SPECTRAL THEORY FOR NEUTRON TRANSPORT
FUNDAMENTALS OF SPECTRAL THEORY
Mustapha Mokhtar-Kharroubi
(In memory of Seiji Ukaï)
Chapter 2
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Abstract
This chapter provides an overview (mostly without proof) of the
fundamental concepts and results on spectral theory of closed linear
operators on complex Banach spaces with a special emphasis on
generators of strongly continuous semigroups.
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Abstract
This chapter provides an overview (mostly without proof) of the
fundamental concepts and results on spectral theory of closed linear
operators on complex Banach spaces with a special emphasis on
generators of strongly continuous semigroups. Because of their
importance in transport theory, the basic spectral properties of positive
operators (i.e. leaving invariant the positive cone of a Banach lattice) are
also given.
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Abstract
This chapter provides an overview (mostly without proof) of the
fundamental concepts and results on spectral theory of closed linear
operators on complex Banach spaces with a special emphasis on
generators of strongly continuous semigroups. Because of their
importance in transport theory, the basic spectral properties of positive
operators (i.e. leaving invariant the positive cone of a Banach lattice) are
also given. Finally, we show the role of peripheral spectral theory of
positive semigroups in their time asymptotic behaviour as t ! +∞.
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Chapter 2
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General references
1
E-B. Davies. One-parameter semigroups, Academic Press, (1980).
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General references
1
E-B. Davies. One-parameter semigroups, Academic Press, (1980).
2
K-J. Engel and R. Nagel. One-parameter semigroups for linear
evolution equations, Springer-Verlag, (2000).
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Chapter 2
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General references
1
E-B. Davies. One-parameter semigroups, Academic Press, (1980).
2
K-J. Engel and R. Nagel. One-parameter semigroups for linear
evolution equations, Springer-Verlag, (2000).
3
I. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of linear
operators, Vol I, Birkhauser-Verlag, (1990).
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Chapter 2
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General references
1
E-B. Davies. One-parameter semigroups, Academic Press, (1980).
2
K-J. Engel and R. Nagel. One-parameter semigroups for linear
evolution equations, Springer-Verlag, (2000).
3
I. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of linear
operators, Vol I, Birkhauser-Verlag, (1990).
4
T. Kato. Perturbation theory of linear operators, Springer-Verlag,
(1984).
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Chapter 2
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General references
1
E-B. Davies. One-parameter semigroups, Academic Press, (1980).
2
K-J. Engel and R. Nagel. One-parameter semigroups for linear
evolution equations, Springer-Verlag, (2000).
3
I. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of linear
operators, Vol I, Birkhauser-Verlag, (1990).
4
T. Kato. Perturbation theory of linear operators, Springer-Verlag,
(1984).
5
R. Nagel (Ed). One-parameter semigroups of positive operators, Lect.
Notes in Math, Vol 1184, Springer-Verlag, (1986).
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Chapter 2
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General references
1
E-B. Davies. One-parameter semigroups, Academic Press, (1980).
2
K-J. Engel and R. Nagel. One-parameter semigroups for linear
evolution equations, Springer-Verlag, (2000).
3
I. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of linear
operators, Vol I, Birkhauser-Verlag, (1990).
4
T. Kato. Perturbation theory of linear operators, Springer-Verlag,
(1984).
5
R. Nagel (Ed). One-parameter semigroups of positive operators, Lect.
Notes in Math, Vol 1184, Springer-Verlag, (1986).
6
A. E. Taylor and D. C. Lay. Introduction to functional analysis,
Krieger publishing company, (1980).
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The …rst words
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The …rst words
Let X be a complex Banach space and let
T : D (T )
X !X
be a closed linear operator with domain D (T ).
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The …rst words
Let X be a complex Banach space and let
T : D (T )
X !X
be a closed linear operator with domain D (T ).
The resolvent set of T
ρ(T ) := fλ 2 C; λ
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T : D (T ) ! X is bijectiveg .
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The …rst words
Let X be a complex Banach space and let
T : D (T )
X !X
be a closed linear operator with domain D (T ).
The resolvent set of T
ρ(T ) := fλ 2 C; λ
T : D (T ) ! X is bijectiveg .
The spectrum of T
σ(T ) := fλ 2 C; λ 2
/ ρ(T )g .
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The …rst words
Let X be a complex Banach space and let
T : D (T )
X !X
be a closed linear operator with domain D (T ).
The resolvent set of T
ρ(T ) := fλ 2 C; λ
T : D (T ) ! X is bijectiveg .
The spectrum of T
σ(T ) := fλ 2 C; λ 2
/ ρ(T )g .
The resolvent operator
(λ
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T)
1
: X ! X (λ 2 ρ(T )).
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BASIC EXAMPLE: If Tx = λx for some x 2 D (T ), x 6= 0, then
λ 2 σ (T ).
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BASIC EXAMPLE: If Tx = λx for some x 2 D (T ), x 6= 0, then
λ 2 σ (T ).
