alpha (x)f(v)g(v^{{\prime}}) }$$where α ∈ L ∞(Ω), f, g are continuous with compact supports. In this case, U 1(t) operates on all L q(Ω × V ) (1 ≤ q ≤ +∞). So, by an interpolation argument, we may restrict ourselves to the case p = 2.
-
Domination arguments: In L p spaces (1 < p < ∞), if O i (i = 1, 2) are two positive operators such that
$$\displaystyle{ O_{1}f \leq O_{2}f\ \ \ \forall f \in L_{+}^{p} }$$and if O 2 is compact then O 1 is also compact; see [3]. So we may assume that V is compact, α = f = g = 1 and σ = 0 .
-
Because of σ = 0
$$\displaystyle{ U(t)\varphi =\varphi (x - tv,v)1_{\left \{t\leq s(x,v)\right \}} }$$where \(s(x,v) =\inf \left \{s > 0;\ x - sv\notin \varOmega \right \}.\) If \(\tilde{\varphi }\in L^{2}(\mathbb{R}^{n} \times V )\) is the trivial extension of \(\varphi\) by zero outside Ω × V then, for nonnegative \(\varphi\)
$$\displaystyle{ U(t)\varphi (x,v) \leq \tilde{\varphi } (x - tv,v)\ \ \forall (x,v) \in \varOmega \times V. }$$so \(U(t)\varphi \leq \mathcal{R}U_{\infty }(t)\mathcal{E}\varphi\) where \(\mathcal{E}: L^{2}(\varOmega \times V ) \rightarrow L^{2}(\mathbb{R}^{n} \times V )\) is the trivial extension operator, \(\mathcal{R}: L^{2}(\mathbb{R}^{n} \times V ) \rightarrow L^{2}(\varOmega \times V )\) is the restriction operator and
$$\displaystyle{ U_{\infty }(t):\psi \in L^{2}(\mathbb{R}^{n} \times V ) \rightarrow \psi (x - tv,v) \in L^{2}(\mathbb{R}^{n} \times V ). }$$ -
Thus
$$\displaystyle\begin{array}{rcl} U_{1}(t)& =& \int _{0}^{t}\mathcal{R}U_{ \infty }(t - s)\mathcal{E}K\mathcal{R}U_{\infty }(s)\mathcal{E}ds {}\\ & =& \mathcal{R}\left (\int _{0}^{t}U_{ \infty }(t - s)\mathcal{E}K\mathcal{R}U_{\infty }(s)ds\right )\mathcal{E} {}\\ &\leq &\mathcal{R}\left (\int _{0}^{t}U_{ \infty }(t - s)KU_{\infty }(s)ds\right )\mathcal{E}. {}\\ \end{array}$$and we are led to deal with the compactness of
$$\displaystyle{ \mathcal{R}\int _{0}^{t}U_{ \infty }(t - s)KU_{\infty }(s)ds: L^{2}(\mathbb{R}^{n} \times V ) \rightarrow L^{2}(\varOmega \times V ). }$$ -
Note that
$$\displaystyle{ \int _{0}^{t}U_{ \infty }(t - s)KU_{\infty }(s)\psi ds =\int _{ 0}^{t}ds\int _{ V }\psi (x - (t - s)v - sv^{{\prime}},v^{{\prime}})\mu (v^{{\prime}}). }$$For any \(\psi (.,.) \in L^{2}(\mathbb{R}^{n} \times V )\), we denote by
$$\displaystyle{ \hat{\psi }(\zeta,v) =\lim _{M\rightarrow \infty } \frac{1} {(2\pi )^{\frac{n} {2} }} \int _{\left \vert \zeta \right \vert \leq M}\psi (x,v)e^{-i\zeta.x}d\zeta }$$its partial Fourier transform with respect to space variable where the limit holds in \(L^{2}(\mathbb{R}^{n} \times V )\) norm. Then
$$\displaystyle{ \left \Vert \psi \right \Vert _{L^{2}(\mathbb{R}^{n}\times V )}^{2} =\int _{ V }\int _{\mathbb{R}^{n}}\left \vert \hat{\psi }(\zeta,v)\right \vert ^{2}d\zeta \mu (\mathit{dv}) }$$and
$$\displaystyle{ \psi (x,v) =\lim _{M\rightarrow \infty } \frac{1} {(2\pi )^{\frac{n} {2} }} \int _{\left \vert \zeta \right \vert \leq M}\hat{\psi }(\zeta,v)e^{i\zeta.x}d\zeta }$$where the limit holds in \(L^{2}(\mathbb{R}^{n} \times V )\) norm.
