SPECTRAL THEORY FOR NEUTRON TRANSPORT
SPECTRA OF WEIGHTED SHIFT SEMIGROUPS
Mustapha Mokhtar-Kharroubi
(In memory of Seiji Ukaï)
Chapter 3
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Abstract
Transport theory is mainly a perturbation theory with respect to
streaming semigroups (weighted shift semigroups)
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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In this chapter, we study the real spectrum of a class of positive
semigroups covering such weighted shift operators.
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In this chapter, we study the real spectrum of a class of positive
semigroups covering such weighted shift operators.
This chapter resumes essentially a paper by M. M-K, Positivity, 10(2)
(2006) 231-249.
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Chapter 3
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In this chapter, we study the real spectrum of a class of positive
semigroups covering such weighted shift operators.
This chapter resumes essentially a paper by M. M-K, Positivity, 10(2)
(2006) 231-249.
We also capture the complex spectrum by exploiting the spectral
invariance by rotations of streaming semigroups given in ( J. Voigt, Acta.
Appl. Math, 2 (1984) 311-331).
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Local quasinilpotence subspace of positive semigroups
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Local quasinilpotence subspace of positive semigroups
For the sake of simplicity, we restrict ourselves to complex Lebesgue spaces
X = Lp (µ) (1 p ∞). Let (S (t ))t >0 be a positive semigroup on
Lp ( µ ) .
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Local quasinilpotence subspace of positive semigroups
For the sake of simplicity, we restrict ourselves to complex Lebesgue spaces
X = Lp (µ) (1 p ∞). Let (S (t ))t >0 be a positive semigroup on
Lp (µ). We de…ne its local quasinilpotence subset by
Y =
f 2 Lp ( µ ) ;
1
lim kU (t ) jf jk t = 0
t !+∞
where jf j is the absolute value of f 2 Lp (µ).
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A lemma
Lemma
Y is a subspace of Lp (µ) invariant under (S (t ))t >0 .
Linearity : Clearly λf 2 Y if f 2 Y . Let ε > 0, f , g 2 Y be given.
There exists t > 0 depending on them such that
kU (t ) jf jk
εt and kU (t ) jg jk
εt 8t > t.
So kU (t ) jf + g jk kU (t ) (jf j + jg j)k 2εt 8t > t
1
1
andkU (t ) jf + g jk t
2 t ε 2ε 8t > max(t, 1).
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A lemma
Lemma
Y is a subspace of Lp (µ) invariant under (S (t ))t >0 .
Linearity : Clearly λf 2 Y if f 2 Y . Let ε > 0, f , g 2 Y be given.
There exists t > 0 depending on them such that
kU (t ) jf jk
εt and kU (t ) jg jk
εt 8t > t.
So kU (t ) jf + g jk kU (t ) (jf j + jg j)k 2εt 8t > t
1
1
andkU (t ) jf + g jk t
2 t ε 2ε 8t > max(t, 1).
Invariance: Let τ > 0, f 2 Y .
1
kU (t ) jU (τ )f jk t
1
kU (t ) (U (τ ) jf j)k t
1
1
= kU (t + τ ) (jf j)k t = kU (t + τ ) (jf j)k t +τ
t +τ
t
as t ! +∞; i.e. U (τ )f 2 Y .
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!
The main assumption
We assume for the sequel that the local quasinilpotence subspace of
(S (t ))t >0 is dense in Lp (µ).
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On the real spectra of positive semigroups
Theorem
Let (S (t ))t >0 be a positive semigroup on Lp (µ) with type ω.
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On the real spectra of positive semigroups
Theorem
Let (S (t ))t >0 be a positive semigroup on Lp (µ) with type ω. If its local
quasinilpotence subspace is dense in Lp (µ) then
0, e ωt
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σap (S (t )).
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Proof
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Proof
Let t > 0 be …xed.
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Proof
Let t > 0 be …xed. Let 0 < µ < e ωt and y 2 Y .
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Proof
Let t > 0 be …xed. Let 0 < µ < e ωt and y 2 Y . The equation
µx
S (t )x = y ; (y 2 Y , ky k = 1)
can be solved by
x=
1 ∞ 1
1 ∞ 1
k
S
(
t
)
y
=
S (kt )y
k
k
µ k∑
µ k∑
=0 µ
=0 µ
provided that this series converges.
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Proof
Let t > 0 be …xed. Let 0 < µ < e ωt and y 2 Y . The equation
S (t )x = y ; (y 2 Y , ky k = 1)
µx
can be solved by
x=
1 ∞ 1
1 ∞ 1
k
S
(
t
)
y
=
S (kt )y
k
k
µ k∑
µ k∑
=0 µ
=0 µ
provided that this series converges. This is the case since
1
S (kt )y
µk
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1
k
=
1
1
kS (kt )y k kt
µ
t
! 0 as k ! +∞.
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Proof
Let t > 0 be …xed. Let 0 < µ < e ωt and y 2 Y . The equation
S (t )x = y ; (y 2 Y , ky k = 1)
µx
can be solved by
x=
1 ∞ 1
1 ∞ 1
k
S
(
t
)
y
=
S (kt )y
k
k
µ k∑
µ k∑
=0 µ
=0 µ
provided that this series converges. This is the case since
1
S (kt )y
µk
1
k
=
1
1
kS (kt )y k kt
µ
t
! 0 as k ! +∞.
In particular x > 0 for y > 0 and
kx k >
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1
µ k +1
S (t )k y
8k 2 N.
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There exists zk 2 Lp+ (µ) such that kzk k = 1 and
S (t )k zk >
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1
S (t )k .
2
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There exists zk 2 Lp+ (µ) such that kzk k = 1 and
S (t )k zk >
1
S (t )k .
2
By the denseness of Y , 9 yk 2 Y such that kyk k = 1
S (t )k yk >
1
S (t )k .
3
We may assume that yk > 0 since S (t )k jyk j > S (t )k yk
jyk j 2 Y .
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and
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There exists zk 2 Lp+ (µ) such that kzk k = 1 and
S (t )k zk >
1
S (t )k .
2
By the denseness of Y , 9 yk 2 Y such that kyk k = 1
S (t )k yk >
1
S (t )k .
3
We may assume that yk > 0 since S (t )k jyk j > S (t )k yk
jyk j 2 Y .
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and
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The solution b
xk of
satis…es
So
xk k >
kb
µb
xk
1
µ k +1
1
S (t )k yk >
lim inf kb
xk k k >
k !+∞
S (t )b
xk = yk
1 1
S (t )k .
3 µ k +1
1
lim S (t )k
µ k !+∞
1
k
=
e ωt
>1
µ
and then limk !+∞ kb
xk k = ∞.
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The solution b
xk of
µb
xk
satis…es
So
xk k >
kb
1
µ k +1
1
1 1
S (t )k .
3 µ k +1
S (t )k yk >
lim inf kb
xk k k >
k !+∞
S (t )b
xk = yk
1
lim S (t )k
µ k !+∞
and then limk !+∞ kb
xk k = ∞.Finally xk :=
kxk k = 1 and kµxk
xbk
kxbk k
1
k
=
e ωt
>1
µ
is such that
S (t )xk k ! 0
i.e. µ 2 σap (S (t )).
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The solution b
xk of
µb
xk
satis…es
So
xk k >
kb
1
µ k +1
1
1 1
S (t )k .