λ is an eigenvalue and
ker(T ) := fx 2 D (T ); (T
λ )x = 0g
is the corresponding eigenspace.
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BASIC EXAMPLE: If Tx = λx for some x 2 D (T ), x 6= 0, then
λ 2 σ (T ).
λ is an eigenvalue and
ker(T ) := fx 2 D (T ); (T
λ )x = 0g
is the corresponding eigenspace.
In contrast to …nite dimensional spaces, in general σ(T ) is not reduced to
eigenvalues !
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BASIC EXAMPLE: If Tx = λx for some x 2 D (T ), x 6= 0, then
λ 2 σ (T ).
λ is an eigenvalue and
ker(T ) := fx 2 D (T ); (T
λ )x = 0g
is the corresponding eigenspace.
In contrast to …nite dimensional spaces, in general σ(T ) is not reduced to
eigenvalues ! For instance, one can show that the spectrum of the
multiplication operator
T : f 2 C [0, 1] ! Tf 2 C [0, 1]
where Tf (x ) = xf (x ) is equal to [0, 1].
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BASIC EXAMPLE: If Tx = λx for some x 2 D (T ), x 6= 0, then
λ 2 σ (T ).
λ is an eigenvalue and
ker(T ) := fx 2 D (T ); (T
λ )x = 0g
is the corresponding eigenspace.
In contrast to …nite dimensional spaces, in general σ(T ) is not reduced to
eigenvalues ! For instance, one can show that the spectrum of the
multiplication operator
T : f 2 C [0, 1] ! Tf 2 C [0, 1]
where Tf (x ) = xf (x ) is equal to [0, 1]. But one can see that
xf (x ) = λf (x ) would imply λ = x with x belonging to some interval !!
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The …rst results
(λ T ) 1 : X ! X is a bounded operator, i.e. (λ
(closed graph theorem).
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T)
1
2 L(X ),
Chapter 2
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The …rst results
(λ T ) 1 : X ! X is a bounded operator, i.e. (λ
(closed graph theorem).
ρ(T ) is an open subset of C
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T)
1
2 L(X ),
Chapter 2
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The …rst results
(λ T ) 1 : X ! X is a bounded operator, i.e. (λ
(closed graph theorem).
ρ(T ) is an open subset of C
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T)
1
2 L(X ),
Chapter 2
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The …rst results
(λ T ) 1 : X ! X is a bounded operator, i.e. (λ
(closed graph theorem).
ρ(T ) is an open subset of C (so σ(T ) is closed)
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T)
1
2 L(X ),
Chapter 2
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The …rst results
(λ T ) 1 : X ! X is a bounded operator, i.e. (λ T )
(closed graph theorem).
ρ(T ) is an open subset of C (so σ(T ) is closed) and
λ 2 ρ (T ) ! ( λ
T)
1
1
2 L(X ),
2 L(X )
is holomorphic.
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The …rst results
(λ T ) 1 : X ! X is a bounded operator, i.e. (λ T )
(closed graph theorem).
ρ(T ) is an open subset of C (so σ(T ) is closed) and
λ 2 ρ (T ) ! ( λ
T)
1
1
2 L(X ),
2 L(X )
is holomorphic. More precisely, if µ 2 ρ(T ) then λ 2 ρ(T ) if
1
and then
jλ µj < (µ T ) 1
(λ
T)
1
+∞
=
∑ (µ
λ )n ( µ
T)
1 n +1
.
0
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The …rst results
(λ T ) 1 : X ! X is a bounded operator, i.e. (λ T )
(closed graph theorem).
ρ(T ) is an open subset of C (so σ(T ) is closed) and
λ 2 ρ (T ) ! ( λ
T)
1
1
2 L(X ),
2 L(X )
is holomorphic. More precisely, if µ 2 ρ(T ) then λ 2 ρ(T ) if
1
and then
jλ µj < (µ T ) 1
(λ
T)
1
+∞
=
∑ (µ
λ )n ( µ
T)
1 n +1
.
0
A consequence: if λ 2 σ(T ) then jλ
µj > (µ
dist (µ, σ(T )) > (µ
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T)
1
T)
1
1
1
so
.
Chapter 2
6 / 57
The …rst results
(λ T ) 1 : X ! X is a bounded operator, i.e. (λ T )
(closed graph theorem).
ρ(T ) is an open subset of C (so σ(T ) is closed) and
λ 2 ρ (T ) ! ( λ
T)
1
1
2 L(X ),
2 L(X )
is holomorphic. More precisely, if µ 2 ρ(T ) then λ 2 ρ(T ) if
1
and then
jλ µj < (µ T ) 1
(λ
T)
1
+∞
=
∑ (µ
λ )n ( µ
T)
1 n +1
.
0
A consequence: if λ 2 σ(T ) then jλ
µj > (µ
dist (µ, σ(T )) > (µ
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T)
1
T)
1
1
1
so
.
Chapter 2
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The …rst results
(λ T ) 1 : X ! X is a bounded operator, i.e. (λ T )
(closed graph theorem).
