SPECTRAL THEORY FOR NEUTRON TRANSPORT
SPECTRA OF PERTURBED OPERATORS AND APPLICATIONS
Mustapha Mokhtar-Kharroubi
(In memory of Seiji Ukaï)
Chapter 4
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Abstract
This chapter deals with two topics:
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Chapter 4
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Abstract
This chapter deals with two topics:
Functional analytic results on stability of essential spectra for
perturbed generators or perturbed semigroups on Banach
spaces
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Abstract
This chapter deals with two topics:
Functional analytic results on stability of essential spectra for
perturbed generators or perturbed semigroups on Banach
spaces
Application to neutron transport theory
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Essentially meromorphic mapping
De…nition
Let X be a complex Banach space and D C be open and connected. A
compact operator valued meromorphic mapping
A : D ! C(X )
(C(X ) 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 zk 2 D where A has poles with Laurent expansions
∞
A(z ) =
∑
(z
zk )m An (zk )
( 0 < mk < ∞ )
n= mk
where An (zk ) (n =
1,
2, ...,
mk ) are …nite rank operators.
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An analytic Fredholm alternative
Theorem
Let X be a complex Banach space and D
C be open and connected. Let
A : D ! C(X )
be essentially meromorphic. Then
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An analytic Fredholm alternative
Theorem
Let X be a complex Banach space and D
C be open and connected. Let
A : D ! C(X )
be essentially meromorphic. Then
(i) Either λ = 1 is an eigenvalue of all A(z )
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An analytic Fredholm alternative
Theorem
Let X be a complex Banach space and D
C be open and connected. Let
A : D ! C(X )
be essentially meromorphic. Then
(i) Either λ = 1 is an eigenvalue of all A(z )
(ii) or [I A(z )] 1 exists except for a discrete set of points and
[I A(z )] 1 is essentially meromorphic on D.
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An analytic Fredholm alternative
Theorem
Let X be a complex Banach space and D
C be open and connected. Let
A : D ! C(X )
be essentially meromorphic. Then
(i) Either λ = 1 is an eigenvalue of all A(z )
(ii) or [I A(z )] 1 exists except for a discrete set of points and
[I A(z )] 1 is essentially meromorphic on D.
See M. Ribaric and I. Vidav, Analytic properties of the inverse A(z ) 1 of
an analytic linear operator valued function A(z ). Arch. Rational Mech.
Anal, 32 (1969) 298–310; Corollary II.
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Essential resolvent set
Let T : D (T ) X ! X be a closed operator. We de…ne its "essential
resolvent set" as
ρe (T ) = ρ(T ) [ σdiscr (T )
where σdiscr (T ) refers to the isolated eigenvalues of T with …nite algebraic
multiplicities.
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Essential resolvent set
Let T : D (T ) X ! X be a closed operator. We de…ne its "essential
resolvent set" as
ρe (T ) = ρ(T ) [ σdiscr (T )
where σdiscr (T ) refers to the isolated eigenvalues of T with …nite algebraic
multiplicities. This set is open.
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Essential resolvent set
Let T : D (T ) X ! X be a closed operator. We de…ne its "essential
resolvent set" as
ρe (T ) = ρ(T ) [ σdiscr (T )
where σdiscr (T ) refers to the isolated eigenvalues of T with …nite algebraic
multiplicities. This set is open.
We note that if T 2 L(X ) then the unbounded component of ρe (T )
coincides with the unbounded component of the Fredholm domain ρF (T )
(see Gohberg, Goldberg, Kaashoek, p. 204).
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A perturbation result
We give …rst a result (and some of its consequences) given in (J. Voigt,
Mh. Math. 90 (1980) 153–161).
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A perturbation result
We give …rst a result (and some of its consequences) given in (J. Voigt,
Mh. Math. 90 (1980) 153–161).
Theorem
Let T : D (T ) X ! X be a closed operator and let Ω be a connected
component of ρe (T ). Let B : D (T ) ! X be T -bounded such that there
exists n 2 N and
B (λ
T)
1 n
is compact (λ 2 Ω \ ρ(T )).
We assume that there exists some λ 2 Ω \ ρ(T ) such that
n
I
B (λ T ) 1 is invertible (i.e. 1 is not an eigenvalue of
n
B (λ T ) 1 ). Then Ω ρe (T + B ) and
B (λ
T
B)
1 n
is compact (λ 2 Ω \ ρ(T + B )).
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Proof
We note that (λ
T)
1
is is essentially meromorphic on Ω.
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Proof
We note that (λ T ) 1 is is essentially meromorphic on Ω. Then
n
Bλ := B (λ T ) 1 and Bλn = B (λ T ) 1 are also essentially
meromorphic on Ω.
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Proof
We note that (λ T ) 1 is is essentially meromorphic on Ω. Then
n
Bλ := B (λ T ) 1 and Bλn = B (λ T ) 1 are also essentially
meromorphic on Ω. Since Bλn is operator compact valued then, by the
analytic Fredholm alternative (I Bλn ) 1 is also essentially meromorphic
on Ω.
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Proof
We note that (λ T ) 1 is is essentially meromorphic on Ω. Then
n
Bλ := B (λ T ) 1 and Bλn = B (λ T ) 1 are also essentially
meromorphic on Ω. Since Bλn is operator compact valued then, by the
analytic Fredholm alternative (I Bλn ) 1 is also essentially meromorphic
on Ω. On the other hand
Bλn = (I
I
Bλ )(I + Bλ + ... + Bλn
1
)
shows that
(I
Bλ )
1
Bλn )
= (I
1
(I + Bλ + ... + Bλn
1
)
is also essentially meromorphic on Ω and then so is
(λ
i.e. Ω
T
B)
1
= (λ
T)
1
(I
Bλ )
1
ρe (T + B ).
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Proof
We note that (λ T ) 1 is is essentially meromorphic on Ω. Then
n
Bλ := B (λ T ) 1 and Bλn = B (λ T ) 1 are also essentially
meromorphic on Ω. Since Bλn is operator compact valued then, by the
analytic Fredholm alternative (I Bλn ) 1 is also essentially meromorphic
on Ω. On the other hand
Bλn = (I
I
Bλ )(I + Bλ + ... + Bλn
1
)
shows that
(I
Bλ )
1
Bλn )
= (I
1
(I + Bλ + ... + Bλn
1
)
is also essentially meromorphic on Ω and then so is
(λ
i.e. Ω
T
B)
1
= (λ
T)
1
1
(I
Bλ )
1 n
= Bλn (I
ρe (T + B ). Finally
B (λ
T
B)
1 n
= Bλ ( I
is also compact on Ω \ ρ(T + B ).
Bλ )
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Bλ )
n
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A corollary
We give now a more precise version of the theorem above under an
additional assumption.
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A corollary
We give now a more precise version of the theorem above under an
additional assumption.
Corollary
Let T : D (T ) X ! X be a closed operator and let Ω be a connected
component of ρe (T ). Let B : D (T ) ! X be T -bounded such that there
exists n 2 N and
B (λ
T)
1 n
is compact (λ 2 Ω \ ρ(T )).
We assume that there exists a sequence (λj )j
B ( λj
T)
1
Ω \ ρ(T ) such that
! 0 (j ! + ∞ ).
Then (λj )j
ρ(T + B ) for j large enough and B (λj T B )
as j ! +∞. Furthermore Ω is a component of ρe (T + B ).
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Proof
We note that for j large enough
B ( λj
T
B)
1
= Bλj (I
Bλj )
1
∞
∑
k =1
Bλj
k
! 0 (j ! + ∞ )
Let Ω0 be the component of ρe (T + B ) which contains Ω. We know that
n
B (λ T B ) 1
is compact (λ 2 Ω \ ρ(T + B )). By analyticity, this
0
extends to Ω \ ρ(T + B ). By considering T as (T + B ) B and
reversing the arguments in the previous theorem one gets Ω0 ρe (T ) and
consequently Ω = Ω0 .
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A remark
Unless B : D (T ) ! X is trivial, the conditions in corollary above impose
that Ω be unbounded
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A remark
Unless B : D (T ) ! X is trivial, the conditions in corollary above impose
that Ω be unbounded because for all x 2 D (T )
kBx k =
B ( λj
B ( λj
T)
1
T)
1
( λj T )x
x j λj j + B ( λj
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T)
1
kTx k .
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Stability of essential radius
Corollary
Let T , B 2 L(X ). We assume that B (λ
unbounded component of ρ(T ), i.e.
B (λ
T)
1 n
T)
1 n
is compact on the
is compact (λ 2 Ω \ ρ(T ))
where Ω is the unbounded connected component of ρe (T ).
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Stability of essential radius
Corollary
Let T , B 2 L(X ). We assume that B (λ
unbounded component of ρ(T ), i.e.
B (λ
T)
1 n
T)
1 n
is compact on the
is compact (λ 2 Ω \ ρ(T ))
where Ω is the unbounded connected component of ρe (T ). The
unbounded components of ρe (T ) and ρe (T + B ) coincide
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Stability of essential radius
Corollary
Let T , B 2 L(X ). We assume that B (λ
unbounded component of ρ(T ), i.e.
B (λ
T)
1 n
T)
1 n
is compact on the
is compact (λ 2 Ω \ ρ(T ))
where Ω is the unbounded connected component of ρe (T ). The
unbounded components of ρe (T ) and ρe (T + B ) coincide and then
re (T ) = re (T + B ).
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Stability of essential radius
Corollary
Let T , B 2 L(X ). We assume that B (λ
unbounded component of ρ(T ), i.e.
B (λ
T)
1 n
T)
1 n
is compact on the
is compact (λ 2 Ω \ ρ(T ))
where Ω is the unbounded connected component of ρe (T ). The
unbounded components of ρe (T ) and ρe (T + B ) coincide and then
re (T ) = re (T + B ).
Proof: Since (λ
T)
1
! 0 as jλj ! ∞ we apply the corollary above.
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Two basic examples
Theorem
Let T , B 2 L(X ).
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Two basic examples
Theorem
Let T , B 2 L(X ).
(i) If B is compact then re (T ) = re (T + B ).
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Two basic examples
Theorem
Let T , B 2 L(X ).
(i) If B is compact then re (T ) = re (T + B ).
(ii) If X = L1 (µ) and if B is a weakly compact operator then
re (T ) = re (T + B ).
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Two basic examples
Theorem
Let T , B 2 L(X ).
(i) If B is compact then re (T ) = re (T + B ).
(ii) If X = L1 (µ) and if B is a weakly compact operator then
re (T ) = re (T + B ).
Proof: The case (i) is clear with n = 1.
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Two basic examples
Theorem
Let T , B 2 L(X ).
(i) If B is compact then re (T ) = re (T + B ).
(ii) If X = L1 (µ) and if B is a weakly compact operator then
re (T ) = re (T + B ).
Proof: The case (i) is clear with n = 1. In the case (ii) B (λ T ) 1 is also
weakly compact on L1 (µ) and consequently its square is compact.
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Strong integral of operator valued mappings
Let (Ω, µ) be a …nite measure space and let X , Y be two Banach spaces.
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Strong integral of operator valued mappings
Let (Ω, µ) be a …nite measure space and let X , Y be two Banach spaces.
Let
G : ω 2 Ω ! G (ω ) 2 L(X , Y )
be bounded and strongly measurable in the sense that for each x 2 X
ω 2 Ω ! G ( ω )x 2 Y
is (Bochner) measurable.
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Strong integral of operator valued mappings
Let (Ω, µ) be a …nite measure space and let X , Y be two Banach spaces.
Let
G : ω 2 Ω ! G (ω ) 2 L(X , Y )
be bounded and strongly measurable in the sense that for each x 2 X
ω 2 Ω ! G ( ω )x 2 Y
is (Bochner) measurable. We de…ne the strong integral of G on Ω as the
bounded operator
Z
Ω
G :x 2X !
Z
Ω
G ( ω )x µ (d ω ) 2 Y .
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Strong integral of operator valued mappings
Let (Ω, µ) be a …nite measure space and let X , Y be two Banach spaces.
Let
G : ω 2 Ω ! G (ω ) 2 L(X , Y )
be bounded and strongly measurable in the sense that for each x 2 X
ω 2 Ω ! G ( ω )x 2 Y
is (Bochner) measurable. We de…ne the strong integral of G on Ω as the
bounded operator
Z
Ω
G :x 2X !
Z
Ω
G ( ω )x µ (d ω ) 2 Y .
Remark: Strongly continuous mappings appear everywhere in semigroup
theory !!
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Compact operator valued mappings
Theorem
Under the conditions above, assume in addition that 8ω 2 Ω,
G (ω ) 2 C(X , Y ) (i.e. G (ω ) is a compact operator).
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Compact operator valued mappings
Theorem
Under the conditions above, assume in addition that 8ω 2 Ω,
G (ω ) 2 C(X , Y ) (i.e. G (ω ) is a compact operator). Then
Z
Ω
G 2 C(X , Y ).
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Proofs
See:
L. W. Weis. (J. Math. Anal. Appl, 129 (1988) 6-23).
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Proofs
See:
L. W. Weis. (J. Math. Anal. Appl, 129 (1988) 6-23).
J. Voigt. (Note di Math XII (1992) 259-269).
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Proofs
See:
L. W. Weis. (J. Math. Anal. Appl, 129 (1988) 6-23).
J. Voigt. (Note di Math XII (1992) 259-269).
M. Kunze and G. Schluchtermann. (Math. Nachr, 191 (1998)
197-214).
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Proofs
See:
L. W. Weis. (J. Math. Anal. Appl, 129 (1988) 6-23).
J. Voigt. (Note di Math XII (1992) 259-269).
M. Kunze and G. Schluchtermann. (Math. Nachr, 191 (1998)
197-214).
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Proofs
See:
L. W. Weis. (J. Math. Anal. Appl, 129 (1988) 6-23).
J. Voigt. (Note di Math XII (1992) 259-269).
M. Kunze and G. Schluchtermann. (Math. Nachr, 191 (1998)
197-214).
In the statement above, we can replace "compact" by "weakly
compact", see
G. Schluchtermann. (Math. Ann, 292 (1992) 263-266).
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Direct proofs in Lebesgue spaces
In Lp spaces spaces, we can provide direct proofs using the Kolmogorov’s
compactness criterion and the Dunford-Pettis criterion of weak
compactness in L1 spaces.
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Direct proofs in Lebesgue spaces
In Lp spaces spaces, we can provide direct proofs using the Kolmogorov’s
compactness criterion and the Dunford-Pettis criterion of weak
compactness in L1 spaces.
See M. M-K. Math Methods. Appl. Sci, 27 (2004) 687-701.
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Spectra of perturbed generators
Theorem
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K .
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Spectra of perturbed generators
Theorem
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K . We assume that K is T -power
compact i.e. there exists n 2 N such that (for λ in some right half-plane
included in ρ(T + K ))
K (λ
T)
1 n
is compact.
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Spectra of perturbed generators
Theorem
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K . We assume that K is T -power
compact i.e. there exists n 2 N such that (for λ in some right half-plane
included in ρ(T + K ))
K (λ
T)
1 n
is compact.
Then the components of ρe (T ) and ρe (T + K ) containing a right
half-plane coincide.
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Spectra of perturbed generators
Theorem
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K . We assume that K is T -power
compact i.e. there exists n 2 N such that (for λ in some right half-plane
included in ρ(T + K ))
K (λ
T)
1 n
is compact.
Then the components of ρe (T ) and ρe (T + K ) containing a right
half-plane coincide. In particular
σ(T + K ) \ fRe λ > s (T )g
consists at most of isolated eigenvalues with …nite algebraic multiplicities.
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Proof
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Proof
It follows from a previous corollary since (λ
Re λ ! +∞.
T)
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1
! 0 as
Chapter 4
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Proof
It follows from a previous corollary since (λ T ) 1 ! 0 as
Re λ ! +∞.
This theorem is a re…nement of a …rst version due to I. Vidav (J. Math.
Anal. Appl, 22 (1968) 144-155).
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On Dyson-Phillips expansions
The perturbed semigroup (V (t ))t >0 is related to the unperturbed one
(U (t ))t >0 by an integral equation (Duhamel equation)
V (t ) = U (t ) +
Z t
U (t )KV (t
s )ds.
0
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On Dyson-Phillips expansions
The perturbed semigroup (V (t ))t >0 is related to the unperturbed one
(U (t ))t >0 by an integral equation (Duhamel equation)
V (t ) = U (t ) +
Z t
U (t )KV (t
s )ds.
0
The integrals are interpreted in a strong sense i.e.
V (t )x = U (t )x +
Z t
0
U (t )KV (t
s )xds (x 2 X ).