3 µ k +1
S (t )k yk >
lim inf kb
xk k k >
k !+∞
S (t )b
xk = yk
1
lim S (t )k
µ k !+∞
and then limk !+∞ kb
xk k = ∞.Finally xk :=
kxk k = 1 and kµxk
xbk
kxbk k
1
k
=
e ωt
>1
µ
is such that
S (t )xk k ! 0
i.e. µ 2 σap (S (t )). The closedness of σap (S (t )) ends the proof.
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On the generator
Lemma
Let (S (t ))t >0 be a positive semigroup on Lp (µ) with generator T .
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On the generator
Lemma
Let (S (t ))t >0 be a positive semigroup on Lp (µ) with generator T . Let Y
be the local quasinilpotence subspace of (S (t ))t >0 .
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On the generator
Lemma
Let (S (t ))t >0 be a positive semigroup on Lp (µ) with generator T . Let Y
be the local quasinilpotence subspace of (S (t ))t >0 . Then, for any λ > ω,
lim
k !∞
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(λ
T)
k
y
1
k
= 0 8y 2 Y .
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Proof
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Proof
For any y 2 Y and any ε > 0 there exists ty ,ε > 0 such that
εt 8t > ty ,ε
kS (t )y k
i.e. (write ε = e
A)
kS (t )y k
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e
At
8t > ty ,ε
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Proof
For any y 2 Y and any ε > 0 there exists ty ,ε > 0 such that
εt 8t > ty ,ε
kS (t )y k
i.e. (write ε = e
A)
kS (t )y k
e
At
8t > ty ,ε
so 9My ,A > 0 such that
kS (t )y k
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My ,A e
At
8t > 0.
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Hence
(λ
T)
k
y
=
Z
Z +∞
0
+∞
0
My ,A
=
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dt1 ...
dt1 ...
Z +∞
0
My ,A
( λ + A)k
Z
Z +∞
0
+∞
0
dt1 ...
dtk e
dtk e
Z +∞
0
λ(t1 +...+tk )
λ(t1 +...+tk )
dtk e
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S (t1 + ... + tk )y
kS (t1 + ... + tk )y k
λ(t1 +...+tk )
e
A (t1 +...+tk )
Chapter 3
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Hence
(λ
T)
k
y
=
Z
Z +∞
0
+∞
0
My ,A
=
dt1 ...
dt1 ...
Z
Z +∞
0
+∞
0
Z +∞
dt1 ...
(λ
T)
0
My ,A
( λ + A)k
k !+∞
λ(t1 +...+tk )
dtk e
Z +∞
0
and
lim sup
λ(t1 +...+tk )
dtk e
k
y
kS (t1 + ... + tk )y k
λ(t1 +...+tk )
dtk e
1
k
S (t1 + ... + tk )y
e
A (t1 +...+tk )
1
λ+A
which ends the proof since A > 0 is arbitrary.
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On the real spectrum of generators
Theorem
Let (S (t ))t >0 be a positive semigroup on Lp (µ) with generator T .
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On the real spectrum of generators
Theorem
Let (S (t ))t >0 be a positive semigroup on Lp (µ) with generator T . Let
s (T ) be the spectral bound of T .
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On the real spectrum of generators
Theorem
Let (S (t ))t >0 be a positive semigroup on Lp (µ) with generator T . Let
s (T ) be the spectral bound of T . If the local quasinilpotence subspace of
(S (t ))t >0 is dense in Lp (µ) then
( ∞, s (T )]
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σap (T ).
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Proof
Let λ < s (T ) < µ be …xed.
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Proof
Let λ < s (T ) < µ be …xed. Consider
1
µ
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λ
x
(µ
T)
1
x = y 2 Y.
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Proof
Let λ < s (T ) < µ be …xed. Consider
1
µ
λ
x
(µ
T)
1
x = y 2 Y.
Arguing as for the semigroup, we show the existence of (xk )k with
kxk k = 1 and
1
xk (µ T ) 1 xk ! 0
µ λ
i.e.
1
µ λ
2 σap ((µ
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T)
1)
or equivalently λ 2 σap (T ).
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Proof
Let λ < s (T ) < µ be …xed. Consider
1
µ
λ
x
(µ
T)
1
x = y 2 Y.
Arguing as for the semigroup, we show the existence of (xk )k with
kxk k = 1 and
1
xk (µ T ) 1 xk ! 0
µ λ
i.e. µ 1 λ 2 σap ((µ T )
σap (T ) ends the proof.
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1)
or equivalently λ 2 σap (T ). The closedness of
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Consequences of spectral invariance
Theorem
Let (S (t ))t >0 be a positive semigroup on Lp (µ) with type ω and
generator T . We assume that the local quasinilpotence subspace of
(S (t ))t >0 is dense in Lp (µ).
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Consequences of spectral invariance
Theorem
Let (S (t ))t >0 be a positive semigroup on Lp (µ) with type ω and
generator T . We assume that the local quasinilpotence subspace of
(S (t ))t >0 is dense in Lp (µ).
(i) If σ(T ) is invariant under translations along the imaginary axis then
σ(T ) = fλ 2 C; Re λ
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ωg .
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Chapter 3
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Consequences of spectral invariance
Theorem
Let (S (t ))t >0 be a positive semigroup on Lp (µ) with type ω and
generator T . We assume that the local quasinilpotence subspace of
(S (t ))t >0 is dense in Lp (µ).
(i) If σ(T ) is invariant under translations along the imaginary axis then
σ(T ) = fλ 2 C; Re λ
ωg .
(ii) If σ(S (t )) is invariant under rotations then
σ(S (t )) = µ 2 C; jµj
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e ωt .
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Chapter 3
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Streaming semigroups
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Chapter 3
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Streaming semigroups
Let Ω Rn be an open subset and let µ be a positive Borel measure on
Rn with support V .
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Streaming semigroups
Let Ω Rn be an open subset and let µ be a positive Borel measure on
Rn with support V .Let
σ : Ω V ! R+
be measurable and such that
lim
Z t
t !0 0
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σ (x
τv , v )d τ = 0 a.e.
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Streaming semigroups
Let Ω Rn be an open subset and let µ be a positive Borel measure on
Rn with support V .Let
σ : Ω V ! R+
be measurable and such that
lim
Z t
t !0 0
σ (x
τv , v )d τ = 0 a.e.
Let
s (x, v ) = inf fs > 0; x
sv 2
/ Ωg
be the so-called exit time function.
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Streaming semigroups
Let Ω Rn be an open subset and let µ be a positive Borel measure on
Rn with support V .Let
σ : Ω V ! R+
be measurable and such that
lim
Z t
σ (x
t !0 0
τv , v )d τ = 0 a.e.
Let
s (x, v ) = inf fs > 0; x
sv 2
/ Ωg
be the so-called exit time function. Then
S (t ) : g ! e
Rt
0
σ(x τv ,v )d τ
g (x
tv , v )1ft
s (x ,v )g
de…nes a positive semigroup on Lp (Ω V ; dx µ(dv )) (for any
1 p +∞), strongly continuous when p < +∞.
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Dual streaming semigroups
0
By duality we have, in the dual Lebesgue space Lp (Ω
S 0 (t ) : f ! e
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Rt
0
σ(x +τv ,v )d τ
f (x + tv , v )1ft
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V ),
s (x , v )g .
Chapter 3
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A invariance property of transport operators
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Chapter 3
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A invariance property of transport operators
Let µ f0g = 0 and
α : (x, v ) 2 Ω
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V !
x.v
jv j2
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.
Chapter 3
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A invariance property of transport operators
Let µ f0g = 0 and
α : (x, v ) 2 Ω
V !
x.v
jv j2
.