ρ(T ) is an open subset of C (so σ(T ) is closed) and
λ 2 ρ (T ) ! ( λ
T)
1
1
2 L(X ),
2 L(X )
is holomorphic. More precisely, if µ 2 ρ(T ) then λ 2 ρ(T ) if
1
and then
jλ µj < (µ T ) 1
(λ
T)
1
+∞
=
∑ (µ
λ )n ( µ
T)
1 n +1
.
0
A consequence: if λ 2 σ(T ) then jλ
µj > (µ
dist (µ, σ(T )) > (µ
In particular (µ
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T)
1
T)
1
T)
1
1
1
so
.
! ∞ as dist (µ, σ(T )) ! 0.
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A remark
For unbounded operators, the spectrum may empty or equal to C !!
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A remark
For unbounded operators, the spectrum may empty or equal to C !! For
example, let X = C ([0, 1] ; C) endowed with the sup-norm and
Tf =
df
,
dx
D (T ) = C 1 ([0, 1]).
Then 8λ 2 C, x 2 [0, 1] ! e λx 2 C is an eigenfunction of T . So
σ(T ) = C.
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A remark
For unbounded operators, the spectrum may empty or equal to C !! For
example, let X = C ([0, 1] ; C) endowed with the sup-norm and
Tf =
df
,
dx
D (T ) = C 1 ([0, 1]).
Then 8λ 2 C, x 2 [0, 1] ! e λx 2 C is an eigenfunction of T . So
σ(T ) = C. If we replace (T , D (T )) by
b f = df ,
T
dx
b ) = f 2 C 1 ([0, 1]); f (0) = 0
D (T
then 8λ 2 C and 8g 2 C 1 ([0, 1]), the equation
λf
is uniquely solvable (f (x ) =
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df
= g , f (0) = 0
dx
R x λ (x s )
b ) = C.
e
g (s )ds) so ρ(T
0
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A remark
For unbounded operators, the spectrum may empty or equal to C !! For
example, let X = C ([0, 1] ; C) endowed with the sup-norm and
Tf =
df
,
dx
D (T ) = C 1 ([0, 1]).
Then 8λ 2 C, x 2 [0, 1] ! e λx 2 C is an eigenfunction of T . So
σ(T ) = C. If we replace (T , D (T )) by
b f = df ,
T
dx
b ) = f 2 C 1 ([0, 1]); f (0) = 0
D (T
then 8λ 2 C and 8g 2 C 1 ([0, 1]), the equation
df
= g , f (0) = 0
dx
R x λ (x s )
b ) = C.
is uniquely solvable (f (x ) =
e
g (s )ds) so ρ(T
0
One sees the key role of "boundary conditions" !
λf
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Bounded operators
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Bounded operators
The spectral radius of T 2 L(X )
1
1
rσ (T ) := sup fjλj ; λ 2 σ (T )g = lim kT n k n = inf kT n k n .
n !∞
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n
Chapter 2
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Bounded operators
The spectral radius of T 2 L(X )
1
1
rσ (T ) := sup fjλj ; λ 2 σ (T )g = lim kT n k n = inf kT n k n .
n !∞
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n
Chapter 2
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Bounded operators
The spectral radius of T 2 L(X )
1
1
rσ (T ) := sup fjλj ; λ 2 σ (T )g = lim kT n k n = inf kT n k n .
n !∞
In particular rσ (T )
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n
kT k .
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Bounded operators
The spectral radius of T 2 L(X )
1
1
rσ (T ) := sup fjλj ; λ 2 σ (T )g = lim kT n k n = inf kT n k n .
n !∞
In particular rσ (T )
kT k .
Laurent’s series
(λ
T)
1
∞
= ∑λ
n
Tn
1
1
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n
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(jλj > rσ (T ))
Chapter 2
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Bounded operators
The spectral radius of T 2 L(X )
1
1
rσ (T ) := sup fjλj ; λ 2 σ (T )g = lim kT n k n = inf kT n k n .
n !∞
In particular rσ (T )
kT k .
Laurent’s series
(λ
T)
1
∞
= ∑λ
n
Tn
1
1
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n
CIMPA School Muizemberg
(jλj > rσ (T ))
Chapter 2
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Bounded operators
The spectral radius of T 2 L(X )
1
1
rσ (T ) := sup fjλj ; λ 2 σ (T )g = lim kT n k n = inf kT n k n .
n !∞
In particular rσ (T )
kT k .
Laurent’s series
(λ
n
T)
1
∞
= ∑λ
n
Tn
1
1
with
T
m
1
=
2i π
Z
C
λm ( λ
(jλj > rσ (T ))
T)
1
dλ
where C is any circle (positively oriented) centered at the origin with
radius > rσ (T ).
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Bounded operators
The spectral radius of T 2 L(X )
1
1
rσ (T ) := sup fjλj ; λ 2 σ (T )g = lim kT n k n = inf kT n k n .
n !∞
In particular rσ (T )
kT k .