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The Duhamel equation is solved by standard iterations
Vj +1 (t )x = U (t )x +
Z t
0
U (t )KVj (t
s )xds (j > 0) V0 = 0
and 8C > 0
sup kVj (t )
t 2[0C ]
V (t )kL(X ) ! 0 as j ! +∞.
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The Duhamel equation is solved by standard iterations
Vj +1 (t )x = U (t )x +
Z t
0
U (t )KVj (t
s )xds (j > 0) V0 = 0
and 8C > 0
sup kVj (t )
t 2[0C ]
V (t )kL(X ) ! 0 as j ! +∞.
In particular V1 (t ) = U (t ),
V2 (t ) = U0 (t ) +
Z t
0
U0 (t )KU0 (t
s )ds := U0 (t ) + U1 (t )
(where U0 (t ) = U (t ))
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The Duhamel equation is solved by standard iterations
Vj +1 (t )x = U (t )x +
Z t
0
s )xds (j > 0) V0 = 0
U (t )KVj (t
and 8C > 0
sup kVj (t )
t 2[0C ]
V (t )kL(X ) ! 0 as j ! +∞.
In particular V1 (t ) = U (t ),
V2 (t ) = U0 (t ) +
Z t
0
U0 (t )KU0 (t
s )ds := U0 (t ) + U1 (t )
(where U0 (t ) = U (t ))
V3 (t ) = U0 (t ) +
Z t
0
U0 ( t ) K [ U0 ( t
= U0 ( t ) + U1 ( t ) +
:
Z t
0
s ) + U1 ( t
U0 ( t ) K [ U1 ( t
s )] ds
s )] ds
= U0 ( t ) + U1 ( t ) + U2 ( t ) .
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Chapter 4
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Finally, (V (t ))t >0 is given by a Dyson-Phillips series
+∞
V (t ) =
∑ Uj ( t )
0
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Chapter 4
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Finally, (V (t ))t >0 is given by a Dyson-Phillips series
+∞
V (t ) =
∑ Uj ( t )
0
where
Uj + 1 ( t ) =
Z t
0
U0 (t )KUj (t
s )ds (j > 0)
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U0 ( t ) = U ( t ) .
Chapter 4
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Remainder terms of Dyson-Phillips expansions
+∞
Rm (t ) :=
∑ Uj ( t )
(m > 0).
j =m
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Chapter 4
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Iterated convolutions
De…nition
Let f , g : R+ ! L(X ) be strongly continuous. We de…ne their
convolution by (the strong integral)
f
g = h (t ) : =
Z t
f (s )g (t
s )ds.
0
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Chapter 4
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Iterated convolutions
De…nition
Let f , g : R+ ! L(X ) be strongly continuous. We de…ne their
convolution by (the strong integral)
f
g = h (t ) : =
Z t
f (s )g (t
s )ds.
0
We note [f ]n the n-fold convolution of f with the convention [f ]1 = f .
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Chapter 4
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Iterated convolutions
De…nition
Let f , g : R+ ! L(X ) be strongly continuous. We de…ne their
convolution by (the strong integral)
f
g = h (t ) : =
Z t
f (s )g (t
s )ds.
0
We note [f ]n the n-fold convolution of f with the convention [f ]1 = f .
We can express Uj (.) for j > 1 as
Uj (.) = [UK ]j
U=U
[KU ]j (j > 1).
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Chapter 4
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A characterization property
The following result is due to M. M-K (Mathematical Topics in neutron
transport theory, World Scienti…c, Vol 46 (1997), Chapter 2).
Theorem
Let n 2 N be given. Then Un (t ) is compact for all t > 0 if and only if
Rm (t ) := ∑j+=∞m Uj (t ) is compact for all t > 0.
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Chapter 4
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A characterization property
The following result is due to M. M-K (Mathematical Topics in neutron
transport theory, World Scienti…c, Vol 46 (1997), Chapter 2).
Theorem
Let n 2 N be given. Then Un (t ) is compact for all t > 0 if and only if
Rm (t ) := ∑j+=∞m Uj (t ) is compact for all t > 0. If X = L1 (µ) then we can
replace "compact" by "weakly compact".
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Chapter 4
24 / 118
Proof
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Chapter 4
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Proof
(i) If Un (t ) is compact for all t > 0 then
Un + 1 ( t ) =
Z t
0
U (s )KUn (t
s )ds
is compact for all t > 0 as a strong integral of "compact operator"
valued mappings.
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Chapter 4
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Proof
(i) If Un (t ) is compact for all t > 0 then
Un + 1 ( t ) =
Z t
0
U (s )KUn (t
s )ds
is compact for all t > 0 as a strong integral of "compact operator"
valued mappings.
By induction Uj (t ) is compact (for all t > 0) for all j > n
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Chapter 4
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Proof
(i) If Un (t ) is compact for all t > 0 then
Un + 1 ( t ) =
Z t
0
U (s )KUn (t
s )ds
is compact for all t > 0 as a strong integral of "compact operator"
valued mappings.
By induction Uj (t ) is compact (for all t > 0) for all j > n and then Rn (t )
is compact for all t > 0 since the series converges in operator norm.
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Chapter 4
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(ii) Conversely, let Rn (t ) be compact for all t > 0.
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Chapter 4
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(ii) Conversely, let Rn (t ) be compact for all t > 0. Then
+∞
+∞
Rn +1 (t ) =
∑
[UK ]j
U
j =n +1
j =n +1
+∞
= [UK ]
∑
Uj ( t ) =
∑
[UK ]j
1
U
j =n +1
+∞
= [UK ]
∑
[UK ]j
j =n
= [UK ] Rn (t ) =
Z t
0
U
U (s )KRn (t
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s )ds
Chapter 4
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(ii) Conversely, let Rn (t ) be compact for all t > 0. Then
+∞
+∞
Rn +1 (t ) =
∑
[UK ]j
U
j =n +1
j =n +1
+∞
= [UK ]
∑
Uj ( t ) =
∑
[UK ]j
1
U
j =n +1
+∞
= [UK ]
∑
[UK ]j
j =n
= [UK ] Rn (t ) =
Z t
0
U
U (s )KRn (t
s )ds
shows that Rn +1 (t ) is also compact for all t > 0 as a strong integral of
"compact operator" valued mappings.
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Chapter 4
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(ii) Conversely, let Rn (t ) be compact for all t > 0. Then
+∞
+∞
Rn +1 (t ) =
∑
[UK ]j
U
j =n +1
j =n +1
+∞
= [UK ]
∑
Uj ( t ) =
∑
[UK ]j
1
U
j =n +1
+∞
= [UK ]
∑
[UK ]j
j =n
= [UK ] Rn (t ) =
Z t
0
U
U (s )KRn (t
s )ds
shows that Rn +1 (t ) is also compact for all t > 0 as a strong integral of
"compact operator" valued mappings. Finally
Un (t ) = Rn (t )
Rn +1 (t )
is also compact for all t > 0.
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Chapter 4
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Upper bound of the essential type of perturbed semigroups
The following result is due to J. Voigt (Proc Steklov Inst Math, 3 (1995)
383-389).
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Chapter 4
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Upper bound of the essential type of perturbed semigroups
The following result is due to J. Voigt (Proc Steklov Inst Math, 3 (1995)
383-389). We give here a slightly di¤erent (and simpler) presentation of
the proof thanks to the boundedness of K .
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Chapter 4
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Upper bound of the essential type of perturbed semigroups
The following result is due to J. Voigt (Proc Steklov Inst Math, 3 (1995)
383-389). We give here a slightly di¤erent (and simpler) presentation of
the proof thanks to the boundedness of K .
Theorem
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K .
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Chapter 4
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Upper bound of the essential type of perturbed semigroups
The following result is due to J. Voigt (Proc Steklov Inst Math, 3 (1995)
383-389). We give here a slightly di¤erent (and simpler) presentation of
the proof thanks to the boundedness of K .
Theorem
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K . If some remainder term
+∞
Rn (t ) :=
∑ Uj ( t )
j =n
is compact for t large enough then ω e (V ) ω e (U ) where ω e (V )
(resp. ω e (U )) is the essential type of (V (t ))t >0 (resp. (U (t ))t >0 ).
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Chapter 4
27 / 118
Upper bound of the essential type of perturbed semigroups
The following result is due to J. Voigt (Proc Steklov Inst Math, 3 (1995)
383-389). We give here a slightly di¤erent (and simpler) presentation of
the proof thanks to the boundedness of K .
Theorem
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K . If some remainder term
+∞
Rn (t ) :=
∑ Uj ( t )
j =n
is compact for t large enough then ω e (V ) ω e (U ) where ω e (V )
(resp. ω e (U )) is the essential type of (V (t ))t >0 (resp. (U (t ))t >0 ). If
X = L1 (µ) then we can replace "compact" by "weakly compact".
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Chapter 4
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Proof
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Chapter 4
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Proof
Let β > ω e (U ) then there exists a …nite range projection Pβ commuting
with (U (t ))t >0 such that for any β0 > β
U (t )(I
Pβ )
0
M β 0 e β t (t > 0).
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Chapter 4
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Proof
Let β > ω e (U ) then there exists a …nite range projection Pβ commuting
with (U (t ))t >0 such that for any β0 > β
U (t )(I
Pβ )
0
M β 0 e β t (t > 0).
(with Pβ = 0 if β > ω (U ) the type of (U (t ))t >0 ).
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Chapter 4
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Proof
Let β > ω e (U ) then there exists a …nite range projection Pβ commuting
with (U (t ))t >0 such that for any β0 > β
U (t )(I
Pβ )
0
M β 0 e β t (t > 0).
(with Pβ = 0 if β > ω (U ) the type of (U (t ))t >0 ). On the other hand, by
stability of essential radius by compact (or weakly compact if
X = L1 (µ)) perturbation
n 1
n 1
j =0
j =0
re (V (t )) = re ( ∑ Uj (t )) = re ( ∑ [UK ]j
U)
for t large enough.
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Chapter 4
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We note that
[UK ]j
=
U (I
P β + P β )K (I
=
U (I
P β )K (I
Pβ + Pβ )
j
j
Pβ ) + Cj (t )
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Chapter 4
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We note that
[UK ]j
=
U (I
P β + P β )K (I
=
U (I
P β )K (I
Pβ + Pβ )
j
j
Pβ ) + Cj (t )
where Cj (t ) is a sum of convolutions where each convolution involves at
one term of the form U (I Pβ )KPβ , UPβ KPβ or UPβ K (I Pβ ).
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Chapter 4
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We note that
[UK ]j
=
U (I
P β + P β )K (I
=
U (I
P β )K (I
Pβ + Pβ )
j
j
Pβ ) + Cj (t )
where Cj (t ) is a sum of convolutions where each convolution involves at
one term of the form U (I Pβ )KPβ , UPβ KPβ or UPβ K (I Pβ ). Such
terms are compact (…nite rank because of Pβ ) so that the convolutions
are compact for all time as strong integrals of "compact operator" valued
mappings.
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Chapter 4
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We note that
[UK ]j
=
U (I
P β + P β )K (I
=
U (I
P β )K (I
Pβ + Pβ )
j
j
Pβ ) + Cj (t )
where Cj (t ) is a sum of convolutions where each convolution involves at
one term of the form U (I Pβ )KPβ , UPβ KPβ or UPβ K (I Pβ ). Such
terms are compact (…nite rank because of Pβ ) so that the convolutions
are compact for all time as strong integrals of "compact operator" valued
mappings.
Thus Cj (t ) is compact for all t > 0.
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Chapter 4
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Once again, the stability of essential radius by compact perturbation gives
for t large enough
n 1
re (V (t )) = re ((I
Pβ )U (t )(I
Pβ ) +
∑
U (I
P β )K (I
Pβ )
j
U)
j =1
n 1
(I
Pβ )U (t )(I
Pβ ) +
∑
U (I
P β )K (I
Pβ )
j
U
j =1
n 1
(I
Pβ )U (t )(I
Pβ ) +
∑
U (I
P β )K (I
Pβ )
j
U
j =1
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Chapter 4
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Observe that
U (I
e (t )
where U
t >0
P β )K (I
Pβ )
h
i
e
e
U = UK
U
is the semigroup U (t )(I
Pβ )
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t >0
.
Chapter 4
31 / 118
Observe that
U (I
e (t )
where U
t >0
P β )K (I
Pβ )
h
i
e
e
U = UK
U
is the semigroup U (t )(I
U (I
P β )K (I
Pβ )
j
Pβ )
t >0
h
ij
e
U = UK
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. More generally
e
U.
Chapter 4
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Observe that
U (I
e (t )
where U
t >0
P β )K (I
h
i
e
e
U = UK
U
Pβ )
is the semigroup U (t )(I
U (I
P β )K (I
Pβ )
t >0
h
ij
e
U = UK
j
By using the estimate
e (t )
U
Pβ )
. More generally
e
U.
0
M β 0 e β t (t > 0),
an elementary calculation shows that
h
e
UK
ij
e
U
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0
cj t j e β t
Chapter 4
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Observe that
U (I
e (t )
where U
t >0
P β )K (I
h
i
e
e
U = UK
U
Pβ )
is the semigroup U (t )(I
U (I
P β )K (I
Pβ )
t >0
h
ij
e
U = UK
j
By using the estimate
e (t )
U
Pβ )
. More generally
e
U.
0
M β 0 e β t (t > 0),
an elementary calculation shows that
re (V (t ))
h
e
UK
ij
0
pn (t )e β t
e
U
0
cj t j e β t so
where pn (t ) is a polynomial of degre n.
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Chapter 4
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End of the proof
Let β00 > β0 . Then there exists a constant Mβ00 such that
re (V (t ))
Mβ00 e β
00
t
for t large enough.
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Chapter 4
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End of the proof
Let β00 > β0 . Then there exists a constant Mβ00 such that
re (V (t ))
Mβ00 e β
00
t
for t large enough.
Let ω e (V ) be the essential type of (V (t ))t >0 .
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Chapter 4
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End of the proof
Let β00 > β0 . Then there exists a constant Mβ00 such that
re (V (t ))
Mβ00 e β
00
t
for t large enough.
Let ω e (V ) be the essential type of (V (t ))t >0 . The fact that
re (V (t )) = e ωe (V )t
implies that ω e (V )
β00 .
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Chapter 4
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End of the proof
Let β00 > β0 . Then there exists a constant Mβ00 such that
re (V (t ))
Mβ00 e β
00
t
for t large enough.
Let ω e (V ) be the essential type of (V (t ))t >0 . The fact that
re (V (t )) = e ωe (V )t
implies that ω e (V ) β00 . Hence ω e (V )
β00 > β0 are chosen arbitrarily.
ω e (U ) since β0 > ω e (U ) and
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Chapter 4
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Remarks
A classical weaker estimate
ω e (V )
ω (U )
(where ω (U ) is the type of (U (t ))t >0 ) is due to I. Vidav (J. Math.
Anal. Appl, 30 (1970) 264-279).
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Chapter 4
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Remarks
A classical weaker estimate
ω e (V )
ω (U )
(where ω (U ) is the type of (U (t ))t >0 ) is due to I. Vidav (J. Math.
Anal. Appl, 30 (1970) 264-279).
The estimate ω e (V ) ω e (U ) is also derived by L. W. Weis in J.
Math. Anal. Appl, 129 (1988) 6-23); (see also G. Schluchtermann in
Op Theory Adv Appl, 103, Birkhauser, (1998) 263-277) by using the
properties of measure of noncompactness of strong integrals.
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Chapter 4
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On stability of essential type
We have seen that if some remainder term
+∞
Rn (t ) :=
∑ Uj ( t )
j =n
is compact (or weakly compact when X = L1 (µ)) for large t then
ω e (V )
ω e (U ).
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Chapter 4
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On stability of essential type
We have seen that if some remainder term
+∞
Rn (t ) :=
∑ Uj ( t )
j =n
is compact (or weakly compact when X = L1 (µ)) for large t then
ω e (V )
ω e (U ).
We show now that if some remainder term is compact (or weakly compact
when X = L1 (µ)) for all t > 0 then
ω e (V ) = ω e (U ).
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Chapter 4
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A preliminary result
Theorem
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K . Let V (t ) = ∑0+∞ Uj (t ) be the
Dyson-Phillips expansion of (V (t ))t >0 .
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Chapter 4
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A preliminary result
Theorem
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K . Let V (t ) = ∑0+∞ Uj (t ) be the
Dyson-Phillips expansion of (V (t ))t >0 . Let U (t ) = ∑0+∞ Vj (t ) be the
Dyson-Phillips expansion of (U (t ))t >0 considered as a perturbation of
(V (t ))t >0 (i.e. T = (T + K ) + ( K )).