For any η > 0
M η : f 2 Lp ( Ω
V) ! e
i ηα(x ,v )
f 2 Lp ( Ω
V)
is an isometric isomorphism.
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Chapter 3
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A similarity property of streaming semigroups
We …nd the following result in (J. Voigt, Acta. Appl. Math, 2 (1984)
311-331).
Theorem
Mη 1 S (t )Mη = e i ηt S (t ).
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Chapter 3
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A similarity property of streaming semigroups
We …nd the following result in (J. Voigt, Acta. Appl. Math, 2 (1984)
311-331).
Theorem
Mη 1 S (t )Mη = e i ηt S (t ). In particular σ(S (t )) is invariant by rotations.
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Chapter 3
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Proof
We have
S (t )Mη f
= e
= e
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Rt
0
Rt
0
σ (x τv ,v )d τ
σ (x τv ,v )d τ
Mη f (x
e
tv , v )1ft
(x tv ).v
iη
jv j2
f (x
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s (x ,v )g
tv , v )1ft
s (x ,v )g
Chapter 3
21 / 52
Proof
We have
S (t )Mη f
= e
= e
Rt
0
Rt
0
σ (x τv ,v )d τ
σ (x τv ,v )d τ
so
Mη 1 S (t )Mη f = e
iη
x .v
jv j2
e
Mη f (x
e
iη
tv , v )1ft
(x tv ).v
iη
jv j2
(x tv ).v
jv j2
f (x
s (x ,v )g
tv , v )1ft
s (x ,v )g
S (t )f = e i ηt S (t )f
so Mη 1 S (t )Mη = e i ηt S (t ). Hence σ(Mη 1 S (t )Mη ) = σ(e i ηt S (t )).
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Chapter 3
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Proof
We have
S (t )Mη f
= e
= e
Rt
0
Rt
0
σ (x τv ,v )d τ
σ (x τv ,v )d τ
so
Mη 1 S (t )Mη f = e
iη
x .v
jv j2
e
Mη f (x
e
iη
tv , v )1ft
(x tv ).v
iη
jv j2
(x tv ).v
jv j2
f (x
s (x ,v )g
tv , v )1ft
s (x ,v )g
S (t )f = e i ηt S (t )f
so Mη 1 S (t )Mη = e i ηt S (t ). Hence σ(Mη 1 S (t )Mη ) = σ(e i ηt S (t )). On
the other hand, by similarity,
σ (Mη 1 S (t )Mη ) = σ (S (t ))
and
σ(e i ηt S (t )) = e i ηt σ (S (t ))
so we are done.
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Chapter 3
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A similarity property of streaming generators
As previously we have:
Theorem
Let T be the generator of a streaming semigroup (S (t ))t >0 .
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Chapter 3
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A similarity property of streaming generators
As previously we have:
Theorem
Let T be the generator of a streaming semigroup (S (t ))t >0 . Then
Mη 1 TMη = T + i ηI .
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Chapter 3
22 / 52
A similarity property of streaming generators
As previously we have:
Theorem
Let T be the generator of a streaming semigroup (S (t ))t >0 . Then
Mη 1 TMη = T + i ηI . In particular σ(T ) is invariant by translation along
the imaginary axis.
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Chapter 3
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Proof
Let f 2 D (T ). Then
S (t )Mη f
t
Mη f
= Mη
Mη 1 S (t )Mη f
f
t
e i ηt S (t )f f
= Mη
t
e i ηt f
e i ηt S (t )f e i ηt f
= Mη
+ Mη
t
t
i
ηt
S (t )f f
e
1
= e i ηt Mη
+
Mη f
t
t
! Mη Tf + i ηMη f
f
so Mη f 2 D (T ) and TMη f = Mη Tf + i ηMη f or Mη 1 TMη = T + i ηI .
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Chapter 3
23 / 52
Proof
Let f 2 D (T ). Then
S (t )Mη f
t
Mη f
= Mη
Mη 1 S (t )Mη f
f
t
e i ηt S (t )f f
= Mη
t
e i ηt f
e i ηt S (t )f e i ηt f
= Mη
+ Mη
t
t
i
ηt
S (t )f f
e
1
= e i ηt Mη
+
Mη f
t
t
! Mη Tf + i ηMη f
f
so Mη f 2 D (T ) and TMη f = Mη Tf + i ηMη f or Mη 1 TMη = T + i ηI .By
similarity, σ(T ) = σ(Mη 1 TMη ) = σ(T ) + i η 8η 2 R.
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Chapter 3
23 / 52
Decomposition of the phase space
We consider the partition of the phase space Ω
E1 = f(x, v ) 2 Ω
E2 = f(x, v ) 2 Ω
E3 = f(x, v ) 2 Ω
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V ; s (x,
V ; s (x,
V ; s (x,
V according to
v ) < +∞g ,
v ) = +∞, s (x, v ) < +∞g ,
v ) = +∞, s (x, v ) = +∞g .
Beamer presentations in SWP and SW
Chapter 3
24 / 52
Decomposition of the phase space
We consider the partition of the phase space Ω
E1 = f(x, v ) 2 Ω
E2 = f(x, v ) 2 Ω
V ; s (x,
E3 = f(x, v ) 2 Ω
V ; s (x,
V ; s (x,
V according to
v ) < +∞g ,
v ) = +∞, s (x, v ) < +∞g ,
v ) = +∞, s (x, v ) = +∞g .
This induces a direct sum
Lp ( Ω
V ; dx
µ(dv )) = Lp (E1 )
L p ( E2 )
L p ( E3 )
where, we identify Lp (Ei ) to the closed subspace of functions
f 2 Lp (Ω V ) vanishing almost everywhere on Ω V n Ei .
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Beamer presentations in SWP and SW
Chapter 3
24 / 52
Decomposition of the phase space
We consider the partition of the phase space Ω
E1 = f(x, v ) 2 Ω
E2 = f(x, v ) 2 Ω
V ; s (x,
E3 = f(x, v ) 2 Ω
V ; s (x,
V ; s (x,
V according to
v ) < +∞g ,
v ) = +∞, s (x, v ) < +∞g ,
v ) = +∞, s (x, v ) = +∞g .
This induces a direct sum
Lp ( Ω
V ; dx
µ(dv )) = Lp (E1 )
L p ( E2 )
L p ( E3 )
where, we identify Lp (Ei ) to the closed subspace of functions
f 2 Lp (Ω V ) vanishing almost everywhere on Ω V n Ei .
If some set Ei has zero measure then we drop out Lp (Ei ) from the
direct sum above.
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Chapter 3
24 / 52
Canonical decomposition of streaming semigroups
Theorem
The subspaces Lp (Ei ) (i = 1, 2, 3) are invariant under (S (t ))t >0. .
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Chapter 3
25 / 52
Canonical decomposition of streaming semigroups
Theorem
The subspaces Lp (Ei ) (i = 1, 2, 3) are invariant under (S (t ))t >0. . For
each i = 1, 2, 3, we denote by (Si (t ))t >0 the part of (S (t ))t >0. on Lp (Ei )
and by Ti its generator.
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Chapter 3
25 / 52
Canonical decomposition of streaming semigroups
Theorem
The subspaces Lp (Ei ) (i = 1, 2, 3) are invariant under (S (t ))t >0. . For
each i = 1, 2, 3, we denote by (Si (t ))t >0 the part of (S (t ))t >0. on Lp (Ei )
and by Ti its generator. Then
σ(S (t )) = σ(S1 (t )) [ σ(S2 (t )) [ σ (S3 (t ))
σ ( T ) = σ ( T1 ) [ σ ( T2 ) [ σ ( T3 ) .