Laurent’s series
(λ
n
T)
1
∞
= ∑λ
n
Tn
1
1
with
T
m
1
=
2i π
Z
C
λm ( λ
(jλj > rσ (T ))
T)
1
dλ
where C is any circle (positively oriented) centered at the origin with
radius > rσ (T ).
If T 2 L(X ) then σ(T ) is bounded and non-empty.
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Classi…cation of the spectrum
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Classi…cation of the spectrum
Let T : D (T )
X ! X be a closed linear operator.
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Classi…cation of the spectrum
Let T : D (T )
X ! X be a closed linear operator.
The point spectrum
σp (T ) = fλ 2 C; λ
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T : D (T ) ! X is not injectiveg .
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Classi…cation of the spectrum
Let T : D (T )
X ! X be a closed linear operator.
The point spectrum
σp (T ) = fλ 2 C; λ
T : D (T ) ! X is not injectiveg .
The approximate point spectrum σap (T )
fλ 2 C; λ
T : D (T ) ! X not injective or (λ
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T )X not closedg .
Chapter 2
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Classi…cation of the spectrum
Let T : D (T )
X ! X be a closed linear operator.
The point spectrum
σp (T ) = fλ 2 C; λ
T : D (T ) ! X is not injectiveg .
The approximate point spectrum σap (T )
fλ 2 C; λ
T : D (T ) ! X not injective or (λ
λ 2 σap (T ) if and only if 9(xn )n
kxn k = 1, kTxn
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T )X not closedg .
D (T ) such that
λxn k ! 0.
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Classi…cation of the spectrum
Let T : D (T )
X ! X be a closed linear operator.
The point spectrum
σp (T ) = fλ 2 C; λ
T : D (T ) ! X is not injectiveg .
The approximate point spectrum σap (T )
fλ 2 C; λ
T : D (T ) ! X not injective or (λ
λ 2 σap (T ) if and only if 9(xn )n
kxn k = 1, kTxn
T )X not closedg .
D (T ) such that
λxn k ! 0.
The residual spectrum σres (T )
fλ 2 C; λ
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T : D (T ) ! X ; ( λ
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T )X is not denseg .
Chapter 2
9 / 57
Classi…cation of the spectrum
Let T : D (T )
X ! X be a closed linear operator.
The point spectrum
σp (T ) = fλ 2 C; λ
T : D (T ) ! X is not injectiveg .
The approximate point spectrum σap (T )
fλ 2 C; λ
T : D (T ) ! X not injective or (λ
λ 2 σap (T ) if and only if 9(xn )n
kxn k = 1, kTxn
T )X not closedg .
D (T ) such that
λxn k ! 0.
The residual spectrum σres (T )
fλ 2 C; λ
T : D (T ) ! X ; ( λ
T )X is not denseg .
σ(T ) = σres (T ) [ σap (T ) (non-disjoint union).
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Duality
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Chapter 2
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Duality
Let T : D (T )
X ! X be a closed densely de…ned linear operator.
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Duality
Let T : D (T ) X ! X be a closed densely de…ned linear operator.
We can de…ne its dual operator
T 0 : D (T 0 )
X0 ! X0
by
hTx, y 0 iX ,X 0 = hx, T 0 y 0 iX ,X 0
with domain
D (T 0 ) = y 0 2 X 0 ; 9c > 0,
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hTx, y 0 i
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c kx k 8x 2 D (T ) .
Chapter 2
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Duality
Let T : D (T ) X ! X be a closed densely de…ned linear operator.
We can de…ne its dual operator
T 0 : D (T 0 )
X0 ! X0
by
hTx, y 0 iX ,X 0 = hx, T 0 y 0 iX ,X 0
with domain
D (T 0 ) = y 0 2 X 0 ; 9c > 0,
hTx, y 0 i
c kx k 8x 2 D (T ) .
(T 0 is closed but not necessarily densely de…ned. If X is re‡exive then T 0
0
is densely de…ned and (T 0 ) = T .)
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Chapter 2
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Duality
Let T : D (T ) X ! X be a closed densely de…ned linear operator.
We can de…ne its dual operator
T 0 : D (T 0 )
X0 ! X0
by
hTx, y 0 iX ,X 0 = hx, T 0 y 0 iX ,X 0
with domain
D (T 0 ) = y 0 2 X 0 ; 9c > 0,
hTx, y 0 i
c kx k 8x 2 D (T ) .
(T 0 is closed but not necessarily densely de…ned. If X is re‡exive then T 0
0
is densely de…ned and (T 0 ) = T .)
σ(T 0 ) = σ(T ) and (λ T 0 )
spectrum is not the same !)
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1
= (λ
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T)
1 0
. (The nature of the
Chapter 2
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Duality
Let T : D (T ) X ! X be a closed densely de…ned linear operator.
We can de…ne its dual operator
T 0 : D (T 0 )
X0 ! X0
by
hTx, y 0 iX ,X 0 = hx, T 0 y 0 iX ,X 0
with domain
D (T 0 ) = y 0 2 X 0 ; 9c > 0,
hTx, y 0 i
c kx k 8x 2 D (T ) .