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Chapter 4
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A preliminary result
Theorem
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K . Let V (t ) = ∑0+∞ Uj (t ) be the
Dyson-Phillips expansion of (V (t ))t >0 . Let U (t ) = ∑0+∞ Vj (t ) be the
Dyson-Phillips expansion of (U (t ))t >0 considered as a perturbation of
(V (t ))t >0 (i.e. T = (T + K ) + ( K )). For any j 2 N , Uj (t ) is
compact for all t > 0 if and only if Vj (t ) is compact for all t > 0.
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Chapter 4
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A preliminary result
Theorem
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K . Let V (t ) = ∑0+∞ Uj (t ) be the
Dyson-Phillips expansion of (V (t ))t >0 . Let U (t ) = ∑0+∞ Vj (t ) be the
Dyson-Phillips expansion of (U (t ))t >0 considered as a perturbation of
(V (t ))t >0 (i.e. T = (T + K ) + ( K )). For any j 2 N , Uj (t ) is
compact for all t > 0 if and only if Vj (t ) is compact for all t > 0. If
X = L1 (µ) then we can replace "compact" by "weakly compact".
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Chapter 4
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A preliminary result
Theorem
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K . Let V (t ) = ∑0+∞ Uj (t ) be the
Dyson-Phillips expansion of (V (t ))t >0 . Let U (t ) = ∑0+∞ Vj (t ) be the
Dyson-Phillips expansion of (U (t ))t >0 considered as a perturbation of
(V (t ))t >0 (i.e. T = (T + K ) + ( K )). For any j 2 N , Uj (t ) is
compact for all t > 0 if and only if Vj (t ) is compact for all t > 0. If
X = L1 (µ) then we can replace "compact" by "weakly compact".
See (M. M-K, Mathematical Topics In neutron transport theory, World
Scienti…c, Vol 46 (1997), Chapter 2).
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By reversing the role of (U (t ))t >0 and (V (t ))t >0 we obtain:
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By reversing the role of (U (t ))t >0 and (V (t ))t >0 we obtain:
Corollary
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K .
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Chapter 4
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By reversing the role of (U (t ))t >0 and (V (t ))t >0 we obtain:
Corollary
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K . Let
Uj + 1 ( t ) =
Z t
0
U0 (t )KUj (t
s )ds (j > 0)
U0 ( t ) = U ( t )
be the terms of the Dyson-Phillips expansion of (V (t ))t >0 .
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Chapter 4
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By reversing the role of (U (t ))t >0 and (V (t ))t >0 we obtain:
Corollary
Let X be a complex Banach space. Let (U (t ))t >0 be a C0 -semigroup with
generator T and let K 2 L(X ). We denote by (V (t ))t >0 the
C0 -semigroup generated by T + K . Let
Uj + 1 ( t ) =
Z t
0
U0 (t )KUj (t
s )ds (j > 0)
U0 ( t ) = U ( t )
be the terms of the Dyson-Phillips expansion of (V (t ))t >0 . If for some
n 2 N Un (t ) is compact (resp. weakly compact if X = L1 (µ)) for all all
t > 0 then
ω e (V ) = ω e (U ).
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Remarks
The stability of essential type appears also in L. W. Weis (J. Math. Anal.
Appl, 129 (1988) 6-23), G. Schluchtermann (Op Theory Adv Appl, 103,
Birkhauser, (1998) 263-277) and in J. Voigt (Proc Steklov Inst Math, 3
(1995) 383-389) under stronger assumptions.
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Chapter 4
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A short digression on resolvent approach
The following "resolvent characterization" is due to S. Brendle (Math.
Nachr, 226 (2001), 35–47).
Theorem
Let n 2 N . Then Un (t ) is compact for all t > 0 if and only if:
(i) t > 0 ! Un (t ) 2 L(X ) is continuous in operator norm
and
(ii) (α + i β T )
and all β 2 R.
1K n
(α + i β
T)
1
is compact for some α > ω (U )
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Chapter 4
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Interest of resolvent approach
In applications, (for example in transport theory with boundary operators
relating the incoming and outgoing ‡uxes) it may happen that the
unperturbed semigroup (U (t ))t >0 is not explicit while the resolvent
(λ T ) 1 is "tractable" !
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Chapter 4
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Interest of resolvent approach
In applications, (for example in transport theory with boundary operators
relating the incoming and outgoing ‡uxes) it may happen that the
unperturbed semigroup (U (t ))t >0 is not explicit while the resolvent
(λ T ) 1 is "tractable" !
The cost of the approach: We have to know a priori that
t > 0 ! Un (t ) 2 L(X )
is continuous in operator norm.
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Chapter 4
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Su¢ cient conditions of continuity in operator norm
The following result is due to S. Brendle (Math. Nachr, 226 (2001),
35–47).
Theorem
Let n 2 N. If X is a Hilbert space and if
(α + i β
T)
1
K
n
(α + i β
T)
1
! 0 as j βj ! +∞
then t > 0 ! Un +2 (t ) 2 L(X ) is continuous in operator norm.
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Chapter 4
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Su¢ cient conditions of continuity in operator norm
The following result is due to S. Brendle (Math. Nachr, 226 (2001),
35–47).
Theorem
Let n 2 N. If X is a Hilbert space and if
(α + i β
T)
1
K
n
(α + i β
T)
1
! 0 as j βj ! +∞
then t > 0 ! Un +2 (t ) 2 L(X ) is continuous in operator norm.
Remark: The continuity of t > 0 ! U1 (t ) 2 L(X ) (if we want to show
the compactness of V (t ) U (t )) is out of reach of this theorem.
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Chapter 4
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Sbihi’s criterion of continuity in operator norm
The following result is due to M. Sbihi (J. Evol. Eq, 7 (1) (2007), 35-58).
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Chapter 4
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Sbihi’s criterion of continuity in operator norm
The following result is due to M. Sbihi (J. Evol. Eq, 7 (1) (2007), 35-58).
(This is actually a corollary of a more general result).
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Chapter 4
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Sbihi’s criterion of continuity in operator norm
The following result is due to M. Sbihi (J. Evol. Eq, 7 (1) (2007), 35-58).
(This is actually a corollary of a more general result).
Theorem
Let X be a Hilbert space and let T be dissipative (i.e.
Re (Tx, x ) 0 8x 2 D (T )). If
K (λ
T)
1
K + K (λ
T)
1
K
! 0 as jIm λj ! +∞
then t > 0 ! U1 (t ) 2 L(X ) is continuous in operator norm.
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Chapter 4
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Sbihi’s criterion of continuity in operator norm
The following result is due to M. Sbihi (J. Evol. Eq, 7 (1) (2007), 35-58).
(This is actually a corollary of a more general result).
Theorem
Let X be a Hilbert space and let T be dissipative (i.e.
Re (Tx, x ) 0 8x 2 D (T )). If
K (λ
T)
1
K + K (λ
T)
1
K
! 0 as jIm λj ! +∞
then t > 0 ! U1 (t ) 2 L(X ) is continuous in operator norm.
Remark: Applications to transport theory are given in (M. Sbihi: J. Evol.
Eq, 7 (1) (2007), 35-58), (B. Lods and M. Sbihi: Math. Meth. Appl. Sci,
29 (2006) 499–523) and in (K. Latrach and B. Lods: Math. Meth. Appl.
Sci, 32 (2009) 1325–1344).
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Chapter 4
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Applications to neutron transport theory
It is time to deal with concrete problems !!
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Chapter 4
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The general philosophy
We have an imperturbed (streaming) semigroup in
Lp (Ω V ; dxdv )
U (t ) : g ! e
Rt
σ (x τv ,v )d τ
g (x
tv , v )1ft
s (x, v ) = inf fs > 0; x
sv 2
/ Ωg
0
s (x ,v )g
where
is the …rst exit time function from the spatial domain Ω.
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Chapter 4
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The general philosophy
We have an imperturbed (streaming) semigroup in
Lp (Ω V ; dxdv )
U (t ) : g ! e
Rt
σ (x τv ,v )d τ
g (x
tv , v )1ft
s (x, v ) = inf fs > 0; x
sv 2
/ Ωg
0
s (x ,v )g
where
is the …rst exit time function from the spatial domain Ω.
We regard the scatterring operator
K :f !
Z
V
k (x, v , v 0 )f (x, v 0 )dv 0
as a perturbation of the generator T of (U (t ))t >0 .
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Two main questions
When is (λ
n2N ?
T)
1K n
compact in Lp (Ω
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V ; dxdv ) for some
Chapter 4
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Two main questions
When is (λ
n2N ?
T)
1K n
compact in Lp (Ω
V ; dxdv ) for some
When is some remainder term Rm (t ) compact in Lp (Ω
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V ; dxdv ) ?
Chapter 4
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Warning !
The resolvent (λ
T)
1
of T cannot be compact
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Chapter 4
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Warning !
The resolvent (λ
T)
1
of T cannot be compact
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Chapter 4
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Warning !
The resolvent (λ T ) 1 of T cannot be compact (in bounded
geometries, σ(T ) is a half-plane when 0 2 V !).
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Chapter 4
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Warning !
The resolvent (λ T ) 1 of T cannot be compact (in bounded
geometries, σ(T ) is a half-plane when 0 2 V !).
The scattering operator
K :f !
Z
V
k (x, v , v 0 )f (x, v 0 )dv 0
is local with respect to the space-varaible x
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Chapter 4
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Warning !
The resolvent (λ T ) 1 of T cannot be compact (in bounded
geometries, σ(T ) is a half-plane when 0 2 V !).
The scattering operator
K :f !
Z
V
k (x, v , v 0 )f (x, v 0 )dv 0
is local with respect to the space-varaible x
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Chapter 4
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Warning !
The resolvent (λ T ) 1 of T cannot be compact (in bounded
geometries, σ(T ) is a half-plane when 0 2 V !).
The scattering operator
K :f !
Z
V
k (x, v , v 0 )f (x, v 0 )dv 0
is local with respect to the space-varaible x so that K cannot be
compact on Lp (Ω V ; dxdv ).
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Chapter 4
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Warning !
The resolvent (λ T ) 1 of T cannot be compact (in bounded
geometries, σ(T ) is a half-plane when 0 2 V !).
The scattering operator
K :f !
Z
V
k (x, v , v 0 )f (x, v 0 )dv 0
is local with respect to the space-varaible x so that K cannot be
compact on Lp (Ω V ; dxdv ).
The good news is that compactness will result from a subtle
combination of (λ T ) 1 and K .
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Chapter 4
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Some standard results on remainder terms
Typically, under quite general assumptions on the scaterring kernel
k (x, v , v 0 ),
for bounded domains Ω and Lebesgue measure dv on Rn as velocity
measure,
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Chapter 4
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Some standard results on remainder terms
Typically, under quite general assumptions on the scaterring kernel
k (x, v , v 0 ),
for bounded domains Ω and Lebesgue measure dv on Rn as velocity
measure, the second order remainder term R2 (t ) is compact in
Lp (Ω V ; dxdv ) (1 < p < ∞) or weakly compact in L1 (Ω V ; dxdv ).
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Chapter 4
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Some standard results on remainder terms
Typically, under quite general assumptions on the scaterring kernel
k (x, v , v 0 ),
for bounded domains Ω and Lebesgue measure dv on Rn as velocity
measure, the second order remainder term R2 (t ) is compact in
Lp (Ω V ; dxdv ) (1 < p < ∞) or weakly compact in L1 (Ω V ; dxdv ).
G. Greiner (Math Z, 185 (1984) 167-177).
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Chapter 4
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Some standard results on remainder terms
Typically, under quite general assumptions on the scaterring kernel
k (x, v , v 0 ),
for bounded domains Ω and Lebesgue measure dv on Rn as velocity
measure, the second order remainder term R2 (t ) is compact in
Lp (Ω V ; dxdv ) (1 < p < ∞) or weakly compact in L1 (Ω V ; dxdv ).
G. Greiner (Math Z, 185 (1984) 167-177).
J. Voigt (J. Math. Anal. Appl, 106 (1985)140-153).
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Chapter 4
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Some standard results on remainder terms
Typically, under quite general assumptions on the scaterring kernel
k (x, v , v 0 ),
for bounded domains Ω and Lebesgue measure dv on Rn as velocity
measure, the second order remainder term R2 (t ) is compact in
Lp (Ω V ; dxdv ) (1 < p < ∞) or weakly compact in L1 (Ω V ; dxdv ).
G. Greiner (Math Z, 185 (1984) 167-177).
J. Voigt (J. Math. Anal. Appl, 106 (1985)140-153).
P. Takak (Transp Theory Stat Phys, 14(5) (1985) 655-667).
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Chapter 4
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Some standard results on remainder terms
Typically, under quite general assumptions on the scaterring kernel
k (x, v , v 0 ),
for bounded domains Ω and Lebesgue measure dv on Rn as velocity
measure, the second order remainder term R2 (t ) is compact in
Lp (Ω V ; dxdv ) (1 < p < ∞) or weakly compact in L1 (Ω V ; dxdv ).
G. Greiner (Math Z, 185 (1984) 167-177).
J. Voigt (J. Math. Anal. Appl, 106 (1985)140-153).
P. Takak (Transp Theory Stat Phys, 14(5) (1985) 655-667).
L. W. Weis (J. Math. Anal. Appl, 129 (1988) 6-23)
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Chapter 4
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Some standard results on remainder terms
Typically, under quite general assumptions on the scaterring kernel
k (x, v , v 0 ),
for bounded domains Ω and Lebesgue measure dv on Rn as velocity
measure, the second order remainder term R2 (t ) is compact in
Lp (Ω V ; dxdv ) (1 < p < ∞) or weakly compact in L1 (Ω V ; dxdv ).
G. Greiner (Math Z, 185 (1984) 167-177).
J. Voigt (J. Math. Anal. Appl, 106 (1985)140-153).
P. Takak (Transp Theory Stat Phys, 14(5) (1985) 655-667).
L. W. Weis (J. Math. Anal. Appl, 129 (1988) 6-23)
M. Mokhtar-Kharroubi. Thèse d’Etat (french habilitation), Paris,
1987.
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Chapter 4
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On velocity measure
Multigroup models
m
∂fi
∂fi
+ v . + σi (x, v )fi (t, x, v ) = ∑
∂t
∂x
j =1
Z
Vj
ki,j (x, v , v 0 )fj (t, x, v 0 )µj (dv 0 ),
where the velocities live on …nitely many spheres
Vj := v 2 R3 , jv j = cj , 1
j
m, (cj > 0),
are not covered by the papers above !
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Chapter 4
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On velocity measure
Multigroup models
m
∂fi
∂fi
+ v . + σi (x, v )fi (t, x, v ) = ∑
∂t
∂x
j =1
Z
Vj
ki,j (x, v , v 0 )fj (t, x, v 0 )µj (dv 0 ),
where the velocities live on …nitely many spheres
Vj := v 2 R3 , jv j = cj , 1
j
m, (cj > 0),
are not covered by the papers above ! (But are covered by the classical
paper by K. Jorgens, Com. Pure. Appl. Math, 11 (1958) 219-242.)
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Chapter 4
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A need for a general formalism covering the various models
We deal a priori with an abstract positive velocity measure µ(dv )
and look for conditions on µ providing answers to the two questions
above.
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Chapter 4
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A need for a general formalism covering the various models
We deal a priori with an abstract positive velocity measure µ(dv )
and look for conditions on µ providing answers to the two questions
above.
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Chapter 4
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A need for a general formalism covering the various models
We deal a priori with an abstract positive velocity measure µ(dv )
and look for conditions on µ providing answers to the two questions
above.i.e. we consider abstract neutron transport equations
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) =
∂t
∂x
Z
V
k (x, v , v 0 )f (t, x, v 0 )µ(dv 0 )
covering for instance both continuous or multigroup models (or even
combinations of the two).
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Chapter 4
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A need for a general formalism covering the various models
We deal a priori with an abstract positive velocity measure µ(dv )
and look for conditions on µ providing answers to the two questions
above.i.e. we consider abstract neutron transport equations
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) =
∂t
∂x
Z
V
k (x, v , v 0 )f (t, x, v 0 )µ(dv 0 )
covering for instance both continuous or multigroup models (or even
combinations of the two).
We obtain di¤erent results according to whether we work in L1
spaces or in Lp spaces with p > 1.
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Chapter 4
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On scattering operators
Let Ω Rn (n > 1) be an open subset and let µ be a positive Radon
measure with support V .
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Chapter 4
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On scattering operators
Let Ω Rn (n > 1) be an open subset and let µ be a positive Radon
measure with support V . Let
X := Lp (Ω
V ; dx
µ(dv ))
with
1
p < +∞.