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Chapter 3
25 / 52
Canonical decomposition of streaming semigroups
Theorem
The subspaces Lp (Ei ) (i = 1, 2, 3) are invariant under (S (t ))t >0. . For
each i = 1, 2, 3, we denote by (Si (t ))t >0 the part of (S (t ))t >0. on Lp (Ei )
and by Ti its generator. Then
σ(S (t )) = σ(S1 (t )) [ σ(S2 (t )) [ σ (S3 (t ))
σ ( T ) = σ ( T1 ) [ σ ( T2 ) [ σ ( T3 ) .
We have also similar results where σ(.) is replaced by σp (.) or σap (.).
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Chapter 3
25 / 52
Canonical decomposition of streaming semigroups
Theorem
The subspaces Lp (Ei ) (i = 1, 2, 3) are invariant under (S (t ))t >0. . For
each i = 1, 2, 3, we denote by (Si (t ))t >0 the part of (S (t ))t >0. on Lp (Ei )
and by Ti its generator. Then
σ(S (t )) = σ(S1 (t )) [ σ(S2 (t )) [ σ (S3 (t ))
σ ( T ) = σ ( T1 ) [ σ ( T2 ) [ σ ( T3 ) .
We have also similar results where σ(.) is replaced by σp (.) or σap (.).In
addition,if σ(., .) is bounded then (S3 (t ))t >0 extends to a positive group.
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Chapter 3
25 / 52
Proof
We have just to check that the direct sum
Lp (Ω V ) = Lp (E1 ) Lp (E2 ) Lp (E3 ) reduces (S (t ))t >0.
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Chapter 3
26 / 52
Proof
We have just to check that the direct sum
Lp (Ω V ) = Lp (E1 ) Lp (E2 ) Lp (E3 ) reduces (S (t ))t >0. We restrict
ourselves to Lp (E1 ).
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Chapter 3
26 / 52
Proof
We have just to check that the direct sum
Lp (Ω V ) = Lp (E1 ) Lp (E2 ) Lp (E3 ) reduces (S (t ))t >0. We restrict
ourselves to Lp (E1 ). Let f 2 Lp (E1 ), i.e. f vanishes almost everywhere on
E2 [ E3 . We have to show that S (t )f 2 Lp (E1 ) i.e. S (t )f vanishes
almost everywhere on E2 [ E3 .
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Chapter 3
26 / 52
Proof
We have just to check that the direct sum
Lp (Ω V ) = Lp (E1 ) Lp (E2 ) Lp (E3 ) reduces (S (t ))t >0. We restrict
ourselves to Lp (E1 ). Let f 2 Lp (E1 ), i.e. f vanishes almost everywhere on
E2 [ E3 . We have to show that S (t )f 2 Lp (E1 ) i.e. S (t )f vanishes
almost everywhere on E2 [ E3 . Since
S (t )f (x, v ) = e
Rt
0
σ(x τv ,v )d τ
f (x
tv , v )1ft
is zero for t > s (x, v ), we assume from the start that t
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Beamer presentations in SWP and SW
s (x ,v )g
s (x, v ).
Chapter 3
26 / 52
Proof
We have just to check that the direct sum
Lp (Ω V ) = Lp (E1 ) Lp (E2 ) Lp (E3 ) reduces (S (t ))t >0. We restrict
ourselves to Lp (E1 ). Let f 2 Lp (E1 ), i.e. f vanishes almost everywhere on
E2 [ E3 . We have to show that S (t )f 2 Lp (E1 ) i.e. S (t )f vanishes
almost everywhere on E2 [ E3 . Since
S (t )f (x, v ) = e
Rt
0
σ(x τv ,v )d τ
f (x
tv , v )1ft
is zero for t > s (x, v ), we assume from the start that t
notes that (x, v ) 2 E2 [ E3 , s (x, v ) = +∞ and
s (x
so that (x
tv ,
tv ,
v ) = t + s (x,
v ) 2 E2 [ E3 and f (x
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s (x ,v )g
s (x, v ). One
v)
tv , v ) = 0.
Beamer presentations in SWP and SW
Chapter 3
26 / 52
Proof
We have just to check that the direct sum
Lp (Ω V ) = Lp (E1 ) Lp (E2 ) Lp (E3 ) reduces (S (t ))t >0. We restrict
ourselves to Lp (E1 ). Let f 2 Lp (E1 ), i.e. f vanishes almost everywhere on
E2 [ E3 . We have to show that S (t )f 2 Lp (E1 ) i.e. S (t )f vanishes
almost everywhere on E2 [ E3 . Since
S (t )f (x, v ) = e
Rt
0
σ(x τv ,v )d τ
f (x
tv , v )1ft
is zero for t > s (x, v ), we assume from the start that t
notes that (x, v ) 2 E2 [ E3 , s (x, v ) = +∞ and
s (x
tv ,
v ) = t + s (x,
s (x ,v )g
s (x, v ). One
v)
so that (x tv , v ) 2 E2 [ E3 and f (x tv , v ) = 0. Since the projection
Pi on Lp (Ei ) along Lp (Ω V n Ei ) commutes with (S (t ))t >0. then the
direct sum above reduces also the generator T .
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Chapter 3
26 / 52
Proof
We have just to check that the direct sum
Lp (Ω V ) = Lp (E1 ) Lp (E2 ) Lp (E3 ) reduces (S (t ))t >0. We restrict
ourselves to Lp (E1 ). Let f 2 Lp (E1 ), i.e. f vanishes almost everywhere on
E2 [ E3 . We have to show that S (t )f 2 Lp (E1 ) i.e. S (t )f vanishes
almost everywhere on E2 [ E3 . Since
S (t )f (x, v ) = e
Rt
0
σ(x τv ,v )d τ
f (x
tv , v )1ft
is zero for t > s (x, v ), we assume from the start that t
notes that (x, v ) 2 E2 [ E3 , s (x, v ) = +∞ and
s (x
tv ,
v ) = t + s (x,
s (x ,v )g
s (x, v ). One
v)
so that (x tv , v ) 2 E2 [ E3 and f (x tv , v ) = 0. Since the projection
Pi on Lp (Ei ) along Lp (Ω V n Ei ) commutes with (S (t ))t >0. then the
direct sum above reduces also the generator T .
Finally, on E3 (if σ(., .) is bounded) (S3 (t ))t >0 extends to a positive group
where
Rt
S3 (t ) 1 g = e 0 σ(x +τv ,v )d τ f (x + tv , v ) (t > 0).
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Beamer presentations in SWP and SW
Chapter 3
26 / 52
Spectra of the …rst reduced streaming semigroup
We start with (S1 (t ))t >0 .
Lemma
Let t > 0 be …xed. For any f 2 Lp (E1 )
p
k S1 ( t ) f k =
Z
ft <s (y , v )g\fs (y , v )<∞g
e
p
Rt
0
σ(y +τv ,v )d τ
jf (y , v )jp dx µ(dv ).
Proof:
We have to compute the norm of S1 (t )f on the set
ft
s (x, v )g \ fs (x,
v ) < +∞g ,
p
so kS1 (t )f k is equal to
Z
ft s (x ,v )g\fs (x , v )<+∞g
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e
p
Rt
0
σ(x τv ,v )d τ
jf (x
Beamer presentations in SWP and SW
tv , v )jp dx µ(dv ).