Hence
where the limit holds in \(L^{2}(\mathbb{R}^{n} \times V )\) norm. For each M > 0, let
We observe that \(\mathcal{R}O_{M}\) is a Hilbert Shmidt operator because Ω has finite volume and V is compact (keep in mind that x ∈ Ω!). It suffices to show that
in \(L^{2}(\mathbb{R}^{n} \times V )\) uniformly in \(\left \Vert \psi \right \Vert _{L^{2}(\mathbb{R}^{n}\times V )} \leq 1\), i.e.
in \(L^{2}(\mathbb{R}^{n} \times V )\) uniformly in \(\left \Vert \psi \right \Vert _{L^{2}(\mathbb{R}^{n}\times V )} \leq 1\). By the Parseval identity, this amounts to
uniformly in \(\left \Vert \psi \right \Vert _{L^{2}(\mathbb{R}^{n}\times V )} \leq 1.\) Using Cauchy-Schwarz inequality, we majorize by
Now
where \(\zeta = \left \vert \zeta \right \vert e,\ (e \in S^{n-1})\ \) and
The first term can made arbitrarily small for \(\varepsilon\) small enough (assumption on the velocity measure μ) and, for \(\varepsilon\) fixed, the second term goes to zero as \(\left \vert \zeta \right \vert \rightarrow \infty \) because of \(\int _{0}^{t}e^{is\left \vert \zeta \right \vert (v-v^{{\prime}}).e }ds\) (Riemann-Lebesgue lemma). This ends the proof. ■
Corollary 11
We assume that Ω has finite Lebesgue measure, the scattering operator is regular in L p (Ω × V;dx ⊗μ(dv)) (1 < p < ∞) and the hyperplanes have zero μ-measure. Then
In particular \(\omega _{e}(V ) =\omega _{e}(U)\) and \(\sigma (V (t)) \cap \left \{\beta;\ \left \vert \beta \right \vert > e^{s(T)t}\right \}\) consists at most of isolated eigenvalues with finite algebraic multiplicities.
Here ω e (U) denotes the essential type of \(\left (U(t)\right )_{t\geqslant 0}\) etc. The assumption on the velocity measure μ is “optimal”:
Theorem 31 ([51])
Let μ be finite, Ω bounded and
Let there exist a hyperplane \(H = \left \{v;v.e = c\right \}(e \in S^{n-1},\ c \in \mathbb{R})\) with positive μ-measure. Then there exists \(\overline{t} > 0\) such that V (t) − U(t) is not compact on L 2 (Ω × V ) for \(0 < t \leq \overline{t}\) .
and
Theorem 32 ([51])
In the general setting above, if every ball centred at zero contains at least a section (by a hyperplane) with positive μ-measure then V (t) − U(t) is not compact on L 2 (Ω × V ) for all t > 0.
The assumption that the scattering operator is regular is “nearly optimal”:
Theorem 33 ([51])
Let μ be an arbitrary positive measure. We assume that its support V is bounded. If V (t) − U(t) is compact on L p (Ω × V ) for all t > 0 then, for any open ball B ⊂Ω, the strong integral
is a compact operator on L p (V ).
Corollary 12
Besides the conditions of Theorem 33, we assume that
is measurable (not simply strongly measurable!) e.g. is piecewise continuous in operator norm. Then K(x) is a compact operator on L p (V ) for almost all x ∈Ω.