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Chapter 4
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On scattering operators
Let Ω Rn (n > 1) be an open subset and let µ be a positive Radon
measure with support V . Let
X := Lp (Ω
V ; dx
µ(dv ))
with
p < +∞.
1
Let
k : (x, v , v 0 ) : Ω
V
V ! k (x, v , v 0 ) 2 R+
be measurable and such that
K : f 2 Lp (Ω
V) !
Z
V
is a bounded operator on Lp (Ω
k (x, v , v 0 )f (x, v 0 )dv 0 2 Lp (Ω
V)
V ).
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Chapter 4
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Since K is local in space variable, we may interprete it as a family of
bounded operators on Lp (V ) indexed by the parameter x 2 Ω i.e. a
mapping
K : x 2 Ω ! K (x ) 2 L(Lp (V )).
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Since K is local in space variable, we may interprete it as a family of
bounded operators on Lp (V ) indexed by the parameter x 2 Ω i.e. a
mapping
K : x 2 Ω ! K (x ) 2 L(Lp (V )).
Then the norm of K as bounded operator on Lp (Ω
kK kL(Lp (Ω
V ))
V ) is given by
= sup kK (x )kL(Lp (V )) .
x 2Ω
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Chapter 4
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Regularity of scattering operators in Lp
In the following we restrict ourselves to 1 < p < ∞.
De…nition
A scattering operator K is called regular if
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Chapter 4
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Regularity of scattering operators in Lp
In the following we restrict ourselves to 1 < p < ∞.
De…nition
A scattering operator K is called regular if
(i) fK (x ); x 2 Ωg is a set of collectively compact operators on
Lp (V ), i.e. the set
n
o
K (x ) ϕ; x 2 Ω, k ϕkLp (V ) 1
is relatively compact in Lp (V ).
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Chapter 4
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Regularity of scattering operators in Lp
In the following we restrict ourselves to 1 < p < ∞.
De…nition
A scattering operator K is called regular if
(i) fK (x ); x 2 Ωg is a set of collectively compact operators on
Lp (V ), i.e. the set
n
o
K (x ) ϕ; x 2 Ω, k ϕkLp (V ) 1
is relatively compact in Lp (V ).
0
(ii) For any ψ0 2 Lp (V ), the set fK 0 (x )ψ0 ; x 2 Ωg is relatively compact
0
in Lp (V ) where p 0 is the conjugate exponent and K 0 (x ) is the dual
operator of K (x ).
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Chapter 4
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Remarks
Roughly speaking, the regularity of K means that for each x 2 Ω,
K (x ) is a compact operator on Lp (V ) and
x 2 Ω ! K (x ) 2 L(Lp (V ))
is "regular".
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Chapter 4
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Remarks
Roughly speaking, the regularity of K means that for each x 2 Ω,
K (x ) is a compact operator on Lp (V ) and
x 2 Ω ! K (x ) 2 L(Lp (V ))
is "regular".
The technical condition (ii) is not apparently a consequence of (i).
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Chapter 4
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Remarks
Roughly speaking, the regularity of K means that for each x 2 Ω,
K (x ) is a compact operator on Lp (V ) and
x 2 Ω ! K (x ) 2 L(Lp (V ))
is "regular".
The technical condition (ii) is not apparently a consequence of (i).
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Chapter 4
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Remarks
Roughly speaking, the regularity of K means that for each x 2 Ω,
K (x ) is a compact operator on Lp (V ) and
x 2 Ω ! K (x ) 2 L(Lp (V ))
is "regular".
The technical condition (ii) is not apparently a consequence of (i). It
0
is of course satis…ed if the dual scattering operator on Lp (Ω V ) is
also regular.
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Chapter 4
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Remarks
Roughly speaking, the regularity of K means that for each x 2 Ω,
K (x ) is a compact operator on Lp (V ) and
x 2 Ω ! K (x ) 2 L(Lp (V ))
is "regular".
The technical condition (ii) is not apparently a consequence of (i). It
0
is of course satis…ed if the dual scattering operator on Lp (Ω V ) is
also regular.
The compactness of K with respect to velocities (at least for
p = 2) is satis…ed by most physical models,
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
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Remarks
Roughly speaking, the regularity of K means that for each x 2 Ω,
K (x ) is a compact operator on Lp (V ) and
x 2 Ω ! K (x ) 2 L(Lp (V ))
is "regular".
The technical condition (ii) is not apparently a consequence of (i). It
0
is of course satis…ed if the dual scattering operator on Lp (Ω V ) is
also regular.
The compactness of K with respect to velocities (at least for
p = 2) is satis…ed by most physical models,
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
52 / 118
Remarks
Roughly speaking, the regularity of K means that for each x 2 Ω,
K (x ) is a compact operator on Lp (V ) and
x 2 Ω ! K (x ) 2 L(Lp (V ))
is "regular".
The technical condition (ii) is not apparently a consequence of (i). It
0
is of course satis…ed if the dual scattering operator on Lp (Ω V ) is
also regular.
The compactness of K with respect to velocities (at least for
p = 2) is satis…ed by most physical models, see Borysiewicz and
Mika, J. Math. Anal. Appl, 26 (1969) 461-478.
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Chapter 4
52 / 118
An approximation property
The follwing result is given in (M. M-K. J. Funct. Anal, 226 (2005)
21-47).
Theorem
The class of regular scattering operators is the closure in the operator
norm of L(Lp (Ω V )) of the class of scattering operators with kernels
k (x, v , v 0 ) =
∑ αj (x )fj (v )gj (v 0 )
j 2J
where αj 2 L∞ (Ω), fj , gj continuous with compact supports and J …nite.
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Chapter 4
53 / 118
The …rst part of the following lemma is given in (M. M-K, Mathematical
Topics in neutron transport theory, World Scienti…c, Vol 46 (1997),
Chapter 3, p.32) and the second part in (M. M-K . J. Funct. Anal, 226
(2005) 21-47).
Lemma
Let µ be a …nite Radon measure on Rn .
(i) If the hyperplanes (through the origin) have zero µ-measure then
sup µ fv ; jv .e j
e 2S n
1
εg ! 0 as ε ! 0.
(ii) If the a¢ ne (i.e. translated) hyperplanes have zero µ-measure then
sup µ
e 2S n 1
µ (v , v 0 ); (v
v 0 ).e
ε ! 0 as ε ! 0.
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Chapter 4
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Our questions
We remind our goal:
When is (λ
n2N ?
T)
1K n
compact in Lp (Ω
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CIMPA School Muizemberg July 22-Aug 4
V ; dxdv ) for some
Chapter 4
55 / 118
Our questions
We remind our goal:
When is (λ
n2N ?
T)
1K n
compact in Lp (Ω
V ; dxdv ) for some
When is some remainder term Rm (t ) compact in Lp (Ω
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V ; dxdv ) ?
Chapter 4
55 / 118
A compactness theorem for the generator
See M. M-K . J. Funct. Anal, 226 (2005) 21-47.
Theorem
We assume that Ω has …nite Lebesgue measure and the scattering
operator is regular in Lp (Ω V ; dx µ(dv )) (1 < p < ∞). If the
hyperplanes have zero µ-measure then (λ T ) 1 K and K (λ T )
compact on Lp (Ω V ; dx µ(dv )).
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Chapter 4
1
are
56 / 118
An averaging lemma
The compactness of K (λ T ) 1 can be expressed as an averaging
lemma in open sets Ω with …nite volume, i.e. if z is a bounded subset of
D (T ) (for the graph norm) then
fK ϕ; ϕ 2 zg is relatively compact in Lp (Ω
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CIMPA School Muizemberg July 22-Aug 4
V ).
Chapter 4
57 / 118
An averaging lemma
The compactness of K (λ T ) 1 can be expressed as an averaging
lemma in open sets Ω with …nite volume, i.e. if z is a bounded subset of
D (T ) (for the graph norm) then
fK ϕ; ϕ 2 zg is relatively compact in Lp (Ω
V ).
See also (F. Golse, P.L. Lions, B. Perthame and R. Sentis. J. Funct. Anal,
76 (1988) 110-125.) for Sobolev regularity of velocity averaging in
bounded convex domains Ω.
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Chapter 4
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Spectral consequence
Theorem
We assume that Ω has …nite Lebesgue measure, the scattering operator is
regular in Lp (Ω V ; dx µ(dv )) (1 < p < ∞) and the hyperplanes have
zero µ-measure. Then
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Chapter 4
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Spectral consequence
Theorem
We assume that Ω has …nite Lebesgue measure, the scattering operator is
regular in Lp (Ω V ; dx µ(dv )) (1 < p < ∞) and the hyperplanes have
zero µ-measure. Then
σ e (T + K ) = σ e (T ).
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Chapter 4
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Spectral consequence
Theorem
We assume that Ω has …nite Lebesgue measure, the scattering operator is
regular in Lp (Ω V ; dx µ(dv )) (1 < p < ∞) and the hyperplanes have
zero µ-measure. Then
σ e (T + K ) = σ e (T ).
In particular
σ(T + K ) \ fRe λ > s (T )g
consists at most of isolated eigenvalues with …nite algebraic multiplicities
where
Z
1 t
s (T ) =
lim inf
σ(y + τv , v )d τ.
t !+∞ (y ,v ) t 0
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Chapter 4
58 / 118
A compactness theorem for the semigroup
See M. M-K . J. Funct. Anal, 226 (2005) 21-47.
Theorem
We assume that Ω has …nite Lebesgue measure and the scattering
operator is regular in Lp (Ω V ; dx µ(dv )) (1 < p < ∞). If the
hyperplanes have zero µ-measure then
V (t )
U (t ) is compact on Lp (Ω
V)
(t > 0),
i.e. the …rst remainder R1 (t ) = ∑j+=∞1 Uj (t ) is compact on Lp (Ω
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
V ).
59 / 118
Strategy of the proof
R1 (t ) = ∑j+=∞m Uj (t ) is compact for all t > 0 if and only if U1 (t ) is.
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Chapter 4
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Strategy of the proof
R1 (t ) = ∑j+=∞m Uj (t ) is compact for all t > 0 if and only if U1 (t ) is.
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Chapter 4
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Strategy of the proof
R1 (t ) = ∑j+=∞m Uj (t ) is compact for all t > 0 if and only if U1 (t ) is.
So we deal with the simpler operator
U1 ( t ) =
Z t
U (t
s )KU (s )ds.
0
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
60 / 118
Strategy of the proof
R1 (t ) = ∑j+=∞m Uj (t ) is compact for all t > 0 if and only if U1 (t ) is.
So we deal with the simpler operator
U1 ( t ) =
Z t
U (t
s )KU (s )ds.
0
U1 (t ) depends (linearly and) continuously on the scattering operator
K.
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Chapter 4
60 / 118
Strategy of the proof
R1 (t ) = ∑j+=∞m Uj (t ) is compact for all t > 0 if and only if U1 (t ) is.
So we deal with the simpler operator
U1 ( t ) =
Z t
U (t
s )KU (s )ds.
0
U1 (t ) depends (linearly and) continuously on the scattering operator
K.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
60 / 118
Strategy of the proof
R1 (t ) = ∑j+=∞m Uj (t ) is compact for all t > 0 if and only if U1 (t ) is.
So we deal with the simpler operator
U1 ( t ) =
Z t
U (t
s )KU (s )ds.
0
U1 (t ) depends (linearly and) continuously on the scattering operator
K.
By approximation, we may assume that
k (x, v , v 0 ) =
∑ αj (x )fj (v )gj (v 0 )
j 2J
where αj 2 L∞ (Ω), fj , gj continuous with compact supports and J is …nite.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
60 / 118
Strategy of the proof
R1 (t ) = ∑j+=∞m Uj (t ) is compact for all t > 0 if and only if U1 (t ) is.
So we deal with the simpler operator
U1 ( t ) =
Z t
U (t
s )KU (s )ds.
0
U1 (t ) depends (linearly and) continuously on the scattering operator
K.
By approximation, we may assume that
k (x, v , v 0 ) =
∑ αj (x )fj (v )gj (v 0 )
j 2J
where αj 2 L∞ (Ω), fj , gj continuous with compact supports and J is …nite.
By linearity, we may even choose
k (x, v , v 0 ) = α(x )f (v )g (v 0 )
where α 2 L∞ (Ω), f , g are continuous with compact supports.
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Chapter 4
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In this case, U1 (t ) operates on all Lq (Ω
V ) (1
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CIMPA School Muizemberg July 22-Aug 4
q
+∞).
Chapter 4
61 / 118
In this case, U1 (t ) operates on all Lq (Ω V ) (1 q +∞).
So, by interpolation argument, we may restrict ourselves to the case
p = 2.
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Chapter 4
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Domination arguments
In Lp spaces (1 < p < ∞), if Oi (i = 1, 2) are two positive
operators such that
O1 f
O2 f
8f 2 Lp+
and if O2 is compact then O1 is also compact. See (D. Aliprantis
and O. Burkinshaw. Math. Z, 174 (1980) 289-298).
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Chapter 4
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Domination arguments
In Lp spaces (1 < p < ∞), if Oi (i = 1, 2) are two positive
operators such that
O1 f
O2 f
8f 2 Lp+
and if O2 is compact then O1 is also compact. See (D. Aliprantis
and O. Burkinshaw. Math. Z, 174 (1980) 289-298).
By domination, we may assume that V is compact, α = f = g = 1
and σ = 0 .
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Chapter 4
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Last domination argument
Because of σ = 0
U (t ) ϕ = ϕ (x
where s (x, v ) = inf fs > 0; x
tv , v )1ft
s (x ,v )g
sv 2
/ Ωg .
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
63 / 118
Last domination argument
Because of σ = 0
U (t ) ϕ = ϕ (x
tv , v )1ft
s (x ,v )g
where s (x, v ) = inf fs > 0; x sv 2
/ Ωg .
If e
ϕ 2 L2 (Rn V ) is the trivial extension of ϕ by zero outside
Ω V then, for nonnegative ϕ
U (t ) ϕ(x, v )
e
ϕ (x
tv , v ) 8(x, v ) 2 Ω
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V.
Chapter 4
63 / 118
Last domination argument
Because of σ = 0
U (t ) ϕ = ϕ (x
tv , v )1ft
s (x ,v )g
where s (x, v ) = inf fs > 0; x sv 2
/ Ωg .
If e
ϕ 2 L2 (Rn V ) is the trivial extension of ϕ by zero outside
Ω V then, for nonnegative ϕ
U (t ) ϕ(x, v )
e
ϕ (x
tv , v ) 8(x, v ) 2 Ω
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CIMPA School Muizemberg July 22-Aug 4
V.
Chapter 4
63 / 118
Last domination argument
Because of σ = 0
U (t ) ϕ = ϕ (x
tv , v )1ft
s (x ,v )g
where s (x, v ) = inf fs > 0; x sv 2
/ Ωg .
If e
ϕ 2 L2 (Rn V ) is the trivial extension of ϕ by zero outside
Ω V then, for nonnegative ϕ
U (t ) ϕ(x, v )
e
ϕ (x
tv , v ) 8(x, v ) 2 Ω
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CIMPA School Muizemberg July 22-Aug 4
V.
Chapter 4
63 / 118
Last domination argument
Because of σ = 0
U (t ) ϕ = ϕ (x
tv , v )1ft
s (x ,v )g
where s (x, v ) = inf fs > 0; x sv 2
/ Ωg .
If e
ϕ 2 L2 (Rn V ) is the trivial extension of ϕ by zero outside
Ω V then, for nonnegative ϕ
U (t ) ϕ(x, v )
so
e
ϕ (x
U (t ) ϕ
tv , v ) 8(x, v ) 2 Ω
V.
RU∞ (t )E ϕ
where E : L2 (Ω V ) ! L2 (Rn V ) is the trivial extension
operator, R : L2 (Rn V ) ! L2 (Ω V ) is the restriction
operator and
U∞ (t ) : ψ 2 L2 (Rn
V ) ! ψ (x
tv , v ) 2 L2 (Rn
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CIMPA School Muizemberg July 22-Aug 4
V ).
Chapter 4
63 / 118
Thus
U1 ( t ) =
Z t
0
= R
R
RU∞ (t
Z t
0
Z t
0
s )E K RU∞ (s )E ds
U∞ (t
s )E K RU∞ (s )ds
U∞ (t
s )KU∞ (s )ds
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CIMPA School Muizemberg July 22-Aug 4
E
E.
Chapter 4
64 / 118
We are led to deal with the compactness of
R
Z t
0
U∞ (t
s )KU∞ (s )ds : L2 (Rn
V ) ! L2 (Ω
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CIMPA School Muizemberg July 22-Aug 4
V ).