(T 0 is closed but not necessarily densely de…ned. If X is re‡exive then T 0
0
is densely de…ned and (T 0 ) = T .)
0
σ(T 0 ) = σ(T ) and (λ T 0 ) 1 = (λ T ) 1 . (The nature of the
spectrum is not the same !)
In particular if T 2 L(X ) then rσ (T 0 ) = rσ (T ).
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Dunford calculus for bounded operators
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Dunford calculus for bounded operators
Let T 2 L(X ) and let Ω 3 λ ! f (λ) 2 C be holomorphic on some open
neigborhood Ω of σ(T ).
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Dunford calculus for bounded operators
Let T 2 L(X ) and let Ω 3 λ ! f (λ) 2 C be holomorphic on some open
neigborhood Ω of σ(T ). Then there exists an open set ω such that
σ(T ) ω ω Ω and ∂ω consists of …nitely many simple closed
curves that do not intersect.
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Dunford calculus for bounded operators
Let T 2 L(X ) and let Ω 3 λ ! f (λ) 2 C be holomorphic on some open
neigborhood Ω of σ(T ). Then there exists an open set ω such that
σ(T ) ω ω Ω and ∂ω consists of …nitely many simple closed
curves that do not intersect. One de…nes
f (T ) =
1
2i π
Z
f (λ)(λ
T)
∂ω
1
d λ 2 L(X )
where ∂ω is properly oriented (the de…nition does not depend on the
choice of ω)
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Chapter 2
11 / 57
Dunford calculus for bounded operators
Let T 2 L(X ) and let Ω 3 λ ! f (λ) 2 C be holomorphic on some open
neigborhood Ω of σ(T ). Then there exists an open set ω such that
σ(T ) ω ω Ω and ∂ω consists of …nitely many simple closed
curves that do not intersect. One de…nes
f (T ) =
1
2i π
Z
f (λ)(λ
T)
∂ω
1
d λ 2 L(X )
where ∂ω is properly oriented (the de…nition does not depend on the
choice of ω) and we have
σ(f (T )) = f (σ(T )) (a spectral mapping theorem).
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Chapter 2
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On reducibility
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Chapter 2
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On reducibility
Let X be a complex Banach space such that
X = X1
X2
(direct sum) where Xi (i = 1, 2) are closed subspaces.
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Chapter 2
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On reducibility
Let X be a complex Banach space such that
X = X1
X2
(direct sum) where Xi (i = 1, 2) are closed subspaces. We denote by
P : x 2 X ! Px the (continuous) projection on X1 along X2 .
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Chapter 2
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On reducibility
Let X be a complex Banach space such that
X = X1
X2
(direct sum) where Xi (i = 1, 2) are closed subspaces. We denote by
P : x 2 X ! Px the (continuous) projection on X1 along X2 . Let
T : D (T )
X !X
be a closed inear operator such that P (D (T ))
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D (T ).
Chapter 2
12 / 57
On reducibility
Let X be a complex Banach space such that
X = X1
X2
(direct sum) where Xi (i = 1, 2) are closed subspaces. We denote by
P : x 2 X ! Px the (continuous) projection on X1 along X2 . Let
T : D (T )
X !X
be a closed inear operator such that P (D (T ))
parts Ti (i = 1, 2) of T on Xi (i = 1, 2) by
D (T ). We de…ne the
D (Ti ) = D (T ) \ Xi , Ti x = Tx (x 2 D (Ti ));
we say that that T is reduced by Xi (i = 1, 2).
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Chapter 2
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On reducibility
Let X be a complex Banach space such that
X = X1
X2
(direct sum) where Xi (i = 1, 2) are closed subspaces. We denote by
P : x 2 X ! Px the (continuous) projection on X1 along X2 . Let
T : D (T )
X !X
be a closed inear operator such that P (D (T ))
parts Ti (i = 1, 2) of T on Xi (i = 1, 2) by
D (T ). We de…ne the
D (Ti ) = D (T ) \ Xi , Ti x = Tx (x 2 D (Ti ));
we say that that T is reduced by Xi (i = 1, 2).
Then Ti (i = 1, 2) are closed operators on Xi (i = 1, 2),
σ(T ) = σ(T1 ) [ σ(T1 ) (not necessarily a disjoint union).,
σp (T ) = σp (T1 ) [ σp (T1 ) and σap (T ) = σap (T1 ) [ σap (T1 ), (see e.g.
Taylor- Lay, Theorem 5.4, p. 289).
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Chapter 2
12 / 57
On reducibility
Let X be a complex Banach space such that
X = X1
X2
(direct sum) where Xi (i = 1, 2) are closed subspaces. We denote by
P : x 2 X ! Px the (continuous) projection on X1 along X2 . Let
T : D (T )
X !X
be a closed inear operator such that P (D (T ))
parts Ti (i = 1, 2) of T on Xi (i = 1, 2) by
D (T ). We de…ne the
D (Ti ) = D (T ) \ Xi , Ti x = Tx (x 2 D (Ti ));
we say that that T is reduced by Xi (i = 1, 2).