Proof
at the Lebesgue points x of \(K:\varOmega \rightarrow \mathcal{L}(L^{p}(V ))\). ■
5.2 L1 Theory
As noted previously, L 1 space is the physical setting for neutron transport because
is the expected number of particles. The L 1 mathematical results are very different from those in L p theory (p > 1) and the analysis is much more involved! Weak compactness is a fundamental tool for spectral theory of neutron transport in L 1 spaces. To this end, we recall first some useful results. Let (E, m) be a σ-finite measure space. A bounded subset B ⊂ L 1(E, m) is relatively weakly compact if
and (if m(E) = ∞) there exists measurable sets E n ⊂ E, m(E n ) < +∞, \(E_{n} \subset E_{n+1}\), \(\cup E_{n} = E\) such that
A bounded subset B ⊂ L 1(E, m) is relatively weakly compact if and only if B is relatively sequentially weakly compact. A bounded operator G on L 1(E, m) is said to be weakly compact if G sends a bounded set into a relatively weakly compact one. If \(G_{i}: L^{1}(E,m) \rightarrow L^{1}(E,m)\) (i = 1, 2) are positive operators and \(G_{1}f \leq G_{2}f\ \forall f \in L_{+}^{1}(E,m)\) then G 1 is weakly compact if G 2 is; this follows easily from the above criterion of weak compactness. If G i (i = 1, 2) are two weakly compact operators on L 1(E, m) then \(G_{1}G_{2}\) is a compact operator [18]. We are going to present (weak) compactness results for neutron transport operators and show their spectral consequences. We give here an overview of [52]. We treat first a model case where
We start with negative results:
Theorem 34 ([52])
Let \(\varOmega \subset \mathbb{R}^{n}\) be an open set with finite Lebesgue measure and μ an arbitrary finite positive Borel measure on \(\mathbb{R}^{n}\) with support V. Let n⩾2 (or n = 1 and 0 ∈ V ). Let
Then K(λ − T) −1 is not weakly compact.
Proof
We can assume without loss of generality that 0 ∈ Ω. Consider just the case n ⩾2. Let \((f_{j})_{j} \subset C_{c}(\varOmega \times V )\) a normalized sequence in L 1(Ω × V ) converging in the weak star topology of measures to the Dirac mass \(\delta _{(0,\overline{v})}\). Then for any ψ ∈ C c (Ω)
i.e. \(K(\lambda -T)^{-1}f_{j}\) tends, in the weak star topology of measures, to a (non-trivial) Radon measure
with support included in a segment. Hence \(m\notin L^{1}(\varOmega )\) and K(λ − T)−1 is not weakly compact. ■
We note that this property was noted for the first time in the whole space in [23]. We have also:
Theorem 35 ([52])
Let n⩾3 and let \(\varOmega \subset \mathbb{R}^{n}\) be an open set with finite Lebesgue measure. Let μ be an arbitrary finite positive Borel measure on \(\mathbb{R}^{n}\) with support V and
Then
-
(i)
(λ − T − K) −1 − (λ − T) −1 is not weakly compact.
-
(ii)
V (t) − U(t) is not weakly compact.
We note that this theorem is false for n = 1 while the case n = 2 is open, see [52]. We recall now a necessary condition on μ.
Theorem 36 ([52])
We assume that the velocity measure is invariant under the symmetry \(v \rightarrow -v.\) Let there exist \(m \in \mathbb{N}\) such that
is compact on L 1 (Ω × V ). Then the hyperplanes (through zero) have zero μ-measure.