Chapter 3
27 / 52
Since s (x tv , v ) = t + s (x, v ) is …nite if and only if s (x,
…nite then the change of variable
y := x
gives s (y ,
v ) is
tv 2 Ω
v ) > t and
k S1 ( t ) f k p =
Z
ft <s (y , v )g\fs (y , v )<∞g
MSI Tech Support (Institute)
e
p
Rt
0
σ (y +τv ,v )d τ
Beamer presentations in SWP and SW
jf (y , v )jp dy µ(dv ).
Chapter 3
28 / 52
The type of (S1 (t ))t >0 is equal to
λ1 = lim
λ1 where
inf
t !+∞ ft <s (y , v )g\fs (y ,
1
v )<∞g t
Z t
0
σ(y + τv , v )d τ.
because
kS1 (t )k =
sup
Rt
e
0
σ(y +τv ,v )d τ
ft <s (y , v )g\fs (y , v )<∞g
Rt
inf ft <s (y , v )g\fs (y , v )<∞g 0 σ(y +τv ,v )d τ
= e
so
ln kS1 (t )k
=
t
inf
ft <s (y , v )g\fs (y ,
1
v )<∞g t
Z t
0
σ(y + τv , v )d τ
and
ω1 =
lim
inf
t !+∞ ft <s (y , v )g\fs (y ,
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1
v )<∞g t
Z t
Beamer presentations in SWP and SW
0
σ(y + τv , v )d τ.
Chapter 3
29 / 52
n
o
σ(S1 (t )) = µ 2 C; jµj e λ1 t , σ(Ti ) = fλ 2 C; Re λ
λ1 g
Indeed, it su¢ ces to show that the local quasinilpotence subspace of
(S1 (t ))t >0 is dense in Lp (E1 ). Let
Om := fx, v ) 2 Ω
V ; s (x,
v)
mg .
We note that [m Lp (Om ) is dense in Lp (E1 ) because of
[m Om = fx, v ) 2 Ω
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V ; s (x,
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v ) < +∞g .
Chapter 3
30 / 52
n
o
σ(S1 (t )) = µ 2 C; jµj e λ1 t , σ(Ti ) = fλ 2 C; Re λ
λ1 g
Indeed, it su¢ ces to show that the local quasinilpotence subspace of
(S1 (t ))t >0 is dense in Lp (E1 ). Let
Om := fx, v ) 2 Ω
V ; s (x,
v)
mg .
We note that [m Lp (Om ) is dense in Lp (E1 ) because of
[m Om = fx, v ) 2 Ω
V ; s (x,
Z
e
v ) < +∞g .
Finally
k S1 ( t ) f k p =
ft <s (y , v )g\fs (y , v )<∞g
p
Rt
0
σ (y +τv ,v )d τ
jf (y , v )jp dy µ(dv )
shows that, for f 2 Lp (Om ), kS1 (t )f k = 0 for t > m so [m Lp (Om ) is
included in the local quasinilpotence subspace of (S1 (t ))t >0 .
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Beamer presentations in SWP and SW
Chapter 3
30 / 52
Spectra of the second reduced streaming semigroup
We deal now with (S2 (t ))t >0 on Lp (E2 ) where
E2 = f(x, v ) 2 Ω
V ; s (x,
v ) = +∞, s (x, v ) < +∞g .
We consider …rst the case
1 < p < +∞.
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Chapter 3
31 / 52
Spectra of the second reduced streaming semigroup
We deal now with (S2 (t ))t >0 on Lp (E2 ) where
E2 = f(x, v ) 2 Ω
V ; s (x,
v ) = +∞, s (x, v ) < +∞g .
We consider …rst the case
1 < p < +∞.
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Beamer presentations in SWP and SW
Chapter 3
31 / 52
Spectra of the second reduced streaming semigroup
We deal now with (S2 (t ))t >0 on Lp (E2 ) where
E2 = f(x, v ) 2 Ω
V ; s (x,
v ) = +∞, s (x, v ) < +∞g .
We consider …rst the case
1 < p < +∞.
We show as previously that the type of (S2 (t ))t >0 is equal to
where
λ2 = lim
inf
t !+∞ ft s (y , v )<∞g\fs (y ,
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1
v )=∞g t
Beamer presentations in SWP and SW
Z t
0
λ2
σ(y + τv , v )d τ.
Chapter 3
31 / 52
Spectra of the second reduced streaming semigroup
We deal now with (S2 (t ))t >0 on Lp (E2 ) where
E2 = f(x, v ) 2 Ω
V ; s (x,
v ) = +∞, s (x, v ) < +∞g .
We consider …rst the case
1 < p < +∞.
We show as previously that the type of (S2 (t ))t >0 is equal to
where
λ2 = lim
inf
t !+∞ ft s (y , v )<∞g\fs (y ,
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1
v )=∞g t
Beamer presentations in SWP and SW
Z t
0
λ2
σ(y + τv , v )d τ.
Chapter 3
31 / 52
Spectra of the second reduced streaming semigroup
We deal now with (S2 (t ))t >0 on Lp (E2 ) where
E2 = f(x, v ) 2 Ω
V ; s (x,
v ) = +∞, s (x, v ) < +∞g .
We consider …rst the case
1 < p < +∞.
We show as previously that the type of (S2 (t ))t >0 is equal to
where
λ2 = lim
inf
t !+∞ ft s (y , v )<∞g\fs (y ,
n
σ(S2 (t )) = µ 2 C; jµj
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e
λ2 t
o
1
v )=∞g t
Z t
0
σ(y + τv , v )d τ.
, σ(T2 ) = fλ 2 C; Re λ
Beamer presentations in SWP and SW
λ2
Chapter 3
λ2 g .
31 / 52
Indeed, by duality σ(S2 (t )) = σ(S20 (t )) where
S20 (t )f = e
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Rt
0
σ (x +τv ,v )d τ
f (x + tv , v ).
Beamer presentations in SWP and SW
Chapter 3
32 / 52
Indeed, by duality σ(S2 (t )) = σ(S20 (t )) where
S20 (t )f = e
Thus kS2 (t )f k
Z
=
Z
p0
is equal to
fs (x , v )=∞, s (x ,v )<∞g
Rt
0
e
fs (y , v )=∞, t s (y ,v )<∞g
MSI Tech Support (Institute)
σ (x +τv ,v )d τ
p0
e
Rt
0
p0
f (x + tv , v ).
σ(y +τv ,v )d τ
Rt
0
jf (x + tv , v )jp dx µ(dv )
σ(y +τv ,v )d τ
Beamer presentations in SWP and SW
jf (y , v )jp dy µ(dv ).
Chapter 3
32 / 52
Indeed, by duality σ(S2 (t )) = σ(S20 (t )) where
S20 (t )f = e
Thus kS2 (t )f k
Z
=
Z
p0
is equal to
fs (x , v )=∞, s (x ,v )<∞g
Rt
0
e
fs (y , v )=∞, t s (y ,v )<∞g
MSI Tech Support (Institute)
σ (x +τv ,v )d τ
p0
e
Rt
0
p0
f (x + tv , v ).
σ(y +τv ,v )d τ
Rt
0
jf (x + tv , v )jp dx µ(dv )
σ(y +τv ,v )d τ
Beamer presentations in SWP and SW
jf (y , v )jp dy µ(dv ).