Then Ti (i = 1, 2) are closed operators on Xi (i = 1, 2),
σ(T ) = σ(T1 ) [ σ(T1 ) (not necessarily a disjoint union).,
σp (T ) = σp (T1 ) [ σp (T1 ) and σap (T ) = σap (T1 ) [ σap (T1 ), (see e.g.
Taylor- Lay, Theorem 5.4, p. 289). Similar results hold for any …nite direct
sum: X = X1 ... Xn .
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Chapter 2
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Spectral decomposition and Riesz projection
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Chapter 2
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Spectral decomposition and Riesz projection
Let T : D (T ) X ! X be closed linear operator such that σ(T ) is a
disjoint union of two non-empty closed subsets σ1 and σ2 .
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Chapter 2
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Spectral decomposition and Riesz projection
Let T : D (T ) X ! X be closed linear operator such that σ(T ) is a
disjoint union of two non-empty closed subsets σ1 and σ2 . Let σ1 be
compact.
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Chapter 2
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Spectral decomposition and Riesz projection
Let T : D (T ) X ! X be closed linear operator such that σ(T ) is a
disjoint union of two non-empty closed subsets σ1 and σ2 . Let σ1 be
compact. Then there exists a …nite number of rectifable simple closed
curves enclosing an open set O which contains σ1 and such that σ2 is
included in the exterior of O.
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Chapter 2
13 / 57
Spectral decomposition and Riesz projection
Let T : D (T ) X ! X be closed linear operator such that σ(T ) is a
disjoint union of two non-empty closed subsets σ1 and σ2 . Let σ1 be
compact. Then there exists a …nite number of rectifable simple closed
curves enclosing an open set O which contains σ1 and such that σ2 is
included in the exterior of O. Then
P :=
Z
Γ
(λ
T)
1
d λ; P 2 = P
and X = X1 X2 (X1 = PX and X2 = (I P )X = KerP) reduces T (i.e.
Xi are T invariant), σ(Ti ) = σi where Ti := TjX i and T1 is bounded.
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Chapter 2
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Spectral decomposition and Riesz projection
Let T : D (T ) X ! X be closed linear operator such that σ(T ) is a
disjoint union of two non-empty closed subsets σ1 and σ2 . Let σ1 be
compact. Then there exists a …nite number of rectifable simple closed
curves enclosing an open set O which contains σ1 and such that σ2 is
included in the exterior of O. Then
P :=
Z
Γ
(λ
T)
1
d λ; P 2 = P
and X = X1 X2 (X1 = PX and X2 = (I P )X = KerP) reduces T (i.e.
Xi are T invariant), σ(Ti ) = σi where Ti := TjX i and T1 is bounded. P
is the spectral projection associated with σ1 .
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Chapter 2
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Isolated singularities
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Chapter 2
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Isolated singularities
If σ1 consists of …nitely many points (λ1 , ..., λn ) then
P = P1 + ... + Pn ,
Pj :=
Z
Γj
(λ
Pj Pk = δjk Pj
T)
1
dλ
(where Γj is e.g. a small circle enclosing λj ). Pj is the spectral projection
associated with λj .
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Chapter 2
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Structure of the resolvent around an isolated singularity
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Chapter 2
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Structure of the resolvent around an isolated singularity
Let µ 2 σ(T ) be an isolated point of σ(T ).
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Chapter 2
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Structure of the resolvent around an isolated singularity
Let µ 2 σ(T ) be an isolated point of σ(T ). A Laurent’s series
(λ
T)
1
+∞
=
∑
(λ
n= ∞
where
An =
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1
2i π
Z
C
(λ
(λ
T) 1
dλ
µ )n +1
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µ)n An
(n 2 Z).
Chapter 2
15 / 57
Structure of the resolvent around an isolated singularity
Let µ 2 σ(T ) be an isolated point of σ(T ). A Laurent’s series
(λ
T)
1
+∞
=
∑
(λ
n= ∞
where
An =
1
2i π
Z
C
T) 1
dλ
µ )n +1
(λ
(λ
µ)n An
(n 2 Z).
In particular, the residues
A
1
1
=
2i π
Z
C
(λ
T)
1
dλ
is the spectral projection P.
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Chapter 2
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Structure of the resolvent around an isolated singularity
Let µ 2 σ(T ) be an isolated point of σ(T ). A Laurent’s series
(λ
T)
1
+∞
=
∑
(λ
n= ∞
where
An =
1
2i π
Z
C
T) 1
dλ
µ )n +1
(λ
(λ
µ)n An
(n 2 Z).
In particular, the residues
A
1
1
=
2i π
Z
C
(λ
T)
1
dλ
is the spectral projection P. In addition
U
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(n +1 )
= ( 1)n ( µ
T )n P (n > 0).