Chapter 4
65 / 118
We are led to deal with the compactness of
R
Z t
0
U∞ (t
s )KU∞ (s )ds : L2 (Rn
V ) ! L2 (Ω
V ).
Note that
Z t
0
U∞ (t
s )KU∞ (s )ψds =
Z t
0
ds
Z
V
ψ (x
(t
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CIMPA School Muizemberg July 22-Aug 4
s )v
sv 0 , v 0 )µ(v 0 ).
Chapter 4
65 / 118
Fourier analysis
For any ψ(x, v ) 2 L2 (Rn
V ), we denote by
b (ζ, v )
ψ
its partial Fourier transform with respect to space variable.
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Chapter 4
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Fourier analysis
For any ψ(x, v ) 2 L2 (Rn
V ), we denote by
b (ζ, v )
ψ
its partial Fourier transform with respect to space variable. Then
kψk2L2 (Rn
V)
=
Z Z
V
Rn
b (ζ, v ) 2 d ζµ(dv )
ψ
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
66 / 118
Fourier analysis
For any ψ(x, v ) 2 L2 (Rn
V ), we denote by
b (ζ, v )
ψ
its partial Fourier transform with respect to space variable. Then
kψk2L2 (Rn
and
V)
=
Z Z
V
Rn
1
n
M !∞ (2π ) 2
ψ(x, v ) = lim
where the limit holds in L2 (Rn
Z
b (ζ, v ) 2 d ζµ(dv )
ψ
jζ j M
b (ζ, v )e i ζ.x d ζ
ψ
V ) norm.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
66 / 118
Hence
Z t
0
=
Z t
U∞ (t
ds
0
Z
V
s )KU∞ (s )ψds
ψ (x
Z
1
n
M !∞ (2π ) 2 jζ j
Z
1
= lim
n
M !∞ (2π ) 2 jζ j
=
lim
(t
e ix .ζ
M
M
sv 0 , v 0 )µ(dv 0 )
s )v
Z
Z t
ds
0
V
Z
V
b (ζ, v )e
ψ
b (ζ, v )
e ix .ζ ψ
where the the limit holds in L2 (Rn
Z t
dse
i ((t s )v +sv 0 ).ζ
µ(dv 0 )
i ((t s )v +sv 0 ).ζ
ds
0
µ(dv 0 )
V ) norm.
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Chapter 4
67 / 118
For each M > 0, let
OM : L2 (Rn
ψ!
Z
jζ j M
Z
e
V
ix .ζ b
ψ(ζ, v )
V ) ! L2 (Rn
Z t
dse
V)
i ((t s )v +sv 0 ).ζ
0
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
ds
µ(dv 0 ).
Chapter 4
68 / 118
For each M > 0, let
OM : L2 (Rn
ψ!
Z
jζ j M
Z
e
V
ix .ζ b
ψ(ζ, v )
V ) ! L2 (Rn
Z t
dse
V)
i ((t s )v +sv 0 ).ζ
0
ds
µ(dv 0 ).
We observe that ROM is a Hilbert Shmidt operator because Ω has
…nite volume and V is compact. ( Keep in mind that x 2 Ω !)
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
68 / 118
It su¢ ces to show that
OM ψ !
in L2 (Rn
Z t
0
ds
Z
V
ψ (x
(t
V ) uniformly in kψkL2 (Rn
s )v
V)
sv 0 , v 0 )µ(dv 0 )
1,
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Chapter 4
69 / 118
It su¢ ces to show that
OM ψ !
in L2 (Rn
Z
0
ds
Z
V
ψ (x
(t
V ) uniformly in kψkL2 (Rn
e ix .ζ
jζ j>M
in L2 (Rn
Z t
Z
V
b (ζ, v 0 )
ψ
Z t
e
s )v
V)
sv 0 , v 0 )µ(dv 0 )
1, i.e.
i ((t s )v +sv 0 ).ζ
0
V ) uniformly in kψkL2 (Rn
V)
ds
µ(dv 0 ) ! 0
1.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
69 / 118
By Parseval identity, this amounts to
Z
V
µ(dv )
Z
jζ j>M
dζ
Z
uniformly in kψkL2 (Rn
V
b (ζ, v 0 )
ψ
V)
Z t
e
i ((t s )v +sv 0 ).ζ
0
2
ds
0
µ(dv )
!0
1.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
70 / 118
By Parseval identity, this amounts to
Z
V
µ(dv )
Z
jζ j>M
dζ
Z
V
uniformly in kψkL2 (Rn
majorize by
sup
jζ j>M
sup
jζ j>M
Z
Z
V
V
V
V
Z t
b (ζ, v 0 )
ψ
V)
e
i ((t s )v +sv 0 ).ζ
2
ds
0
0
µ(dv )
!0
1. Using Cauchy-Schwarz inequality, we
0
µ(dv )µ(dv )
µ(dv 0 )µ(dv )
Z t
e
i ((t s )v +sv 0 ).ζ
2
ds
0
Z t
e
i ((t s )v +sv 0 ).ζ
0
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
2
ds
Z
Rn
V
b (ζ, v 0 )
ψ
8 kψk
Chapter 4
2
1.
70 / 118
Now
Z
=
=
Z
Z
V
V
V
V
V
V
µ(dv 0 )µ(dv )
µ(dv 0 )µ(dv )
µ(dv 0 )µ(dv )
where ζ = jζ j e, (e 2 S n
Z t
e
i ((t s )v +sv 0 ).ζ
2
ds
0
Z t
e is (v
v 0 ).ζ
2
ds
0
Z t
0
e is jζ j(v
v 0 ).e
2
ds
1)
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
71 / 118
and
Z
=
Z
V
V
µ(dv 0 )µ(dv )
fj(v v 0 ).e j εg
+
t2
+
Z
Z
0
fj(v v 0 ).e j εg
fj(v v 0 ).e j>εg
v 0 ).e
e is jζ j(v
µ(dv 0 )µ(dv )
fj(v v 0 ).e j>εg
Z
Z t
Z t
µ(dv 0 )µ(dv )
0
2
ds
e is jζ j(v
Z t
0
v 0 ).e
e is jζ j(v
2
ds
v 0 ).e
2
ds
µ(dv 0 )µ(dv )
0
µ(dv )µ(dv )
Z t
e
0
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CIMPA School Muizemberg July 22-Aug 4
is jζ j(v v 0 ).e
2
ds
.
Chapter 4
72 / 118
and
Z
=
Z
V
V
µ(dv 0 )µ(dv )
fj(v v 0 ).e j εg
+
t2
+
Z
Z
0
fj(v v 0 ).e j εg
fj(v v 0 ).e j>εg
v 0 ).e
e is jζ j(v
µ(dv 0 )µ(dv )
fj(v v 0 ).e j>εg
Z
Z t
Z t
µ(dv 0 )µ(dv )
0
2
ds
e is jζ j(v
Z t
0
v 0 ).e
e is jζ j(v
2
ds
v 0 ).e
2
ds
µ(dv 0 )µ(dv )
0
µ(dv )µ(dv )
Z t
e
0
is jζ j(v v 0 ).e
2
ds
.
The …rst term can made arbitrarily small for ε small enough (assumption
on the velocity measure µ)
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
72 / 118
and
Z
=
Z
V
V
µ(dv 0 )µ(dv )
fj(v v 0 ).e j εg
+
t2
+
Z
Z
0
fj(v v 0 ).e j εg
fj(v v 0 ).e j>εg
v 0 ).e
e is jζ j(v
µ(dv 0 )µ(dv )
fj(v v 0 ).e j>εg
Z
Z t
Z t
µ(dv 0 )µ(dv )
0
2
ds
e is jζ j(v
Z t
0
v 0 ).e
e is jζ j(v
2
ds
v 0 ).e
2
ds
µ(dv 0 )µ(dv )
0
µ(dv )µ(dv )
Z t
e
0
is jζ j(v v 0 ).e
2
ds
.
The …rst term can made arbitrarily small for ε small enough (assumption
on the velocity measure µ)R and, for ε …xed, the second term goes to
0
t
zero as jζ j ! ∞ because of 0 e is jζ j(v v ).e ds (Riemann-Lebesgue lemma).
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Chapter 4
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and
Z
=
Z
V
V
µ(dv 0 )µ(dv )
fj(v v 0 ).e j εg
+
t2
+
Z
Z
0
fj(v v 0 ).e j εg
fj(v v 0 ).e j>εg
v 0 ).e
e is jζ j(v
µ(dv 0 )µ(dv )
fj(v v 0 ).e j>εg
Z
Z t
Z t
µ(dv 0 )µ(dv )
0
2
ds
e is jζ j(v
Z t
0
v 0 ).e
e is jζ j(v
2
ds
v 0 ).e
2
ds
µ(dv 0 )µ(dv )
0
µ(dv )µ(dv )
Z t
e
0
is jζ j(v v 0 ).e
2
ds
.
The …rst term can made arbitrarily small for ε small enough (assumption
on the velocity measure µ)R and, for ε …xed, the second term goes to
0
t
zero as jζ j ! ∞ because of 0 e is jζ j(v v ).e ds (Riemann-Lebesgue lemma).
This ends the proof !!
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Chapter 4
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Spectral consequence
Theorem
We assume that Ω has …nite Lebesgue measure, the scattering operator is
regular in Lp (Ω V ; dx µ(dv )) (1 < p < ∞) and the hyperplanes have
zero µ-measure. Then
σe (V (t )) = σe (U (t )).
In particular ω e (V ) = ω e (U ) and
n
o
σ(V (t )) \ β; j βj > e s (T )t
consists at most of isolated eigenvalues with …nite algebraic multiplicities.
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Chapter 4
73 / 118
On the assumption on the velocity measure
The assumption on the velocity measure µ is "optimal".
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Chapter 4
74 / 118
On the assumption on the velocity measure
The assumption on the velocity measure µ is "optimal". For instance, we
can …nd in (M. M-K . J. Funct. Anal, 226 (2005) 21-47) the following:
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Chapter 4
74 / 118
On the assumption on the velocity measure
The assumption on the velocity measure µ is "optimal". For instance, we
can …nd in (M. M-K . J. Funct. Anal, 226 (2005) 21-47) the following:
Theorem
Let µ be …nite, Ω bounded and
K : ϕ 2 L2 (Ω
V) !
Z
V
ϕ(x, v )µ(dv ).
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Chapter 4
74 / 118
On the assumption on the velocity measure
The assumption on the velocity measure µ is "optimal". For instance, we
can …nd in (M. M-K . J. Funct. Anal, 226 (2005) 21-47) the following:
Theorem
Let µ be …nite, Ω bounded and
K : ϕ 2 L2 (Ω
V) !
Z
V
ϕ(x, v )µ(dv ).
Let there exist a hyperplane H = fv ; v .e = c g (e 2 S n
positive µ-measure.
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1,
c 2 R) with
Chapter 4
74 / 118
On the assumption on the velocity measure
The assumption on the velocity measure µ is "optimal". For instance, we
can …nd in (M. M-K . J. Funct. Anal, 226 (2005) 21-47) the following:
Theorem
Let µ be …nite, Ω bounded and
K : ϕ 2 L2 (Ω
V) !
Z
V
ϕ(x, v )µ(dv ).
Let there exist a hyperplane H = fv ; v .e = c g (e 2 S n 1 , c 2 R) with
positive µ-measure. Then there exists t > 0 such that V (t ) U (t ) is not
compact on L2 (Ω V ) for 0 < t t.
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Chapter 4
74 / 118
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Chapter 4
75 / 118
We can …nd also:
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Chapter 4
75 / 118
We can …nd also:
Theorem
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 L2 (Ω V ) for all t > 0.
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Chapter 4
75 / 118
On the assumption on the scattering operator
The assumption that the scatterig operator is regular is "nearly optimal".
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Chapter 4
76 / 118
On the assumption on the scattering operator
The assumption that the scatterig operator is regular is "nearly optimal".
For instance, we can …nd in (M. M-K . J. Funct. Anal, 226 (2005) 21-47)
the following:
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Chapter 4
76 / 118
On the assumption on the scattering operator
The assumption that the scatterig operator is regular is "nearly optimal".
For instance, we can …nd in (M. M-K . J. Funct. Anal, 226 (2005) 21-47)
the following:
Theorem
Let µ be an arbitrary positive measure. We assume that its support V is
bounded.
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Chapter 4
76 / 118
On the assumption on the scattering operator
The assumption that the scatterig operator is regular is "nearly optimal".
For instance, we can …nd in (M. M-K . J. Funct. Anal, 226 (2005) 21-47)
the following:
Theorem
Let µ be an arbitrary positive measure. We assume that its support V is
bounded. If V (t ) U (t ) is compact on Lp (Ω V ) for all t > 0 then, for
any open ball B Ω, the strong integral
Z
K (x )dx
B
is a compact operator on Lp (V ).
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Chapter 4
76 / 118
We deduce then
Corollary
Besides the conditions above, we assume that
x 2 Ω ! K (x ) 2 L(Lp (V ))
is measurable (not simply strongly measurable !) e.g. is piecewise
continuous in operator norm.
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Chapter 4
77 / 118
We deduce then
Corollary
Besides the conditions above, we assume that
x 2 Ω ! K (x ) 2 L(Lp (V ))
is measurable (not simply strongly measurable !) e.g. is piecewise
continuous in operator norm. Then K (x ) is a compact operator on
Lp (V ) for almost all x 2 Ω.
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Chapter 4
77 / 118
We deduce then
Corollary
Besides the conditions above, we assume that
x 2 Ω ! K (x ) 2 L(Lp (V ))
is measurable (not simply strongly measurable !) e.g. is piecewise
continuous in operator norm. Then K (x ) is a compact operator on
Lp (V ) for almost all x 2 Ω.
Proof:
1
jB j
Z
B
K (x )dx ! K (x ) in L(Lp (V )) as jB j ! 0
at the Lebesgue points x of K : Ω ! L(Lp (V )).
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Chapter 4
77 / 118
What about L1 theory ?
L1 space is the physical setting for neutron transport because
Z Z
Ω
V
f (x, v , t )dx µ(dv )
is the expected number of particles.
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Chapter 4
78 / 118
What about L1 theory ?
L1 space is the physical setting for neutron transport because
Z Z
Ω
V
f (x, v , t )dx µ(dv )
is the expected number of particles.
The mathematical results are very di¤erent from those in Lp
theory (p > 1).
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Chapter 4
78 / 118
What about L1 theory ?
L1 space is the physical setting for neutron transport because
Z Z
Ω
V
f (x, v , t )dx µ(dv )
is the expected number of particles.
The mathematical results are very di¤erent from those in Lp
theory (p > 1).
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Chapter 4
78 / 118
What about L1 theory ?
L1 space is the physical setting for neutron transport because
Z Z
Ω
V
f (x, v , t )dx µ(dv )
is the expected number of particles.
The mathematical results are very di¤erent from those in Lp
theory (p > 1). (And the analysis is much more involved !)
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Chapter 4
78 / 118
Reminder on weakly compact sets
Let (E , m ) be a σ-…nite measure space. A bounded subset
B L1 (E , m ) is relatively weakly compact if
sup
f 2B
Z
A
jf j dm ! 0 as m(A) ! 0
and (if m (E ) = ∞) there exists measurable sets En
, m (En ) < +∞, En En +1 , [En = E such that
sup
f 2B
Z
E nc
E
jf j dm ! 0 as n ! ∞.
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Chapter 4
79 / 118
Reminder on weakly compact sets
Let (E , m ) be a σ-…nite measure space. A bounded subset
B L1 (E , m ) is relatively weakly compact if
sup
f 2B
Z
A
jf j dm ! 0 as m(A) ! 0
and (if m (E ) = ∞) there exists measurable sets En
, m (En ) < +∞, En En +1 , [En = E such that
sup
f 2B
Z
E nc
E
jf j dm ! 0 as n ! ∞.
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Chapter 4
79 / 118
Reminder on weakly compact sets
Let (E , m ) be a σ-…nite measure space. A bounded subset
B L1 (E , m ) is relatively weakly compact if
sup
f 2B
Z
A
jf j dm ! 0 as m(A) ! 0
and (if m (E ) = ∞) there exists measurable sets En
, m (En ) < +∞, En En +1 , [En = E such that
sup
f 2B
Z
E nc
E
jf j dm ! 0 as n ! ∞.
A bounded subset B L1 (E , m ) is relatively weakly compact if and
only if B is relatively sequentially weakly compact.
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Chapter 4
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On weakly compact operators
A bounded operator G : L1 (E , m ) ! L1 (E , m ) is weakly compact if
G sends a bounded set into a relatively weakly compact one.
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Chapter 4
80 / 118
On weakly compact operators
A bounded operator G : L1 (E , m ) ! L1 (E , m ) is weakly compact if
G sends a bounded set into a relatively weakly compact one.