We note that for any \(m \in \mathbb{N}\)
where
is the trivial extension operator,
is the restriction operator and T ∞ is the generator of
We start with a fundamental observation:
Lemma 9
Let μ be an arbitrary finite positive measure on \(\mathbb{R}^{n}\) with support V and
For any λ > 0 there exists a finite positive measure β on \(\mathbb{R}^{n}\) (depending on λ) such that
Moreover,
Proof
where μ t is the image of μ under the dilation \(v \rightarrow tv\). So
where \(\beta:=\int _{ 0}^{\infty }e^{-\lambda t}\mu _{t}\mathit{dt}\) (strong integral). Finally \(\hat{\beta }(\zeta )\) is given by
This ends the proof. ■
Remark 4
\(\hat{\beta }(\zeta ) =\int _{\mathbb{R}^{n}}\frac{\mu (\mathit{dv})} {\lambda +i\zeta.v} \rightarrow 0\ \ \) for\(\ \ \left \vert \zeta \right \vert \rightarrow \infty \) if and only if the hyperplanes (through zero) have zero μ-measure; see [52]. We are going to show that the compactness results rely on how fast \(\hat{\beta }(\zeta )\) goes to zero as\(\ \ \left \vert \zeta \right \vert \rightarrow \infty.\)
Theorem 37 ([52])
We assume that \(\varOmega \subset \mathbb{R}^{n}\) is an open set with finite Lebesgue measure. Let the velocity measure μ be finite and such that
for some \(m \in \mathbb{N}.\) Then \(\left [K(\lambda -T)^{-1}K\right ]^{m}\) is weakly compact in L 1 (Ω × V ) and \(\left [K(\lambda -T)^{-1}K\right ]^{m+1}\) is compact in L 1 (Ω × V ).
Proof
We start from
where
Since
then our assumption amounts to \(\hat{\beta ^{(m)}} \in L^{2}(\mathbb{R}^{n}).\) In particular \(\beta ^{(m)} \in L^{2}(\mathbb{R}^{n})\) (β (m) is now a function!). It follows that
and then \(\left [K(\lambda -T_{\infty })^{-1}K\right ]^{m}\) maps continuously\(L^{1}(\mathbb{R}^{n} \times V )\) into \(L^{2}(\mathbb{R}^{n}).\) Hence
is weakly compact because the imbedding of L 2(Ω) into L 1(Ω) is weakly compact since Ω has finite Lebesgue measure (a bounded subset of L 2(Ω) is equi-integrable). Finally \(\left [K(\lambda -T)^{-1}K\right ]^{m}\) is also weakly compact by a domination argument. It follows that \(\left [K(\lambda -T)^{-1}K\right ]^{2m}\) is compact as a product of two weakly compact operators. Actually
is compact because K(λ − T)−1 is a Dunford-Pettis operator, see below. ■
We give now a geometrical condition on μ implying the compactness results.
Theorem 38 ([52])
Let \(\varOmega \subset \mathbb{R}^{n}\) be an open set with finite Lebesgue measure. Let the velocity measure μ be finite and there exist α > 0, c > 0 such that
Then \(\left [K(\lambda -T)^{-1}K\right ]^{m+1}\) is compact in L 1 (Ω × V ) for \(m > \frac{n(\alpha +1)} {2\alpha }.\)
Proof
Note that \(\hat{\beta }(\zeta )\) is a continuous function. According to the preceding theorem, we need just check the integrability of \(\left \vert \hat{\beta }(\zeta )\right \vert ^{2m}\) at infinity. Up to a factor \((2\pi )^{-\frac{n} {2} }\)
where \(e = \frac{\zeta }{\left \vert \zeta \right \vert }.\) So, for any \(\varepsilon > 0,\)
Optimizing with respect to \(\varepsilon\) yields
for some positive constant C depending on λ. We are done if \(\frac{2m\alpha } {\alpha +1} > n\ \) i.e. if \(m > \frac{n(\alpha +1)} {2\alpha }.\blacksquare\)
In the same spirit (but with more involved estimates) we can show:
Theorem 39 ([52])
Let \(\varOmega \subset \mathbb{R}^{n}\) be an open set with finite Lebesgue measure. Let the velocity measure μ be finite and there exist α > 0, c > 0 such that
Then U m (t) is weakly compact in L 1 (Ω × V ) for all t⩾0 and for m⩾m 0 where m 0 is the smallest odd integer greater than \(\ \frac{n(\alpha +1)} {2\alpha } + 1.\)
We point out that in Theorems 38 and 39, the condition on m does not depend on the constant c in the statement; this fact is fundamental if we want to pass from model cases to more general models. We show now how to treat by approximation more general velocity measures and scattering kernels. Indeed, in the approximation procedure, a general (a priori infinite) velocity measure μ is approximated, by truncation, by a sequence of finite measures μ j such that
where α is independent of j.