Chapter 3
32 / 52
Indeed, by duality σ(S2 (t )) = σ(S20 (t )) where
S20 (t )f = e
Thus kS2 (t )f k
Z
=
Z
p0
is equal to
fs (x , v )=∞, s (x ,v )<∞g
Rt
0
e
fs (y , v )=∞, t s (y ,v )<∞g
σ (x +τv ,v )d τ
p0
e
Rt
0
p0
f (x + tv , v ).
σ(y +τv ,v )d τ
Rt
0
jf (x + tv , v )jp dx µ(dv )
σ(y +τv ,v )d τ
jf (y , v )jp dy µ(dv ).
Introducing the sets
p0
Om0 := fx, v ) 2 E2 ; s (y , v )
p0
mg
one sees that [m L (Om0 ) is dense in L (E2 ) because of
[m Om = E2 .
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Beamer presentations in SWP and SW
Chapter 3
32 / 52
Indeed, by duality σ(S2 (t )) = σ(S20 (t )) where
S20 (t )f = e
Thus kS2 (t )f k
Z
=
Z
p0
is equal to
fs (x , v )=∞, s (x ,v )<∞g
Rt
0
e
fs (y , v )=∞, t s (y ,v )<∞g
σ (x +τv ,v )d τ
p0
e
Rt
0
p0
f (x + tv , v ).
σ(y +τv ,v )d τ
Rt
0
jf (x + tv , v )jp dx µ(dv )
σ(y +τv ,v )d τ
jf (y , v )jp dy µ(dv ).
Introducing the sets
p0
Om0 := fx, v ) 2 E2 ; s (y , v )
p0
mg
one sees that [m L (Om0 ) is dense in L (E2 ) because of
0
[m Om = E2 .
Since in Lp (Om0 ), kS2 (t )f k = 0 for t > m then the local quasinilpotence
subspace of (S20 (t ))t >0 is dense.
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Beamer presentations in SWP and SW
Chapter 3
32 / 52
Indeed, by duality σ(S2 (t )) = σ(S20 (t )) where
S20 (t )f = e
Thus kS2 (t )f k
Z
=
Z
p0
is equal to
fs (x , v )=∞, s (x ,v )<∞g
Rt
0
e
fs (y , v )=∞, t s (y ,v )<∞g
σ (x +τv ,v )d τ
p0
e
Rt
0
p0
f (x + tv , v ).
σ(y +τv ,v )d τ
Rt
0
jf (x + tv , v )jp dx µ(dv )
σ(y +τv ,v )d τ
jf (y , v )jp dy µ(dv ).
Introducing the sets
p0
Om0 := fx, v ) 2 E2 ; s (y , v )
p0
mg
one sees that [m L (Om0 ) is dense in L (E2 ) because of
0
[m Om = E2 .
Since in Lp (Om0 ), kS2 (t )f k = 0 for t > m then the local quasinilpotence
subspace of (S20 (t ))t >0 is dense. This ends the proof because
σ(S2 (t )) = σ(S20 (t )) and σ(T2 ) = σ(T20 ).
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Chapter 3
32 / 52
Spectra of the third reduced streaming (semi)group
Theorem
Let S := σ(T3 ) \ R be the real spectrum of T3 .
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Chapter 3
33 / 52
Spectra of the third reduced streaming (semi)group
Theorem
Let S := σ(T3 ) \ R be the real spectrum of T3 . Then
σ(T3 ) = S + iR and σ(S3 (t )) = e t σ(T 3 ) .
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Beamer presentations in SWP and SW
Chapter 3
33 / 52
Spectra of the third reduced streaming (semi)group
Theorem
Let S := σ(T3 ) \ R be the real spectrum of T3 . Then
σ(T3 ) = S + iR and σ(S3 (t )) = e t σ(T 3 ) .
Moreover, sup S =
λ3 and inf S =
λ3 = lim
inf
λ3 = lim
sup
t !+∞ f s (y , v )=∞,
t !+∞ f s (y , v )=∞,
MSI Tech Support (Institute)
λ3 where
1
s (y ,v )=∞g t
Z t
σ(y + τv , v )d τ
1
s (y ,v )=∞g t
Z t
σ(y + τv , v )d τ.
0
0
Beamer presentations in SWP and SW
Chapter 3
33 / 52
Proof
The fact that σ(T3 ) is invariant by translation along the imaginary axis
and that e t σ(T 3 ) is invariant under the rotations is a general feature of
streaming semigroups in arbitrary geometry.
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Beamer presentations in SWP and SW
Chapter 3
34 / 52
Proof
The fact that σ(T3 ) is invariant by translation along the imaginary axis
and that e t σ(T 3 ) is invariant under the rotations is a general feature of
streaming semigroups in arbitrary geometry. The spectral mapping property
for the real spectrum is due to the fact that (S3 (t ))t 2R is a positive
C0 -group (see G. Greiner, Quart. J. Math. Oxford, 35(2) (1984) 37-47).
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Proof
The fact that σ(T3 ) is invariant by translation along the imaginary axis
and that e t σ(T 3 ) is invariant under the rotations is a general feature of
streaming semigroups in arbitrary geometry. The spectral mapping property
for the real spectrum is due to the fact that (S3 (t ))t 2R is a positive
C0 -group (see G. Greiner, Quart. J. Math. Oxford, 35(2) (1984) 37-47).
The type λ3 of (S3 (t ))t >0 is obtained as for (S1 (t ))t >0 or (S2 (t ))t >0 .
Finally, λ3 is the spectral bound of the generator of (S3 ( t ))t >0 , (i.e.
T3 ) and is obtained similarly.
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Chapter 3
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Space homogeneous cross-sections
Theorem
If σ : Ω V ! R+ is space-homogeneous then S := σ(T3 ) \ R is
nothing but the essential range of σ(.).
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Chapter 3
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Space homogeneous cross-sections
Theorem
If σ : Ω V ! R+ is space-homogeneous then S := σ(T3 ) \ R is
nothing but the essential range of σ(.).
See the details in (M. M-K, Positivity, 10(2) (2006) 231-249).
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Chapter 3
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Space homogeneous cross-sections
Theorem
If σ : Ω V ! R+ is space-homogeneous then S := σ(T3 ) \ R is
nothing but the essential range of σ(.).
See the details in (M. M-K, Positivity, 10(2) (2006) 231-249).
In particular, S := σ(T3 ) \ R need not be connected.
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Chapter 3
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Open question
The description of σ(T3 ) \ R for general σ : Ω
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V ! R+ .
Chapter 3
36 / 52
Open question
The description of σ(T3 ) \ R for general σ : Ω
V ! R+ .
When Ω = Rn , the situation is well understod for bounded and
compactly supported (in space) collision frequency σ : Ω V ! R+ ;
see A. Huber, Int Eq Op Theory, 6 (1983) 357-371.
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Chapter 3
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On the description of σ (S2 (t)) for p = 1
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Chapter 3
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Reminders on sun-dual theory
Let X be a complex Banach space and let (S (t ))t >0 be a C0 -semigroup
on X with generator T .
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Chapter 3
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Reminders on sun-dual theory
Let X be a complex Banach space and let (S (t ))t >0 be a C0 -semigroup
on X with generator T . Let (S 0 (t ))t >0 be the dual semigroup on the
dual space X 0 .
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Chapter 3
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Reminders on sun-dual theory
Let X be a complex Banach space and let (S (t ))t >0 be a C0 -semigroup
on X with generator T . Let (S 0 (t ))t >0 be the dual semigroup on the
dual space X 0 . If X is not re‡exive then a priori (S 0 (t ))t >0 is not
strongly continuous.