If Gi : L1 (E , m ) ! L1 (E , m ) (i = 1, 2) are positive operators and
G1 f
G2 f 8f 2 L1+ (E , m ) then G1 is weakly compact if G2 is.
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Chapter 4
80 / 118
On weakly compact operators
A bounded operator G : L1 (E , m ) ! L1 (E , m ) is weakly compact if
G sends a bounded set into a relatively weakly compact one.
If Gi : L1 (E , m ) ! L1 (E , m ) (i = 1, 2) are positive operators and
G1 f
G2 f 8f 2 L1+ (E , m ) then G1 is weakly compact if G2 is.
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Chapter 4
80 / 118
On weakly compact operators
A bounded operator G : L1 (E , m ) ! L1 (E , m ) is weakly compact if
G sends a bounded set into a relatively weakly compact one.
If Gi : L1 (E , m ) ! L1 (E , m ) (i = 1, 2) are positive operators and
G1 f
G2 f 8f 2 L1+ (E , m ) then G1 is weakly compact if G2 is. (This
follows easily from the above criterion of weak compactness.)
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Chapter 4
80 / 118
On weakly compact operators
A bounded operator G : L1 (E , m ) ! L1 (E , m ) is weakly compact if
G sends a bounded set into a relatively weakly compact one.
If Gi : L1 (E , m ) ! L1 (E , m ) (i = 1, 2) are positive operators and
G1 f
G2 f 8f 2 L1+ (E , m ) then G1 is weakly compact if G2 is. (This
follows easily from the above criterion of weak compactness.)
If Gi : L1 (E , m ) ! L1 (E , m ) (i = 1, 2) are two weakly compact
operators then G1 G2 is a compact operator.
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Chapter 4
80 / 118
On weakly compact operators
A bounded operator G : L1 (E , m ) ! L1 (E , m ) is weakly compact if
G sends a bounded set into a relatively weakly compact one.
If Gi : L1 (E , m ) ! L1 (E , m ) (i = 1, 2) are positive operators and
G1 f
G2 f 8f 2 L1+ (E , m ) then G1 is weakly compact if G2 is. (This
follows easily from the above criterion of weak compactness.)
If Gi : L1 (E , m ) ! L1 (E , m ) (i = 1, 2) are two weakly compact
operators then G1 G2 is a compact operator.
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Chapter 4
80 / 118
On weakly compact operators
A bounded operator G : L1 (E , m ) ! L1 (E , m ) is weakly compact if
G sends a bounded set into a relatively weakly compact one.
If Gi : L1 (E , m ) ! L1 (E , m ) (i = 1, 2) are positive operators and
G1 f
G2 f 8f 2 L1+ (E , m ) then G1 is weakly compact if G2 is. (This
follows easily from the above criterion of weak compactness.)
If Gi : L1 (E , m ) ! L1 (E , m ) (i = 1, 2) are two weakly compact
operators then G1 G2 is a compact operator. See N. Dunford and J.
T. Schwartz, Linear operators, Part I, Interscience Pub, 1958.
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Chapter 4
80 / 118
Weak compactness is a fundamental tool for spectral theory of
neutron transport in L1 spaces!
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Chapter 4
81 / 118
We are going to present (weak-)compactness results for neutron
transport operators
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Chapter 4
82 / 118
We are going to present (weak-)compactness results for neutron
transport operators and show their spectral consequences in general
cases.
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Chapter 4
82 / 118
We are going to present (weak-)compactness results for neutron
transport operators and show their spectral consequences in general
cases.
We give an overview of : M. M-K. Di¤. Int. Eq, 18(11) (2005)
1221-1242 ).
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Chapter 4
82 / 118
We treat …rst model cases where
σ(x, v ) = 0, µ is …nite and k (x, v , v 0 ) = 1.
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Chapter 4
83 / 118
Negative results
Theorem
Let Ω Rn be an open set with …nite volume and µ an arbitrary …nite
positive Borel measure on Rn with support V . Let n > 2 (or n = 1 and
0 2 V ). Let
K : ϕ 2 L (Ω
1
V) !
Z
V
ϕ(x, v )µ(dv ) 2 L1 (Ω).
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Chapter 4
84 / 118
Negative results
Theorem
Let Ω Rn be an open set with …nite volume and µ an arbitrary …nite
positive Borel measure on Rn with support V . Let n > 2 (or n = 1 and
0 2 V ). Let
K : ϕ 2 L (Ω
1
Then K (λ
T)
1
V) !
Z
V
ϕ(x, v )µ(dv ) 2 L1 (Ω).
is not weakly compact.
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Chapter 4
84 / 118
Proof
We can assume without loss of generality that 0 2 Ω. Consider just the
case n > 2.
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Chapter 4
85 / 118
Proof
We can assume without loss of generality that 0 2 Ω. Consider just the
case n > 2. Let (fj )j
Cc (Ω V ) a sequence normalized in L1 (Ω V )
converging in the weak star topology of measures to the Dirac mass δ(0,v ) .
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Chapter 4
85 / 118
Proof
We can assume without loss of generality that 0 2 Ω. Consider just the
case n > 2. Let (fj )j
Cc (Ω V ) a sequence normalized in L1 (Ω V )
converging in the weak star topology of measures to the Dirac mass δ(0,v ) .
Then for any ψ 2 Cc (Ω)
hK ( λ
T)
1
fj , ψ i
=
=
!
Z
Ω
ψ(x )dx
Z Z
Ω
V
µ(dv )
V
fj ( y , v )
Z s (0, v )
0
Z
e
λt
Z
Z s (x ,v )
e
0
s (y , v )
e
0
λt
λt
fj ( x
tv , v )dt
ψ(y + tv )dt dy µ(dv
ψ(tv )dt
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Chapter 4
85 / 118
i.e. K (λ T ) 1 fj tends, in the weak star topology of measures, to a
(non-trivial) Radon measure
m : ψ 2 Cc (Ω) !
Z s (0, v )
0
e
λt
ψ(tv )dt
with support included in a segment.
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Chapter 4
86 / 118
i.e. K (λ T ) 1 fj tends, in the weak star topology of measures, to a
(non-trivial) Radon measure
m : ψ 2 Cc (Ω) !
Z s (0, v )
0
e
λt
ψ(tv )dt
/ L1 (Ω) and
with support included in a segment. Hence m 2
1
K (λ T ) is not weakly compact.
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Chapter 4
86 / 118
i.e. K (λ T ) 1 fj tends, in the weak star topology of measures, to a
(non-trivial) Radon measure
m : ψ 2 Cc (Ω) !
Z s (0, v )
0
e
λt
ψ(tv )dt
/ L1 (Ω) and
with support included in a segment. Hence m 2
1
K (λ T ) is not weakly compact.
Remark: This property was noted for the …rst time (in the whole space) in
(F. Golse, P.L. Lions, B. Perthame and R. Sentis. J. Funct. Anal, 76
(1988) 110-125.)
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Chapter 4
86 / 118
By using similar techniques, we can show (see the details in (M. M-K, Di¤.
Int. Eq, 18(11) (2005) 1221-1242 ):
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Chapter 4
87 / 118
By using similar techniques, we can show (see the details in (M. M-K, Di¤.
Int. Eq, 18(11) (2005) 1221-1242 ):
Theorem
Let n > 3 and let Ω
Rn be an open set with …nite volume. Let µ be an
arbitrary …nite positive Borel measure on Rn with support V and
K : ϕ 2 L (Ω
1
V) !
Z
V
ϕ(x, v )µ(dv ) 2 L1 (Ω).
Then
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Chapter 4
87 / 118
By using similar techniques, we can show (see the details in (M. M-K, Di¤.
Int. Eq, 18(11) (2005) 1221-1242 ):
Theorem
Let n > 3 and let Ω
Rn be an open set with …nite volume. Let µ be an
arbitrary …nite positive Borel measure on Rn with support V and
K : ϕ 2 L (Ω
1
Then
(i) (λ
T
K)
1
(λ
V) !
T)
1
Z
V
ϕ(x, v )µ(dv ) 2 L1 (Ω).
is not weakly compact.
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Chapter 4
87 / 118
By using similar techniques, we can show (see the details in (M. M-K, Di¤.
Int. Eq, 18(11) (2005) 1221-1242 ):
Theorem
Let n > 3 and let Ω
Rn be an open set with …nite volume. Let µ be an
arbitrary …nite positive Borel measure on Rn with support V and
K : ϕ 2 L (Ω
1
V) !
Z
V
ϕ(x, v )µ(dv ) 2 L1 (Ω).
Then
(i) (λ T K ) 1 (λ T ) 1 is not weakly compact.
(ii) V (t ) U (t ) is not weakly compact.
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Chapter 4
87 / 118
A necessary assumption
Theorem
We assume that the velocity measure is invariant under the symmetry
v ! v . Let there exist m 2 N such that
K (λ
is compact on L1 (Ω
T)
1 m
V ).
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Chapter 4
88 / 118
A necessary assumption
Theorem
We assume that the velocity measure is invariant under the symmetry
v ! v . Let there exist m 2 N such that
K (λ
is compact on L1 (Ω
µ-measure.
T)
1 m
V ). Then the hyperplanes (through zero) have zero
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Chapter 4
88 / 118
A necessary assumption
Theorem
We assume that the velocity measure is invariant under the symmetry
v ! v . Let there exist m 2 N such that
K (λ
is compact on L1 (Ω
µ-measure.
T)
1 m
V ). Then the hyperplanes (through zero) have zero
See M. M-K, Di¤. Int. Eq, 18(11) (2005) 1221-1242.
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Chapter 4
88 / 118
Domination argument
For any m 2 N
K (λ
T)
1
K
m
R K (λ
T∞ )
1
K
m
E
where
E : L1 (Ω
V ) ! L1 (Rn
V)
V ) ! L1 (Ω
V)
is the trivial extension operator,
R : L1 (Rn
is the restriction operator and T∞ is the generator of
U∞ (t ) : ψ 2 L1 (Rn
V ) ! ψ (x
tv , v ) 2 L1 (Rn
University of FrancheComté Besançon France (Institute)
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V ).
Chapter 4
89 / 118
A fundamental observation
Lemma
Let µ be an arbitrary …nite positive measure on Rn with support V and
K : ϕ 2 L1 (Ω
V) !
Z
V
ϕ(x, v )µ(dv ) 2 L1 (Ω).
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Chapter 4
90 / 118
A fundamental observation
Lemma
Let µ be an arbitrary …nite positive measure on Rn with support V and
K : ϕ 2 L1 (Ω
V) !
Z
V
ϕ(x, v )µ(dv ) 2 L1 (Ω).
Then there exists a …nite positive measure β on Rn such that for any
λ>0
K (λ T∞ ) 1 K ϕ = β K ϕ.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
90 / 118
A fundamental observation
Lemma
Let µ be an arbitrary …nite positive measure on Rn with support V and
K : ϕ 2 L1 (Ω
V) !
Z
V
ϕ(x, v )µ(dv ) 2 L1 (Ω).
Then there exists a …nite positive measure β on Rn such that for any
λ>0
K (λ T∞ ) 1 K ϕ = β K ϕ.
Moreover,
b
β(ζ ) =
1
n
(2π ) 2
Z
Rn
µ(dv )
.
λ + i ζ.v
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
90 / 118
Proof
K (λ
T∞ )
1
Kϕ =
=
Z
n
ZR∞
µ(dv )
e
λt
e
λt
e
λt
Z ∞
dt
0
=
Z ∞
0
=
Z ∞
0
dt
e
λt
0
Z
Z
[ µt
Rn
Rn
(K ϕ)(x
tv )dt
(K ϕ)(x
tv )µ(dv )
(K ϕ)(x
z )µt (dz )
K ϕ] dt
where µt is the image of µ under the dilation v ! tv .
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
91 / 118
So
K (λ
T∞ )
where
β :=
1
Z ∞
0
Kϕ = β Kϕ
e
λt
µt dt
(strong integral).
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
92 / 118
So
K (λ
T∞ )
where
β :=
1
Z ∞
Kϕ = β Kϕ
e
λt
0
(strong integral). Finally b
β(ζ ) is given by
Z ∞
0
e
λt
µbt (ζ )dt =
=
=
Z
µt dt
Z
∞
1
e λt
e
n
(2π ) 2 0
Rn
Z ∞
Z
1
λt
e
e
n
(2π ) 2 0
Rn
Z
1
µ(dv )
.
n
n
2
λ
(2π ) R + i ζ.v
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
i ζ.v
µt (dv ) dt
it ζ.v
µ(dv ) dt
Chapter 4
92 / 118
So
K (λ
T∞ )
where
β :=
1
Z ∞
Kϕ = β Kϕ
e
λt
0
(strong integral). Finally b
β(ζ ) is given by
Z ∞
0
e
λt
µbt (ζ )dt =
=
=
Z
µt dt
Z
∞
1
e λt
e
n
(2π ) 2 0
Rn
Z ∞
Z
1
λt
e
e
n
(2π ) 2 0
Rn
Z
1
µ(dv )
.
n
n
2
λ
(2π ) R + i ζ.v
i ζ.v
µt (dv ) dt
it ζ.v
µ(dv ) dt
This ends the proof.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
92 / 118
A remark
b
β(ζ ) =
Z
Rn
µ(dv )
! 0 for jζ j ! ∞
λ + i ζ.v
if and only if the hyperplanes (through zero) have zero µ-measure;
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
93 / 118
A remark
b
β(ζ ) =
Z
Rn
µ(dv )
! 0 for jζ j ! ∞
λ + i ζ.v
if and only if the hyperplanes (through zero) have zero µ-measure; see M.
M-K, Di¤. Int. Eq, 18(11) (2005) 1221-1242.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
93 / 118
Smoothing e¤ects of iterated convolution of measures
We are going to show that the compactness results rely on how fast
b
β(ζ ) ! 0 as jζ j ! ∞ ?
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
94 / 118
Smoothing e¤ects of iterated convolution of measures
We are going to show that the compactness results rely on how fast
b
β(ζ ) ! 0 as jζ j ! ∞ ?
Theorem
We assume that Ω Rn is an open set with …nite volume. Let the
velocity measure µ be …nite and such that
Z
for some m 2 N.
Rn
b
β(ζ )
2m
d ζ < +∞
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
94 / 118
Smoothing e¤ects of iterated convolution of measures
We are going to show that the compactness results rely on how fast
b
β(ζ ) ! 0 as jζ j ! ∞ ?
Theorem
We assume that Ω Rn is an open set with …nite volume. Let the
velocity measure µ be …nite and such that
Z
Rn
for some m 2 N. Then K (λ
L1 (Ω V )
b
β(ζ )
T)
2m
d ζ < +∞
1K m
is weakly compact in
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
94 / 118
Smoothing e¤ects of iterated convolution of measures
We are going to show that the compactness results rely on how fast
b
β(ζ ) ! 0 as jζ j ! ∞ ?
Theorem
We assume that Ω Rn is an open set with …nite volume. Let the
velocity measure µ be …nite and such that
Z
Rn
b
β(ζ )
2m
d ζ < +∞
m
is weakly compact in
for some m 2 N. Then K (λ T ) 1 K
m +1
1
1
is compact in L1 (Ω V ).
L (Ω V ) and K (λ T ) K
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
94 / 118
Proof
We start from
K (λ
T∞ )
1
K
m
ϕ = β (m )
ϕ
where
β(m ) = β .... β (m times).
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
95 / 118
Proof
We start from
K (λ
T∞ )
1
K
m
ϕ = β (m )
ϕ
where
β(m ) = β .... β (m times).
Since
d
β (m ) ( ζ ) = b
β(ζ )
m
then our assumption amounts to
d
β(m ) 2 L2 (Rn ).
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
95 / 118
In particular β(m ) 2 L2 (Rn ) (β(m ) is now a function !).
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
96 / 118
In particular β(m ) 2 L2 (Rn ) (β(m ) is now a function !). It follows that
K (λ
and then K (λ
L2 (Rn ).
T∞ )
T∞ )
1K m
1
K
m
ϕ = β (m )
ϕ 2 L2 (Rn )
maps continuously L1 (Rn
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
V ) into
Chapter 4
96 / 118
In particular β(m ) 2 L2 (Rn ) (β(m ) is now a function !). It follows that
K (λ
T∞ )
T∞ )
1K m
R K (λ
T∞ )
and then K (λ
L2 (Rn ). Hence
1
K
m
ϕ = β (m )
ϕ 2 L2 (Rn )
maps continuously L1 (Rn
1
K
m
: L1 (Rn
V ) into
V ) ! L1 (Ω)
is weakly compact because the imbedding of L2 (Ω) into L1 (Ω) is weakly
compact since Ω has …nite volume (a bounded subset of L2 (Ω) is
equi-integrable).