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Chapter 2
15 / 57
Isolated singularity with …nite rank spectral projection
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Chapter 2
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Isolated singularity with …nite rank spectral projection
We have
U
(n +1 ) U (m +1 )
=U
(n +m +1 )
so µ is a pole of the resolvent (i.e. there exists k > 0 such that
U k 6= 0 and U n = 0 8n > k) if and only if there exists k > 0 such that
U k 6= 0 and U (k +1 ) = 0.
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Chapter 2
16 / 57
Isolated singularity with …nite rank spectral projection
We have
U
(n +1 ) U (m +1 )
=U
(n +m +1 )
so µ is a pole of the resolvent (i.e. there exists k > 0 such that
U k 6= 0 and U n = 0 8n > k) if and only if there exists k > 0 such that
U k 6= 0 and U (k +1 ) = 0. Then k is the order of the pole.
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Chapter 2
16 / 57
Isolated singularity with …nite rank spectral projection
We have
U
(n +1 ) U (m +1 )
=U
(n +m +1 )
so µ is a pole of the resolvent (i.e. there exists k > 0 such that
U k 6= 0 and U n = 0 8n > k) if and only if there exists k > 0 such that
U k 6= 0 and U (k +1 ) = 0. Then k is the order of the pole. In this case,
µ is an eigenvalue of T and PX = Ker (µ T )k .
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Chapter 2
16 / 57
Isolated singularity with …nite rank spectral projection
We have
U
(n +1 ) U (m +1 )
=U
(n +m +1 )
so µ is a pole of the resolvent (i.e. there exists k > 0 such that
U k 6= 0 and U n = 0 8n > k) if and only if there exists k > 0 such that
U k 6= 0 and U (k +1 ) = 0. Then k is the order of the pole. In this case,
µ is an eigenvalue of T and PX = Ker (µ T )k . The algebraic
multiplicity ma +∞ of µ is the dimension of PX .
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Chapter 2
16 / 57
Isolated singularity with …nite rank spectral projection
We have
U
(n +1 ) U (m +1 )
=U
(n +m +1 )
so µ is a pole of the resolvent (i.e. there exists k > 0 such that
U k 6= 0 and U n = 0 8n > k) if and only if there exists k > 0 such that
U k 6= 0 and U (k +1 ) = 0. Then k is the order of the pole. In this case,
µ is an eigenvalue of T and PX = Ker (µ T )k . The algebraic
multiplicity ma +∞ of µ is the dimension of PX . Conversely, if
ma < +∞, i.e. P is of …nite rank (or equivalently if P is compact), then
(µ T )m a P = 0 and then µ is a pole of the resolvent of order ma .
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Chapter 2
16 / 57
Isolated singularity with …nite rank spectral projection
We have
U
(n +1 ) U (m +1 )
=U
(n +m +1 )
so µ is a pole of the resolvent (i.e. there exists k > 0 such that
U k 6= 0 and U n = 0 8n > k) if and only if there exists k > 0 such that
U k 6= 0 and U (k +1 ) = 0. Then k is the order of the pole. In this case,
µ is an eigenvalue of T and PX = Ker (µ T )k . The algebraic
multiplicity ma +∞ of µ is the dimension of PX . Conversely, if
ma < +∞, i.e. P is of …nite rank (or equivalently if P is compact), then
(µ T )m a P = 0 and then µ is a pole of the resolvent of order ma .
Actually, the order k of the pole is the smallest j 2 N such that
(µ T )j P = 0.
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Chapter 2
16 / 57
Isolated singularity with …nite rank spectral projection
We have
U
(n +1 ) U (m +1 )
=U
(n +m +1 )
so µ is a pole of the resolvent (i.e. there exists k > 0 such that
U k 6= 0 and U n = 0 8n > k) if and only if there exists k > 0 such that
U k 6= 0 and U (k +1 ) = 0. Then k is the order of the pole. In this case,
µ is an eigenvalue of T and PX = Ker (µ T )k . The algebraic
multiplicity ma +∞ of µ is the dimension of PX . Conversely, if
ma < +∞, i.e. P is of …nite rank (or equivalently if P is compact), then
(µ T )m a P = 0 and then µ is a pole of the resolvent of order ma .
Actually, the order k of the pole is the smallest j 2 N such that
(µ T )j P = 0. The subspace Ker (µ T )k contains the generalized
eigenvectors;
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Chapter 2
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Isolated singularity with …nite rank spectral projection
We have
U
(n +1 ) U (m +1 )
=U
(n +m +1 )
so µ is a pole of the resolvent (i.e. there exists k > 0 such that
U k 6= 0 and U n = 0 8n > k) if and only if there exists k > 0 such that
U k 6= 0 and U (k +1 ) = 0. Then k is the order of the pole. In this case,
µ is an eigenvalue of T and PX = Ker (µ T )k . The algebraic
multiplicity ma +∞ of µ is the dimension of PX . Conversely, if
ma < +∞, i.e. P is of …nite rank (or equivalently if P is compact), then
(µ T )m a P = 0 and then µ is a pole of the resolvent of order ma .