A scattering operator K in L 1(Ω × V ) is said to be regular if\(\ \left \{K(x);\ x \in \varOmega \right \}\) is a set of collectively weakly compact operators on L 1(V ), i.e. the set
is relatively weakly compact in L 1(V ). This class of scattering kernels appears (with Lebesgue measure dv on \(\mathbb{R}^{n})\) in P. Takak [72] and L. W. Weis [82]. See B. Lods [37] for the extension of P. Takak’s construction to abstract velocity measures μ.
Theorem 40 ([37])
Let K be a regular scattering operator in L 1 (Ω × V ). Then there exists a sequence \((K_{j})_{j}\) of scattering operators such that:
-
(i)
0 ≤ K j ≤ K
-
(ii)
\(\left \Vert K - K_{j}\right \Vert _{\mathcal{L}(L^{1}(\varOmega \times V ))} \rightarrow 0\) as \(j \rightarrow +\infty.\)
-
(iii)
For each K j there exists f j ∈ L 1 (V ) such that
$$\displaystyle{ K_{j}\varphi \leq f_{j}(v)\int \varphi (x,v^{{\prime}})\mu (\mathit{dv}^{{\prime}})\ \ \forall \varphi \in L_{ +}^{1}(\varOmega \times V ). }$$
We are ready to show:
Theorem 41 ([52])
Let \(\varOmega \subset \mathbb{R}^{n}\) be an open set with finite Lebesgue measure and let K be a regular scattering operator in L 1 (Ω × V ). We assume that the velocity measure μ is such that: There exists α > 0 such that for any c 1 > 0 there exists c 2 > 0 such that
Then the components of ρ e (T) and ρ e (T + K) containing a right half-plane coincide. In particular
consists at most of isolated eigenvalues with finite algebraic multiplicities where
Proof
Let us show that \(\left [K(\lambda -T)^{-1}K\right ]^{m+1}\) is weakly compact in L 1(Ω × V ) for \(m > \frac{n(\alpha +1)} {2\alpha }\). We fix \(m > \frac{n(\alpha +1)} {2\alpha }\). It suffices to show the weak compactness of \(\left [K_{j}(\lambda -T)^{-1}K_{j}\right ]^{m+1}\) for all j (K j from Theorem 40). By domination, we may replace K j by \(\hat{K_{j}}\) where
By approximation again we may suppose that f j is continuous with compact support. Let j be fixed and denote by V j the support of f j . We note that \(\left [\widehat{K_{j}}(\lambda -T)^{-1}\widehat{K_{j}}\right ]^{m}\) leaves invariant \(L^{1}(\varOmega \times V _{j})\) and\(\ \widehat{K_{j}}\) maps L 1(Ω × V ) into \(L^{1}(\varOmega \times V _{j})\). By replacing f j (v) by its supremum, the model case (dealt with previously) in \(L^{1}(\varOmega \times V _{j})\) insures that \(\left [K_{j}(\lambda -T)^{-1}K_{j}\right ]^{m}\) is weakly compact on \(L^{1}(\varOmega \times V _{j})\) so
is weakly compact on L 1(Ω × V ). Finally some power of K(λ − T)−1 is compact on L 1(Ω × V ) and we conclude by the general theory. ■
Theorem 42 ([52])
Let \(\varOmega \subset \mathbb{R}^{n}\) be an open set with finite Lebesgue measure and let K be a regular scattering operator in L 1 (Ω × V ). We assume that the velocity measure μ is such that: There exists α > 0 such that for any c 1 > 0 there exists c 2 > 0 such that
Then \(\left (V (t)\right )_{t\geqslant 0}\) and \(\left (U(t)\right )_{t\geqslant 0}\) have the same essential type. In particular
consists at most of isolated eigenvalues with finite algebraic multiplicities.