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
96 / 118
In particular β(m ) 2 L2 (Rn ) (β(m ) is now a function !). It follows that
K (λ
T∞ )
T∞ )
1K m
R K (λ
T∞ )
and then K (λ
L2 (Rn ). Hence
1
K
m
ϕ = β (m )
ϕ 2 L2 (Rn )
maps continuously L1 (Rn
1
K
m
: L1 (Rn
V ) into
V ) ! L1 (Ω)
is weakly compact because the imbedding of L2 (Ω) into L1 (Ω) is weakly
compact since Ω has …nite volume (a bounded subset of L2 (Ω) is
m
equi-integrable). Finally K (λ T ) 1 K
is also weakly compact by
domination.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
96 / 118
In particular β(m ) 2 L2 (Rn ) (β(m ) is now a function !). It follows that
K (λ
T∞ )
T∞ )
1K m
R K (λ
T∞ )
and then K (λ
L2 (Rn ). Hence
1
K
m
ϕ = β (m )
ϕ 2 L2 (Rn )
maps continuously L1 (Rn
1
K
m
: L1 (Rn
V ) into
V ) ! L1 (Ω)
is weakly compact because the imbedding of L2 (Ω) into L1 (Ω) is weakly
compact since Ω has …nite volume (a bounded subset of L2 (Ω) is
m
equi-integrable). Finally K (λ T ) 1 K
is also weakly compact by
2m
1
is compact as a product
domination. It follows that K (λ T ) K
of two weakly compact operators.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
96 / 118
In particular β(m ) 2 L2 (Rn ) (β(m ) is now a function !). It follows that
K (λ
1
T∞ )
T∞ )
1K m
R K (λ
T∞ )
and then K (λ
L2 (Rn ). Hence
m
K
ϕ = β (m )
ϕ 2 L2 (Rn )
maps continuously L1 (Rn
1
K
m
: L1 (Rn
V ) into
V ) ! L1 (Ω)
is weakly compact because the imbedding of L2 (Ω) into L1 (Ω) is weakly
compact since Ω has …nite volume (a bounded subset of L2 (Ω) is
m
equi-integrable). Finally K (λ T ) 1 K
is also weakly compact by
2m
1
is compact as a product
domination. It follows that K (λ T ) K
of two weakly compact operators. Actually
K (λ
T)
1
K
m +1
is compact because K (λ
below.
= K (λ
T)
1
T)
1
K K (λ
T)
1
K
m
is a Dunford-Pettis operator, see
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
96 / 118
A practical criterion
We give now a geometrical condition on µ implying our compactness
results.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
97 / 118
A practical criterion
We give now a geometrical condition on µ implying our compactness
results.
Theorem
Let Ω Rn be an open set with …nite volume. Let the velocity measure µ
be …nite and there exist α > 0, c > 0 such that
sup µ fv ; jv .e j
e 2S n
1
εg
cεα .
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
97 / 118
A practical criterion
We give now a geometrical condition on µ implying our compactness
results.
Theorem
Let Ω Rn be an open set with …nite volume. Let the velocity measure µ
be …nite and there exist α > 0, c > 0 such that
sup µ fv ; jv .e j
e 2S n
Then K (λ
T)
1 K m +1
1
εg
cεα .
is compact in L1 (Ω
V ) for m >
n ( α +1 )
2α .
Warning: The condition on m does not depend on the constant c above
!
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
97 / 118
Proof
Note that b
β(ζ ) is a continuous function. According to the preceeding
2m
at in…nity.
β(ζ )
theorem, we need just check the integrability of b
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
98 / 118
Proof
Note that b
β(ζ ) is a continuous function. According to the preceeding
2m
at in…nity. Up to
β(ζ )
theorem, we need just check the integrability of b
a factor (2π )
n
2
b
β(ζ ) =
where e =
Z
Rn
µ(dv )
λ + i ζ.v
Z
Rn
q
µ(dv )
λ2 + jζ j2 je.v j2
ζ
.
jζ j
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
98 / 118
Proof
Note that b
β(ζ ) is a continuous function. According to the preceeding
2m
at in…nity. Up to
β(ζ )
theorem, we need just check the integrability of b
a factor (2π )
n
2
b
β(ζ ) =
where e =
b
β(ζ )
ζ
.
jζ j
Z
Rn
Z
µ(dv )
λ + i ζ.v
Rn
q
µ(dv )
λ2 + jζ j2 je.v j2
So, for any ε > 0,
Z
fje.v j<εg
λ
1
q
µ(dv )
λ2 + jζ j2 je.v j2
µ fje.v j < εg +
kµk
jζ j ε
+
λ
Z
fje.v j>εg
1
cεα +
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
q
µ(dv )
λ2 + jζ j2 je.v j2
kµk
.
jζ j ε
Chapter 4
98 / 118
We choose ε such that λ
1
cεα =
ε=
λc
kµk
jζ jε
1
i.e.
kµk
jζ j
1
α +1
.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
99 / 118
We choose ε such that λ
1
kµk
jζ jε
cεα =
ε=
1
λc
i.e.
1
α +1
kµk
jζ j
.
Then
λ
1
α
1
λc
kµk
jζ j
cε = λ
c
λc
1
and
b
β(ζ )
2λ
1
c
1
α
α +1
=
α
α +1
kµk
jζ j
2λ
1
c λc
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
1
α
j ζ j α +1
kµk
α
α +1
.
Chapter 4
99 / 118
We choose ε such that λ
1
kµk
jζ jε
cεα =
ε=
1
λc
i.e.
1
α +1
kµk
jζ j
.
Then
λ
1
α
1
λc
kµk
jζ j
cε = λ
c
λc
1
and
Hence
b
β(ζ )
2λ
1
c
1
b
β(ζ )
2m
α
α +1
=
α
α +1
kµk
jζ j
2λ
C
2mα
1
c λc
1
α
j ζ j α +1
kµk
α
α +1
.
.
j ζ j α +1
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
99 / 118
We choose ε such that λ
1
kµk
jζ jε
cεα =
ε=
1
λc
i.e.
1
α +1
kµk
jζ j
.
Then
λ
1
α
1
λc
kµk
jζ j
cε = λ
c
λc
1
and
Hence
b
β(ζ )
We are done if
2λ
2mα
α +1
1
c
1
b
β(ζ )
2m
> n i.e. if m >
α
α +1
=
α
α +1
kµk
jζ j
2λ
C
2mα
1
c λc
1
α
j ζ j α +1
kµk
α
α +1
.
.
j ζ j α +1
n ( α +1 )
2α .
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
99 / 118
On remainder terms of Dyson-Phillips expansions
In the same spirit (but with more involved estimates) we can show:
Theorem
Let Ω Rn be an open set with …nite volume. Let the velocity measure µ
be …nite and there exist α > 0, c > 0 such that
sup µ
µ (v , v 0 ); (v
v 0 ).e
ε
cεα .
e 2S n 1
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
100 / 118
On remainder terms of Dyson-Phillips expansions
In the same spirit (but with more involved estimates) we can show:
Theorem
Let Ω Rn be an open set with …nite volume. Let the velocity measure µ
be …nite and there exist α > 0, c > 0 such that
sup µ
µ (v , v 0 ); (v
v 0 ).e
ε
cεα .
e 2S n 1
Then Um (t ) is weakly compact in L1 (Ω V ) for all t > 0 and for
n ( α +1 )
m > m0 where m0 is the smallest odd integer greater than 2α + 1.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
100 / 118
On remainder terms of Dyson-Phillips expansions
In the same spirit (but with more involved estimates) we can show:
Theorem
Let Ω Rn be an open set with …nite volume. Let the velocity measure µ
be …nite and there exist α > 0, c > 0 such that
sup µ
µ (v , v 0 ); (v
v 0 ).e
ε
cεα .
e 2S n 1
Then Um (t ) is weakly compact in L1 (Ω V ) for all t > 0 and for
n ( α +1 )
m > m0 where m0 is the smallest odd integer greater than 2α + 1.
see M. M-K, Di¤. Int. Eq, 18(11) (2005) 1221-1242.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
100 / 118
On remainder terms of Dyson-Phillips expansions
In the same spirit (but with more involved estimates) we can show:
Theorem
Let Ω Rn be an open set with …nite volume. Let the velocity measure µ
be …nite and there exist α > 0, c > 0 such that
sup µ
µ (v , v 0 ); (v
v 0 ).e
ε
cεα .
e 2S n 1
Then Um (t ) is weakly compact in L1 (Ω V ) for all t > 0 and for
n ( α +1 )
m > m0 where m0 is the smallest odd integer greater than 2α + 1.
see M. M-K, Di¤. Int. Eq, 18(11) (2005) 1221-1242.
Warning: The condition on m does not depend on the constant c above
!
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
100 / 118
The general case
We show now how to treat by approximation more general velocity
measures and scattering kernels.
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CIMPA School Muizemberg July 22-Aug 4
Chapter 4
101 / 118
Regular scattering operators in L1 spaces
De…nition
A scattering operator K in L1 (Ω
V ) is regular if
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
102 / 118
Regular scattering operators in L1 spaces
De…nition
A scattering operator K in L1 (Ω V ) is regular if
fK (x ); x 2 Ωg is a set of collectively weakly compact operators on
L1 (V ), i.e. the set
n
o
K (x ) ϕ; x 2 Ω, k ϕkL1 (V ) 1
is relatively weakly compact in L1 (V ).
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
102 / 118
Regular scattering operators in L1 spaces
De…nition
A scattering operator K in L1 (Ω V ) is regular if
fK (x ); x 2 Ωg is a set of collectively weakly compact operators on
L1 (V ), i.e. the set
n
o
K (x ) ϕ; x 2 Ω, k ϕkL1 (V ) 1
is relatively weakly compact in L1 (V ).
Remark: This class appears (with Lebesgue measure dv on Rn ) in P.
Takak (Transp Theory Stat Phys, 14(5) (1985) 655-667) and L. W. Weis
(J. Math. Anal. Appl, 129 (1988) 6-23). See B. Lods (Math. Meth.
Appl. Sci. 27 (2004) 1049–1075) for the extension of P. Takak’s
construction to abstract velocity measures µ.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
102 / 118
An approximation property
The following result is given in (B. Lods, Math. Meth. Appl. Sci. 27
(2004) 1049–1075).
Theorem
Let K be a regular scattering operator in L1 (Ω V ). Then there exists a
sequence (Kj )j of scattering operators such that:
(i) 0 Kj
K
(ii) kK Kj kL(L1 (Ω V )) ! 0 as j ! +∞.
(iii) For each Kj there exists fj 2 L1 (V ) such that
Kj ϕ
fj ( v )
Z
ϕ(x, v 0 )µ(dv 0 ) 8 ϕ 2 L1+ (Ω
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
V ).
Chapter 4
103 / 118
General spectral results in L1
Theorem
Let Ω Rn be an open set with …nite volume and let K be a regular
scattering operator in L1 (Ω V ). We assume that the velocity measure µ
is such that: There exists α > 0 such that for any c1 > 0 there
exists c2 > 0 such that
sup µ fv ; jv j
e 2S n
1
c1 , jv .e j
εg
c2 ε α .
Then the components of ρe (T ) and ρe (T + K ) containing a right
half-plane coincide.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
104 / 118
General spectral results in L1
Theorem
Let Ω Rn be an open set with …nite volume and let K be a regular
scattering operator in L1 (Ω V ). We assume that the velocity measure µ
is such that: There exists α > 0 such that for any c1 > 0 there
exists c2 > 0 such that
sup µ fv ; jv j
e 2S n
1
c1 , jv .e j
εg
c2 ε α .
Then the components of ρe (T ) and ρe (T + K ) containing a right
half-plane coincide. In particular
σ(T + K ) \ fRe λ > s (T )g
consists at most of isolated eigenvalues with …nite algebraic multiplicities
where
Z
1 t
s (T ) =
lim inf
σ(y + τv , v )d τ.
t !+∞ (y ,v ) t 0
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
104 / 118
Proof
Let us show that K (λ
for m >
Kj ( λ
T)
1 K m +1
is weakly compact in L1 (Ω
n ( α +1 )
n ( α +1 )
2α . We …x m >
2α . It su¢ ces to
m +1
1
T ) Kj
is weakly compact in L1 (Ω
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
V)
show that
V ) for all j.
Chapter 4
105 / 118
Proof
Let us show that K (λ
T)
1 K m +1
is weakly compact in L1 (Ω
n ( α +1 )
n ( α +1 )
2α . We …x m >
2α . It su¢ ces to
m +1
1
T ) Kj
is weakly compact in L1 (Ω
for m >
Kj ( λ
domination, we may replace Kj by Kbj where
Kbj ϕ = fj (v )
Z
V)
show that
V ) for all j. By
ϕ(x, v 0 )µ(dv 0 ).
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
105 / 118
Proof
Let us show that K (λ
T)
1 K m +1
is weakly compact in L1 (Ω
n ( α +1 )
n ( α +1 )
2α . We …x m >
2α . It su¢ ces to
m +1
1
T ) Kj
is weakly compact in L1 (Ω
for m >
Kj ( λ
domination, we may replace Kj by Kbj where
Kbj ϕ = fj (v )
Z
V)
show that
V ) for all j. By
ϕ(x, v 0 )µ(dv 0 ).
By approximation again we may suppose that fj is continuous with
compact support.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
105 / 118
Proof
Let us show that K (λ
T)
1 K m +1
is weakly compact in L1 (Ω
n ( α +1 )
n ( α +1 )
2α . We …x m >
2α . It su¢ ces to
m +1
1
T ) Kj
is weakly compact in L1 (Ω
for m >
Kj ( λ
domination, we may replace Kj by Kbj where
Kbj ϕ = fj (v )
Z
V)
show that
V ) for all j. By
ϕ(x, v 0 )µ(dv 0 ).
By approximation again we may suppose that fj is continuous with
Let j be i…xed and denote by Vj the support of fj . We
compact support.
h
m
note that Kbj (λ T ) 1 Kbj
leaves invariant L1 (Ω Vj ) and Kbj maps
L1 (Ω V ) into L1 (Ω Vj ). By replacing fj (v ) by its supremum, the
model case (dealt with previously) in L1 (Ω Vj ) insures that
m
Kj ( λ T ) 1 Kj
is weakly compact on L1 (Ω Vj ) so
Kj ( λ
T)
1
Kj
m +1
L1 (Ω
= Kj ( λ
T)
1
Kj
m
Kj ( λ
T)
1
Kj
is weakly compact on
V ). Finally some power of K (λ T ) 1 is
compact on L1 (Ω V ) CIMPA
and School
we conclude
by the general theory.
University of FrancheComté Besançon France (Institute)
Muizemberg July 22-Aug 4
Chapter 4
105 / 118
Theorem
Let Ω Rn be an open set with …nite volume and let K be a regular
scattering operator in L1 (Ω V ). We assume that the velocity measure µ
is such that: There exists α > 0 such that for any c1 > 0 there
exists c2 > 0 such that
sup µ
e 2S n
1
µ (v , v 0 ); jv j , v 0
c1 , ( v
v 0 ).e
ε
c2 ε α .
Then (V (t ))t >0 and (U (t ))t >0 have the same essential type. In particular
n
o
σ(V (t )) \ α 2 C; jαj > e s (T )t
consists at most of isolated eigenvalues with …nite algebraic multiplicities.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
106 / 118
Proof
Let us show that Rm +1 (t ) is weakly compact in L1 (Ω V ) for all t > 0
and for m > m0 where m0 is the smallest odd integer greater
n ( α +1 )
than 2α + 1.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
107 / 118
Proof
Let us show that Rm +1 (t ) is weakly compact in L1 (Ω V ) for all t > 0
and for m > m0 where m0 is the smallest odd integer greater
n ( α +1 )
than 2α + 1. We …x m > m0 .
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
107 / 118
Proof
Let us show that Rm +1 (t ) is weakly compact in L1 (Ω V ) for all t > 0
and for m > m0 where m0 is the smallest odd integer greater
n ( α +1 )
than 2α + 1. We …x m > m0 . It su¢ ces to show that Rmj +1 (t ) is
weakly compact in L1 (Ω V ) for all j where Rmj +1 (t ) is the remainder
term of order m + 1 corresponding to a perturbation Kj in place of K
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
107 / 118
Proof
Let us show that Rm +1 (t ) is weakly compact in L1 (Ω V ) for all t > 0
and for m > m0 where m0 is the smallest odd integer greater
n ( α +1 )
than 2α + 1. We …x m > m0 . It su¢ ces to show that Rmj +1 (t ) is
weakly compact in L1 (Ω V ) for all j where Rmj +1 (t ) is the remainder
term of order m + 1 corresponding to a perturbation Kj in place of K By
domination, we may replace Kj by Kbj where
Kbj ϕ = fj (v )
Z
ϕ(x, v 0 )µ(dv 0 ).