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Chapter 3
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Reminders on sun-dual theory
Let X be a complex Banach space and let (S (t ))t >0 be a C0 -semigroup
on X with generator T . Let (S 0 (t ))t >0 be the dual semigroup on the
dual space X 0 . If X is not re‡exive then a priori (S 0 (t ))t >0 is not
strongly continuous. Let
L := x 0 2 X 0 ;
S 0 (t )x 0
x 0 ! 0 as t ! 0
the subspace of strong continuity of (S 0 (t ))t >0 .
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Chapter 3
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L
is a closed subspace of X 0 invariant under (S 0 (t ))t >0 .
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Chapter 3
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L
is a closed subspace of X 0 invariant under (S 0 (t ))t >0 .
L = D (T 0 ) (the closure in X 0 ).
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Chapter 3
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L
is a closed subspace of X 0 invariant under (S 0 (t ))t >0 .
L = D (T 0 ) (the closure in X 0 ).
We denote by (S (t ))t >0 the restriction of (S 0 (t ))t >0 to L
(sun-dual C0 -semigroup).
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Chapter 3
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L
is a closed subspace of X 0 invariant under (S 0 (t ))t >0 .
L = D (T 0 ) (the closure in X 0 ).
We denote by (S (t ))t >0 the restriction of (S 0 (t ))t >0 to L
(sun-dual C0 -semigroup).
The generator T of (S (t ))t >0 is given
D (T ) = fx 0 2 D (T 0 ), T 0 x 0 2 L g and T x 0 = T 0 x 0 .
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Chapter 3
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L
is a closed subspace of X 0 invariant under (S 0 (t ))t >0 .
L = D (T 0 ) (the closure in X 0 ).
We denote by (S (t ))t >0 the restriction of (S 0 (t ))t >0 to L
(sun-dual C0 -semigroup).
The generator T of (S (t ))t >0 is given
D (T ) = fx 0 2 D (T 0 ), T 0 x 0 2 L g and T x 0 = T 0 x 0 .
σ(T ) = σ(T 0 ) = σ(T ) and σ(S (t )) = σ(S 0 (t )) = σ(S (t )).
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Chapter 3
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L
is a closed subspace of X 0 invariant under (S 0 (t ))t >0 .
L = D (T 0 ) (the closure in X 0 ).
We denote by (S (t ))t >0 the restriction of (S 0 (t ))t >0 to L
(sun-dual C0 -semigroup).
The generator T of (S (t ))t >0 is given
D (T ) = fx 0 2 D (T 0 ), T 0 x 0 2 L g and T x 0 = T 0 x 0 .
σ(T ) = σ(T 0 ) = σ(T ) and σ(S (t )) = σ(S 0 (t )) = σ(S (t )).
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Chapter 3
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L
is a closed subspace of X 0 invariant under (S 0 (t ))t >0 .
L = D (T 0 ) (the closure in X 0 ).
We denote by (S (t ))t >0 the restriction of (S 0 (t ))t >0 to L
(sun-dual C0 -semigroup).
The generator T of (S (t ))t >0 is given
D (T ) = fx 0 2 D (T 0 ), T 0 x 0 2 L g and T x 0 = T 0 x 0 .
σ(T ) = σ(T 0 ) = σ(T ) and σ(S (t )) = σ(S 0 (t )) = σ(S (t )).
(See e.g. Nagel-Engel, Chapter 4)
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Chapter 3
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Sun-dual theory for streaming semigroups
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Chapter 3
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Sun-dual theory for streaming semigroups
We consider (S2 (t ))t >0 in L1 (E2 ) where
E2 = f(x, v ) 2 Ω
V ; s (x,
v ) = +∞, s (x, v ) < +∞g
and assume that
σ:Ω
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V ! R+ is bounded.
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Chapter 3
40 / 52
Sun-dual theory for streaming semigroups
We consider (S2 (t ))t >0 in L1 (E2 ) where
E2 = f(x, v ) 2 Ω
V ; s (x,
v ) = +∞, s (x, v ) < +∞g
and assume that
σ:Ω
Since
V ! R+ is bounded.
σ(S2 (t )) = σ(S20 (t )) = σ(S2 (t )),
it su¢ ces to identify σ(S2 (t )).
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Chapter 3
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Sun-dual theory for streaming semigroups
We consider (S2 (t ))t >0 in L1 (E2 ) where
E2 = f(x, v ) 2 Ω
V ; s (x,
v ) = +∞, s (x, v ) < +∞g
and assume that
σ:Ω
Since
V ! R+ is bounded.
σ(S2 (t )) = σ(S20 (t )) = σ(S2 (t )),
it su¢ ces to identify σ(S2 (t )). Because of the boundedness of
σ : Ω V ! R+ ,
(
L =
f 2 L∞ (E2 ), sup jf (x + tv , v )
(x ,v )
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)
f (x, v )j ! 0 as t ! 0 .
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Chapter 3
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Sun-dual theory for streaming semigroups
We consider (S2 (t ))t >0 in L1 (E2 ) where
E2 = f(x, v ) 2 Ω
V ; s (x,
v ) = +∞, s (x, v ) < +∞g
and assume that
σ:Ω
Since
V ! R+ is bounded.
σ(S2 (t )) = σ(S20 (t )) = σ(S2 (t )),
it su¢ ces to identify σ(S2 (t )). Because of the boundedness of
σ : Ω V ! R+ ,
(
L =
f 2 L∞ (E2 ), sup jf (x + tv , v )
(x ,v )
)
f (x, v )j ! 0 as t ! 0 .
Actually, we are going to work in the smaller closed subspace
(
)
L0 : =
f 2L ,
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sup
f(y ,v ), s (y ,v )>r g
jf (y , v )j ! 0 as r ! ∞ .
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Chapter 3
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Lemma
(S 0 (t ))t >0 leaves invariant L0 .
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Chapter 3
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Lemma
(S 0 (t ))t >0 leaves invariant L0 .
Proof:
S 0 (t )f (y , v )
jf (y + tv , v )j
and s (y + tv , v ) = s (y , v ) + t ! ∞ if and only if s (y , v ) ! ∞ so that
S 0 (t )f 2 L0 if f 2 L0 .
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Chapter 3
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Let S0 (t )
generator.
t >0
be the restriction of S2 (t )
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t >0
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to L0 and let T0 be its
Chapter 3
42 / 52
Let S0 (t )
generator.
t >0
be the restriction of S2 (t )
σap (S0 (t ))
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t >0
σap (S2 (t )) and σap (T0 )
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to L0 and let T0 be its
σap (T2 ).
Chapter 3
42 / 52
Let S0 (t )
generator.
t >0
be the restriction of S2 (t )
σap (S0 (t ))
σap (S2 (t )) and σap (T0 )
In particular, σap (S0 (t ))
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t >0
to L0 and let T0 be its
σap (T2 ).
σ(S2 (t )) and σap (T0 )
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σ ( T2 ) .
Chapter 3
42 / 52
Let
L00 := f 2 L , 9r > 0, f (y , v ) = 0 for s (y , v ) > r .
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Chapter 3
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Let
L00 := f 2 L , 9r > 0, f (y , v ) = 0 for s (y , v ) > r .
Theorem
L00 is dense in L0 .
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Chapter 3
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Let
L00 := f 2 L , 9r > 0, f (y , v ) = 0 for s (y , v ) > r .
Theorem
L00 is dense in L0 .
Corollary
The local quasinilpotence subspace of S0 (t )
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t >0
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is dense in L0 .
Chapter 3
43 / 52
Let
L00 := f 2 L , 9r > 0, f (y , v ) = 0 for s (y , v ) > r .