Proof
Let us show that R m+1(t) is weakly compact in L 1(Ω × V ) for all t ⩾0 and for m ⩾ m 0 where m 0 is the smallest odd integer greater than\(\ \frac{n(\alpha +1)} {2\alpha } + 1.\) We fix m ⩾ m 0. It suffices to show that \(R_{m+1}^{j}(t)\) is weakly compact in L 1(Ω × V ) for all j where \(R_{m+1}^{j}(t)\) is the remainder term of order m + 1 corresponding to a perturbation K j in place of K (K j from Theorem 40). By domination, we may replace K j by \(\widehat{K_{j}}\) where
By approximation again we may suppose that f j is continuous with compact support. Let j be fixed and denote by V j the support of f j . We note that \(R_{m}^{j}(t)\) leaves invariant \(L^{1}(\varOmega \times V _{j})\) and \(\widehat{K_{j}}\) maps L 1(Ω × V ) into \(L^{1}(\varOmega \times V _{j}).\) Note also that
By dominating f j by its supremum, the previous model case (and a domination argument) shows that \(R_{m}^{j}(t - s)\) is weakly compact in \(L^{1}(\varOmega \times V _{j})\). Thus \(R_{m}^{j}(t - s)\widehat{K_{j}}U(s)\) is weakly compact in L 1(Ω × V ) and then so is \(R_{m+1}^{j}\) as a strong integral of “weakly compact operator” valued mapping. We conclude by the general theory. ■
Remark 5
The conditions on μ in Theorems 41 and 42 are satisfied e.g. for Lebesgue measure on \(\mathbb{R}^{n}\) or on spheres (multigroup models).
5.3 Dunford-Pettis Operators in Transport Theory
A bounded operator \(G \in \mathcal{L}(L^{1}(\nu ))\) is called a Dunford-Pettis (or a completely continuous) operator if G maps a weakly compact subset into a (norm) compact subset. For instance, a weakly compact operator on L 1(ν) is Dunford-Pettis; this explains why the product of two weakly compact operators on L 1(ν) is compact. More generally, if G 2 is weakly compact on L 1(ν) and G 1 is Dunford-Pettis on L 1(ν) then \(G_{1}G_{2}\) is compact on L 1(ν). We have seen that various relevant operators for neutron transport are not weakly compact in L 1(Ω × V ), for instance:
We can show however that they are all Dunford-Pettis, (see [50] for more information). This explains why we claimed in the proof of Theorem 37 that
is compact since \(\left [K(\lambda -T)^{-1}K\right ]^{m}\) is weakly compact and K(λ − T)−1 is Dunford-Pettis. We restrict ourselves to:
Theorem 43 ([50])
Let \(\varOmega \subset \mathbb{R}^{n}\) be an open set with finite Lebesgue measure and let K be a regular scattering operator in L 1 (Ω × V ). If the affine hyperplanes have zero μ-measure then V (t) − U(t) is a Dunford-Pettis operator on L 1 (Ω × V ).
Proof
We note first that \(V (t) - U(t) =\sum _{ 1}^{\infty }U_{j}(t)\ \) is Dunford-Pettis for all t ⩾0 if and only if U 1(t) is [50]. By approximation, we may assume that their exists f continuous with support in \(\left \{\left \vert v\right \vert < \overline{c}\right \}\) such that
Let E ⊂ L 1(Ω × V ) be relatively weakly compact. In particular
We decompose \(\varphi \in E\) as
so
uniformly in \(\varphi \in E.\) On the other hand \(\varphi \chi _{\left \{\left \vert v\right \vert <c\right \}}\) is zero for \(\left \vert v\right \vert > c\) and (for \(c > \overline{c}\)) for any ψ ∈ L 1(Ω × V ), K ψ is zero for \(\left \vert v\right \vert > c\). So we may assume from the beginning that V is bounded and then K maps also L 2(Ω × V ) into itself. We decompose \(\varphi \in E\) as
and note that
so
The equi-integrability of E implies that
uniformly in \(\varphi \in E\) and finally \(\left \Vert U_{1}(t)\varphi _{\alpha }^{2}\right \Vert _{L^{1}} \rightarrow 0\ \ \) as\(\ \ c \rightarrow +\infty \) uniformly in \(\varphi \in E\). Since, \(\left \{\varphi _{\alpha }^{1}\right \}\) is bounded in L 1 and in L ∞ then the interpolation inequality
shows that \(\left \{\varphi _{\alpha }^{1}\right \}\) is also bounded in L 2(Ω × V ). We know (from the L 2 theory) that U 1(t) is compact in L 2(Ω × V ) so\(\ \left \{U_{1}(t)\varphi _{\alpha }^{1}\right \}\) is relatively compact in L 2(Ω × V ) and then relatively compact in L 1(Ω × V ) since Ω × V has a finite measure. Thus \(\left \{U_{1}(t)\varphi;\ \varphi \in E\right \}\) is as close to a relatively compact subset of L 1(Ω × V ) as we want and finally \(\left \{U_{1}(t)\varphi;\ \varphi \in E\right \}\) is relatively compact. ■
6 Comments
6.1 Measure Convolution Operators in Transport Theory
The compactness results for neutron transport operators in L p spaces with 1 < p < ∞ (see Subsection 5.1) can also be derived from the analysis of just two particular measure convolution operators on \(\mathbb{R}^{n}\), see [59].