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
107 / 118
Proof
Let us show that Rm +1 (t ) is weakly compact in L1 (Ω V ) for all t > 0
and for m > m0 where m0 is the smallest odd integer greater
n ( α +1 )
than 2α + 1. We …x m > m0 . It su¢ ces to show that Rmj +1 (t ) is
weakly compact in L1 (Ω V ) for all j where Rmj +1 (t ) is the remainder
term of order m + 1 corresponding to a perturbation Kj in place of K By
domination, we may replace Kj by Kbj where
Kbj ϕ = fj (v )
Z
ϕ(x, v 0 )µ(dv 0 ).
By approximation again we may suppose that fj is continuous with
compact support.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
107 / 118
Proof
Let us show that Rm +1 (t ) is weakly compact in L1 (Ω V ) for all t > 0
and for m > m0 where m0 is the smallest odd integer greater
n ( α +1 )
than 2α + 1. We …x m > m0 . It su¢ ces to show that Rmj +1 (t ) is
weakly compact in L1 (Ω V ) for all j where Rmj +1 (t ) is the remainder
term of order m + 1 corresponding to a perturbation Kj in place of K By
domination, we may replace Kj by Kbj where
Kbj ϕ = fj (v )
Z
ϕ(x, v 0 )µ(dv 0 ).
By approximation again we may suppose that fj is continuous with
compact support. Let j be …xed and denote by Vj the support of fj .
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
107 / 118
We note that Rmj (t ) leaves invariant L1 (Ω
into L1 (Ω Vj ).
Vj ) and Kbj maps L1 (Ω
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
V)
108 / 118
We note that Rmj (t ) leaves invariant L1 (Ω
into L1 (Ω Vj ). Note also that
Rmj +1 = Rmj
h
Vj ) and Kbj maps L1 (Ω
i Z t
Kbj U =
Rmj (t
0
V)
s )Kbj U (s )ds.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
108 / 118
We note that Rmj (t ) leaves invariant L1 (Ω
into L1 (Ω Vj ). Note also that
Rmj +1 = Rmj
h
Vj ) and Kbj maps L1 (Ω
i Z t
Kbj U =
Rmj (t
0
V)
s )Kbj U (s )ds.
By dominating fj by its supremum, the previous model case (and a
domination argument) shows that Rmj (t s ) is weakly compact in
L1 (Ω Vj ).
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
108 / 118
We note that Rmj (t ) leaves invariant L1 (Ω
into L1 (Ω Vj ). Note also that
Rmj +1 = Rmj
h
Vj ) and Kbj maps L1 (Ω
i Z t
Kbj U =
Rmj (t
0
V)
s )Kbj U (s )ds.
By dominating fj by its supremum, the previous model case (and a
domination argument) shows that Rmj (t s ) is weakly compact in
L1 (Ω Vj ). Thus Rmj (t s )Kbj U (s ) is weakly compact in L1 (Ω V ) and
then so is Rmj +1 as a strong integral of "weakly compact operator" valued
mapping. We conclude by the general theory.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
108 / 118
Dunford-Pettis operators in transport theory
G 2 L(L1 (ν)) is called a Dunford-Pettis (or a completely
continuous) operator if G maps a weakly compact subset into a
(norm) compact subset.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
109 / 118
Dunford-Pettis operators in transport theory
G 2 L(L1 (ν)) is called a Dunford-Pettis (or a completely
continuous) operator if G maps a weakly compact subset into a
(norm) compact subset.
Exemple: Weakly compact operators on L1 (ν) are Dunford-Pettis.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
109 / 118
Dunford-Pettis operators in transport theory
G 2 L(L1 (ν)) is called a Dunford-Pettis (or a completely
continuous) operator if G maps a weakly compact subset into a
(norm) compact subset.
Exemple: Weakly compact operators on L1 (ν) are Dunford-Pettis.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
109 / 118
Dunford-Pettis operators in transport theory
G 2 L(L1 (ν)) is called a Dunford-Pettis (or a completely
continuous) operator if G maps a weakly compact subset into a
(norm) compact subset.
Exemple: Weakly compact operators on L1 (ν) are Dunford-Pettis.
(This explains why the product of two weakly compact operators on
L1 (ν) is compact).
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
109 / 118
Dunford-Pettis operators in transport theory
G 2 L(L1 (ν)) is called a Dunford-Pettis (or a completely
continuous) operator if G maps a weakly compact subset into a
(norm) compact subset.
Exemple: Weakly compact operators on L1 (ν) are Dunford-Pettis.
(This explains why the product of two weakly compact operators on
L1 (ν) is compact).
If G2 is weakly compact on L1 (ν) and G1 is Dunford-Pettis on
L1 (ν) then G1 G2 is compact.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
109 / 118
We have seen that various relevant operators for neutron transport
are not weakly compact in L1 (Ω V ), for instance:
K (λ
T)
1
, (λ
T
K)
1
(λ
T)
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
1
and V (t )
U (t ).
Chapter 4
110 / 118
We have seen that various relevant operators for neutron transport
are not weakly compact in L1 (Ω V ), for instance:
K (λ
T)
1
, (λ
T
K)
1
(λ
T)
1
and V (t )
U (t ).
We show however that they are all Dunford-Pettis.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
110 / 118
We have seen that various relevant operators for neutron transport
are not weakly compact in L1 (Ω V ), for instance:
K (λ
T)
1
, (λ
T
K)
1
(λ
T)
1
and V (t )
U (t ).
We show however that they are all Dunford-Pettis.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
110 / 118
We have seen that various relevant operators for neutron transport
are not weakly compact in L1 (Ω V ), for instance:
K (λ
T)
1
, (λ
T
K)
1
(λ
T)
1
and V (t )
U (t ).
We show however that they are all Dunford-Pettis. See (M. M-K.
Math Methods. Appl. Sci, 27 (2004) 687-701 for more information.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
110 / 118
We have seen that various relevant operators for neutron transport
are not weakly compact in L1 (Ω V ), for instance:
K (λ
T)
1
, (λ
T
K)
1
(λ
T)
1
and V (t )
U (t ).
We show however that they are all Dunford-Pettis. See (M. M-K.
Math Methods. Appl. Sci, 27 (2004) 687-701 for more information.
This explains why we claimed previously that
K (λ
T)
1
K
m +1
= K (λ
is compact since K (λ T ) 1 K
K (λ T ) 1 is Dunford-Pettis.
m
T)
1
K K (λ
T)
1
K
m
is weakly compact and
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
110 / 118
We restrict ourselves to:
Theorem
Let Ω Rn be an open set with …nite volume and let K be a regular
scattering operator in L1 (Ω V ). If the a¢ ne hyperplanes have zero
µ-measure then V (t ) U (t ) is a Dunford-Pettis operator on L1 (Ω V ).
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
111 / 118
Proof
We note …rst (as for compactness) that V (t ) U (t ) = ∑1∞ Uj (t ) is
Dunford-Pettis for all t > 0 if and only if U1 (t ) is.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
112 / 118
Proof
We note …rst (as for compactness) that V (t ) U (t ) = ∑1∞ Uj (t ) is
Dunford-Pettis for all t > 0 if and only if U1 (t ) is. By approximation, we
may assume that ther exists f continuous with support in fjv j < c g such
that
Z
K ϕ f (v ) ϕ(x, v 0 )µ(dv 0 ) 8 ϕ 2 L1+ (Ω V ).
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
112 / 118
Proof
We note …rst (as for compactness) that V (t ) U (t ) = ∑1∞ Uj (t ) is
Dunford-Pettis for all t > 0 if and only if U1 (t ) is. By approximation, we
may assume that ther exists f continuous with support in fjv j < c g such
that
Z
K ϕ f (v ) ϕ(x, v 0 )µ(dv 0 ) 8 ϕ 2 L1+ (Ω V ).
Let E
L1 (Ω
V ) be relatively weakly compact.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
112 / 118
Proof
We note …rst (as for compactness) that V (t ) U (t ) = ∑1∞ Uj (t ) is
Dunford-Pettis for all t > 0 if and only if U1 (t ) is. By approximation, we
may assume that ther exists f continuous with support in fjv j < c g such
that
Z
K ϕ f (v ) ϕ(x, v 0 )µ(dv 0 ) 8 ϕ 2 L1+ (Ω V ).
Let E
L1 (Ω
sup
ϕ 2E
V ) be relatively weakly compact. In particular
Z
jv j>c
µ(dv )
Z
Ω
j ϕ(x, v )j dx ! 0 as c ! +∞.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
112 / 118
We decompose ϕ 2 E as
ϕ = ϕχfjv j<c g + ϕχfjv j>c g
so
U1 (t )( ϕχfjv j>c g )
kU1 (t )k ϕχfjv j>c g ! 0 as c ! +∞
uniformly in ϕ 2 E .
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
113 / 118
We decompose ϕ 2 E as
ϕ = ϕχfjv j<c g + ϕχfjv j>c g
so
U1 (t )( ϕχfjv j>c g )
kU1 (t )k ϕχfjv j>c g ! 0 as c ! +∞
uniformly in ϕ 2 E . On the other hand ϕχfjv j<c g is zero for jv j > c and
(for c > c) for any ψ 2 L1 (Ω V ), K ψ is zero for jv j > c.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
113 / 118
We decompose ϕ 2 E as
ϕ = ϕχfjv j<c g + ϕχfjv j>c g
so
U1 (t )( ϕχfjv j>c g )
kU1 (t )k ϕχfjv j>c g ! 0 as c ! +∞
uniformly in ϕ 2 E . On the other hand ϕχfjv j<c g is zero for jv j > c and
(for c > c) for any ψ 2 L1 (Ω V ), K ψ is zero for jv j > c. So we may
assume from the beginning that V is bounded
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
113 / 118
We decompose ϕ 2 E as
ϕ = ϕχfjv j<c g + ϕχfjv j>c g
so
U1 (t )( ϕχfjv j>c g )
kU1 (t )k ϕχfjv j>c g ! 0 as c ! +∞
uniformly in ϕ 2 E . On the other hand ϕχfjv j<c g is zero for jv j > c and
(for c > c) for any ψ 2 L1 (Ω V ), K ψ is zero for jv j > c. So we may
assume from the beginning that V is bounded and then K maps also
L2 (Ω V ) into itself.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
113 / 118
We decompose ϕ 2 E as
ϕ = ϕ1α + ϕ2α := ϕχfj ϕj<αg + ϕχfj ϕj>αg
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
114 / 118
We decompose ϕ 2 E as
ϕ = ϕ1α + ϕ2α := ϕχfj ϕj<αg + ϕχfj ϕj>αg
and note that
Z
j ϕj >
so
dx
Z
χfj ϕj>αg
j ϕj > α (dx
µ fj ϕj > αg)
µ fj ϕj > αg ! 0 as α ! +∞.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
114 / 118
We decompose ϕ 2 E as
ϕ = ϕ1α + ϕ2α := ϕχfj ϕj<αg + ϕχfj ϕj>αg
and note that
Z
j ϕj >
so
dx
Z
χfj ϕj>αg
j ϕj > α (dx
µ fj ϕj > αg)
µ fj ϕj > αg ! 0 as α ! +∞.
The equi-integrability of E implies that
ϕ2α
L1
! 0 as α ! +∞
uniformly in ϕ 2 E and …nally U1 (t ) ϕ2α
in ϕ 2 E .
L1
! 0 as c ! +∞ uniformly
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
114 / 118
Since, ϕ1α is bounded in L1 and in L∞ then the interpolation inequality
ϕ1α
shows that
L2
ϕ1α
1
2
L1
ϕ1α is also bounded in L2 (Ω
ϕ1α
1
2
L∞
V ).
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
115 / 118
Since, ϕ1α is bounded in L1 and in L∞ then the interpolation inequality
ϕ1α
L2
ϕ1α
1
2
L1
ϕ1α
1
2
L∞
shows that ϕ1α is also bounded in L2 (Ω V ). We know (from the L2
theory ) that U1 (t ) is compact in L2 (Ω V )
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
115 / 118
Since, ϕ1α is bounded in L1 and in L∞ then the interpolation inequality
ϕ1α
L2
ϕ1α
1
2
L1
ϕ1α
1
2
L∞
shows that ϕ1α is also bounded in L2 (Ω V ). We know (from the L2
theory ) that U1 (t ) is compact in L2 (Ω V ) so U1 (t ) ϕ1α is relatively
compact in L2 (Ω V ) and then relatively compact in L1 (Ω V ) since
Ω V has a …nite measure.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
115 / 118
Since, ϕ1α is bounded in L1 and in L∞ then the interpolation inequality
ϕ1α
L2
ϕ1α
1
2
L1
ϕ1α
1
2
L∞
shows that ϕ1α is also bounded in L2 (Ω V ). We know (from the L2
theory ) that U1 (t ) is compact in L2 (Ω V ) so U1 (t ) ϕ1α is relatively
compact in L2 (Ω V ) and then relatively compact in L1 (Ω V ) since
Ω V has a …nite measure.
Thus fU1 (t ) ϕ; ϕ 2 E g is as close to a relatively compact subset of
L1 (Ω V ) as we want
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
115 / 118
Since, ϕ1α is bounded in L1 and in L∞ then the interpolation inequality
ϕ1α
L2
ϕ1α
1
2
L1
ϕ1α
1
2
L∞
shows that ϕ1α is also bounded in L2 (Ω V ). We know (from the L2
theory ) that U1 (t ) is compact in L2 (Ω V ) so U1 (t ) ϕ1α is relatively
compact in L2 (Ω V ) and then relatively compact in L1 (Ω V ) since
Ω V has a …nite measure.
Thus fU1 (t ) ϕ; ϕ 2 E g is as close to a relatively compact subset of
L1 (Ω V ) as we want and …nally fU1 (t ) ϕ; ϕ 2 E g is relatively
compact.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
115 / 118
On partly elastic scattering operators
Such models were Introduced by Larsen and Zweifel (J. Math. Phys, 15
(1974) 1987-1997).
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
116 / 118
On partly elastic scattering operators
Such models were Introduced by Larsen and Zweifel (J. Math. Phys, 15
(1974) 1987-1997).
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) = Ke f + Kin f
∂t
∂x
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
116 / 118
On partly elastic scattering operators
Such models were Introduced by Larsen and Zweifel (J. Math. Phys, 15
(1974) 1987-1997).
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) = Ke f + Kin f
∂t
∂x
where
Kin f =
Z
R3
k (x, v , v 0 )f (x, v 0 )dv 0 (inelastic operator)
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
116 / 118
On partly elastic scattering operators
Such models were Introduced by Larsen and Zweifel (J. Math. Phys, 15
(1974) 1987-1997).
∂f
∂f
+ v . + σ(x, v )f (t, x, v ) = Ke f + Kin f
∂t
∂x
where
Kin f =
Z
R3
k (x, v , v 0 )f (x, v 0 )dv 0 (inelastic operator)
and
Ke f =
Z
S2
k (x, ρ, ω, ω 0 )f (x, ρω 0 )dS (ω 0 ) (elastic operator)
where v = ρω.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
116 / 118
σ(T + Ke ) consists of a half-plane and various "curves".
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
117 / 118
σ(T + Ke ) consists of a half-plane and various "curves".
Perturbation theory:
A = ( T + Ke ) + Ki
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
117 / 118
σ(T + Ke ) consists of a half-plane and various "curves".
Perturbation theory:
A = ( T + Ke ) + Ki
Compactness results and stability of essential type by M. Sbihi:
Spectral theory of neutron transport semigroups with partly elastic
collision operators. J. Math. Phys, 47 (2006).
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
117 / 118
On the leading eigenvalue of neutron transport
This is a part of peripheral spectral analysis of neutron transport.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
118 / 118
On the leading eigenvalue of neutron transport
This is a part of peripheral spectral analysis of neutron transport. See:
M. M-K: Mathematical Topics in neutron transport theory, World
Scienti…c, Vol 46 (1997), Chapter 5 and references therein.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
118 / 118
On the leading eigenvalue of neutron transport
This is a part of peripheral spectral analysis of neutron transport. See:
M. M-K: Mathematical Topics in neutron transport theory, World
Scienti…c, Vol 46 (1997), Chapter 5 and references therein.
M. M-K, J. Math. Anal. Appl, 315 (2006) 263-275.
University of FrancheComté Besançon France (Institute)
CIMPA School Muizemberg July 22-Aug 4
Chapter 4
118 / 118