Theorem
L00 is dense in L0 .
Corollary
The local quasinilpotence subspace of S0 (t )
t >0
is dense in L0 .
Proof of the corollary: The local quasinilpotence subspace of S0 (t )
contains L00 .
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Chapter 3
t >0
43 / 52
Proof of the theorem
Lemma
L is an algebra.
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Chapter 3
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A troncation procedure
For each m 2 N, let γm : [0, +∞[ ! [0, 1] be smooth (say C 1 ) and such
that
1 if s m
γm (s ) =
0 if s > 2m.
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Chapter 3
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A troncation procedure
For each m 2 N, let γm : [0, +∞[ ! [0, 1] be smooth (say C 1 ) and such
that
1 if s m
γm (s ) =
0 if s > 2m.
Lemma
8m 2 N, (x, v ) ! γm (s (x, v )) belongs to L00 .
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Chapter 3
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A troncation procedure
For each m 2 N, let γm : [0, +∞[ ! [0, 1] be smooth (say C 1 ) and such
that
1 if s m
γm (s ) =
0 if s > 2m.
Lemma
8m 2 N, (x, v ) ! γm (s (x, v )) belongs to L00 .
Proof: We have just to show that (x, v ) ! γm (s (x, v )) belongs to
L .
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Chapter 3
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A troncation procedure
For each m 2 N, let γm : [0, +∞[ ! [0, 1] be smooth (say C 1 ) and such
that
1 if s m
γm (s ) =
0 if s > 2m.
Lemma
8m 2 N, (x, v ) ! γm (s (x, v )) belongs to L00 .
Proof: We have just to show that (x, v ) ! γm (s (x, v )) belongs to
L . Since γm is Lipschitz then
jγm (s (x + tv , v ))
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γm (s (x, v ))j = jγm (s (x, v ) + t )
γm (s (x, v ))j
Ct 8(x, v ).
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Chapter 3
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Proof of the theorem:
Let f 2 L0 then 8m 2 N, (x, v ) ! γm (s (x, v ))f (x, v ) belongs to L00 .
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Chapter 3
46 / 52
Proof of the theorem:
Let f 2 L0 then 8m 2 N, (x, v ) ! γm (s (x, v ))f (x, v ) belongs to L00 .
jγm (s (x, v ))f (x, v )
f (x, v )j = j(1
γm (s (x, v )))f (x, v )j
sup
s (x ,v )>2m
jf (x, v )j ! 0 as m ! ∞
since f 2 L0 .
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Chapter 3
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Consequences
Let ω 0 be the type of S0 (t )
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t >0
.
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Chapter 3
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Consequences
Let ω 0 be the type of S0 (t )
t >0
.
By the general theory
σ(T0 ) = Re λ
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ω0
n
, σ(S0 (t )) = µ; jµj
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o
e ω0 t .
Chapter 3
47 / 52
Consequences
Let ω 0 be the type of S0 (t )
t >0
.
By the general theory
σ(T0 ) = Re λ
ω0
n
, σ(S0 (t )) = µ; jµj
o
e ω0 t .
We have to identify ω 0 !
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Chapter 3
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Consequences
Let ω 0 be the type of S0 (t )
t >0
.
By the general theory
σ(T0 ) = Re λ
ω0
n
, σ(S0 (t )) = µ; jµj
o
e ω0 t .
We have to identify ω 0 !
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Chapter 3
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Consequences
Let ω 0 be the type of S0 (t )
t >0
.
By the general theory
σ(T0 ) = Re λ
ω0
n
, σ(S0 (t )) = µ; jµj
o
e ω0 t .
We have to identify ω 0 !
S0 ( t )
kS20 (t )k = kS2 (t )k so
ω0
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ω 2 = type of (S2 (t ))t >0 .
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Chapter 3
47 / 52
On the other hand,
gm : (x, v ) ! γm (s (x, v ))
belongs to L0 and kgm k
S0 ( t )
>
=
1 so that
S0 (t )gm = sup e
(y ,v )
sup e
(y ,v )
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Rt
0
σ(y +τv ,v )d τ
Rt
0
σ (y +τv ,v )d τ
γm (s (y + tv , v ))
γm (s (y , v ) + t )
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8m 2 N.
Chapter 3
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On the other hand,
gm : (x, v ) ! γm (s (x, v ))
belongs to L0 and kgm k
S0 ( t )
>
=
1 so that
Rt
S0 (t )gm = sup e
0
(y ,v )
sup e
(y ,v )
Rt
0
σ(y +τv ,v )d τ
But γm (s (y , v ) + t ) = 1 if s (y , v )
S0 ( t ) >
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sup
fs (y ,v ) m t g
γm (s (y + tv , v ))
γm (s (y , v ) + t )
m
e
σ (y +τv ,v )d τ
Rt
0
8m 2 N.
t so
σ(y +τv ,v )d τ
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8m 2 N.
Chapter 3
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Finally
S0 ( t ) >
sup
e
fs (y ,v )<+∞g
Rt
0
σ(y +τv ,v )d τ
= kS2 (t )k
whence
ω2
ω0
and we are done.
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Chapter 3
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General vector …elds
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Chapter 3
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General vector …elds
We consider
F : Rn ! Rn
be a Lipschitz vector …eld.
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Chapter 3
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General vector …elds
We consider
F : Rn ! Rn
be a Lipschitz vector …eld. We denote by
Φ(x, t )
the unique globale solution to
d
X (t ) = F (X (t )), t 2 R
dt
X (0) = x.
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Let Ω
Rn be an open set and let
s := inf fs > 0; Φ(x,
s) 2
/ Ωg
be the exit times from Ω (with the convention that inf ∅ = +∞).
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Let Ω
Rn be an open set and let
s := inf fs > 0; Φ(x,
s) 2
/ Ωg
be the exit times from Ω (with the convention that inf ∅ = +∞). We
de…ne the weighted shift semigroup
U (t ) : f ! U (t )f
where
U (t )f = e
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Rt
0
ν(Φ(x , s ))ds
f (Φ(x,
t ))χft <s
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(x )g (x ).
Chapter 3
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We introduce the sets
Ω1 = fx 2 Ω; s+ (x ) < ∞g , Ω2 = fx 2 Ω; s+ (x ) = ∞, s (x ) < ∞g
and
Ω3 = fx 2 Ω; s+ (x ) = ∞, s (x ) = ∞g .
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Chapter 3
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We introduce the sets
Ω1 = fx 2 Ω; s+ (x ) < ∞g , Ω2 = fx 2 Ω; s+ (x ) = ∞, s (x ) < ∞g
and
Ω3 = fx 2 Ω; s+ (x ) = ∞, s (x ) = ∞g .
Then Lp (Ωi ) (i = 1, 2, 3) are invariant under (U (t ))t >0 and we can
extend the previous spectral theory of streaming semigroups.
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Chapter 3
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We introduce the sets
Ω1 = fx 2 Ω; s+ (x ) < ∞g , Ω2 = fx 2 Ω; s+ (x ) = ∞, s (x ) < ∞g
and
Ω3 = fx 2 Ω; s+ (x ) = ∞, s (x ) = ∞g .
Then Lp (Ωi ) (i = 1, 2, 3) are invariant under (U (t ))t >0 and we can
extend the previous spectral theory of streaming semigroups. see: B. Lods,
M. M-K and M. Sbihi, Comm Pure Appl Anal, 8 (5) (2009) 1-24.
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Chapter 3
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