6.2 Unbounded Geometries
The compactness results given in this lecture in spatial domains Ω with finite Lebesgue measure need no be true in general domains, e.g. the results are false in the whole space (i.e. \(\varOmega =\mathbb{R} ^{n}\)) and space homogeneous cross-sections. However, under suitable assumptions on the cross-sections, we can recover the compactness results above in unbounded geometries [60]. Actually, for general geometries and cross-sections, the relevant perturbation theory does not concern the essential spectra (and the essential types) but rather the critical spectra (and the critical types); we refer to [10, 53, 54, 59, 62, 67] for the abstract theory and how to use it in the context of neutron transport theory.
6.3 Leading Eigenvalue of Neutron Transport
The time asymptotic behaviour of neutron transport semigroup is meaningful if the latter has a spectral gap or equivalently if its generator has a leading eigenvalue. This topic relies on peripheral spectral analysis of neutron transport. The first relevant question concerns irreducibility criteria of neutron transport semigroups for which we refer e.g. to [26, 47, 78] and [48, Chapter 5]. The second relevant question concerns the effective existence of leading eigenvalues; besides the isotropic case dealt with by [29], we refer to ([48, Chapter 5]) for general tools. Variational characterizations of the leading eigenvalue in L p spaces (InfSup or SupInf criteria) and lower bounds of this eigenvalue are given in [55]. The criticality eigenvalue problem is dealt with in [41, 66] and [48, Chapter 5]. Finally, we refer to [40] for variational characterizations of the criticality eigenvalue.
6.4 Partly Elastic Scattering Operators
Most of the literature on spectral theory of neutron transport is devoted either to one speed models or to inelastic models. Despite their apparent difference, these two models can be covered by a unique general formalism as we did in Sect. 5. However, more complex models which take into account of both elastic and inelastic scatterings appear e.g. in [35]:
where
and
where v = ρ ω. The peculiarity of the elastic scattering operator is that it is not compact “in velocities” in contrast to usual inelastic scattering operators. This explains the complexity of σ(T + K e ) which consists of a half-plane and various “curves”[35]. We find in [68] various compactness results (due to K i ) and spectral results. In particular the semigroups generated by T + K e and by \(T + K_{e} + K_{i}\) have the same essential type.
6.5 Generalized Boundary Conditions
We point out that “zero incoming flux” is the natural (i.e. physical) boundary condition for neutron transport. However, various boundary conditions (relating e.g. the incoming and outgoing fluxes) were also considered in kinetic theory [11] or in structured cell population models (see e.g. [5]). The literature on the subject is considerable and we do not try to comment on it. We just note that the corresponding advection semigroup is no longer explicit and, for this reason, spectral analysis of such kinetic models with non local boundary conditions is much more technical. We refer for instance to [38] and references therein.
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Mokhtar-Kharroubi, M. (2015). Spectral Theory for Neutron Transport. In: Banasiak, J., Mokhtar-Kharroubi, M. (eds) Evolutionary Equations with Applications in Natural Sciences. Lecture Notes in Mathematics, vol 2126. Springer, Cham. https://doi.org/10.1007/978-3-319-11322-7_7
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