Actually, the order k of the pole is the smallest j 2 N such that
(µ T )j P = 0. The subspace Ker (µ T )k contains the generalized
eigenvectors; it coincides with the eigenspace if and only if
PX = Ker (µ T ), i.e. k = 1 (simple pole); µ is said to be a semi-simple
eigenvalue.
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Chapter 2
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Isolated singularity with …nite rank spectral projection
We have
U
(n +1 ) U (m +1 )
=U
(n +m +1 )
so µ is a pole of the resolvent (i.e. there exists k > 0 such that
U k 6= 0 and U n = 0 8n > k) if and only if there exists k > 0 such that
U k 6= 0 and U (k +1 ) = 0. Then k is the order of the pole. In this case,
µ is an eigenvalue of T and PX = Ker (µ T )k . The algebraic
multiplicity ma +∞ of µ is the dimension of PX . Conversely, if
ma < +∞, i.e. P is of …nite rank (or equivalently if P is compact), then
(µ T )m a P = 0 and then µ is a pole of the resolvent of order ma .
Actually, the order k of the pole is the smallest j 2 N such that
(µ T )j P = 0. The subspace Ker (µ T )k contains the generalized
eigenvectors; it coincides with the eigenspace if and only if
PX = Ker (µ T ), i.e. k = 1 (simple pole); µ is said to be a semi-simple
eigenvalue. We say that µ is algebraically simple if ma = 1.
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Chapter 2
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Application to Riesz-Schauder theory
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Chapter 2
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Application to Riesz-Schauder theory
Let T : X ! X a compact operator (i.e. maps bounded sets into
relatively compact ones).
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Chapter 2
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Application to Riesz-Schauder theory
Let T : X ! X a compact operator (i.e. maps bounded sets into
relatively compact ones). Then σ(T )/ f0g consists at most of isolated
eigenvalues.
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Chapter 2
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Application to Riesz-Schauder theory
Let T : X ! X a compact operator (i.e. maps bounded sets into
relatively compact ones). Then σ(T )/ f0g consists at most of isolated
eigenvalues. Let α 2 σ(T ) with α 6= 0.
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Chapter 2
17 / 57
Application to Riesz-Schauder theory
Let T : X ! X a compact operator (i.e. maps bounded sets into
relatively compact ones). Then σ(T )/ f0g consists at most of isolated
eigenvalues. Let α 2 σ(T ) with α 6= 0. De…ne Tλ (in the neighborhood
of α) by
( λ T ) 1 = λ 1 + Tλ .
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Chapter 2
17 / 57
Application to Riesz-Schauder theory
Let T : X ! X a compact operator (i.e. maps bounded sets into
relatively compact ones). Then σ(T )/ f0g consists at most of isolated
eigenvalues. Let α 2 σ(T ) with α 6= 0. De…ne Tλ (in the neighborhood
of α) by
( λ T ) 1 = λ 1 + Tλ .
Then (λ T )(λ
compact.
1
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+ Tλ ) = I implies that Tλ = T (λ
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1
Tλ + λ
2
I ) is
Chapter 2
17 / 57
Application to Riesz-Schauder theory
Let T : X ! X a compact operator (i.e. maps bounded sets into
relatively compact ones). Then σ(T )/ f0g consists at most of isolated
eigenvalues. Let α 2 σ(T ) with α 6= 0. De…ne Tλ (in the neighborhood
of α) by
( λ T ) 1 = λ 1 + Tλ .
Then (λ T )(λ 1 + Tλ ) = I implies that Tλ = T (λ 1 Tλ + λ 2 I ) is
compact. So (C being a small circle around α) the spectral projection
A
1
=
=
Z
1
(λ T )
2i π C
Z
1
Tλ d λ
2i π C
1
1
dλ =
2i π
Z
C
λ
1
1
dλ +
2i π
Z
C
Tλ d λ
is compact too and then α has a …nite algebraic multiplicity.
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Chapter 2
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Extension to power compact operators
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Chapter 2
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Extension to power compact operators
Let T 2 L(X ) and n 2 N (n > 2) such that T n is compact.
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Chapter 2
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Extension to power compact operators
Let T 2 L(X ) and n 2 N (n > 2) such that T n is compact. The spectral
n
mapping theorem σ(T n ) = (σ(T )) implies that σ(T )/ f0g consists at
most of isolated points.
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Chapter 2
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Extension to power compact operators
Let T 2 L(X ) and n 2 N (n > 2) such that T n is compact. The spectral
n
mapping theorem σ(T n ) = (σ(T )) implies that σ(T )/ f0g consists at
most of isolated points. Let α 2 σ(T ) with α 6= 0.
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Chapter 2
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Extension to power compact operators
Let T 2 L(X ) and n 2 N (n > 2) such that T n is compact. The spectral
n
mapping theorem σ(T n ) = (σ(T )) implies that σ(T )/ f0g consists at
most of isolated points. Let α 2 σ(T ) with α 6= 0. Then, for λ close to
α, (λn T n ) = (λn 1 I + λn 2 T + ... + T n 1 )(λ T ) implies
(λ
T)
1
= (λn T n ) 1 (λn 1 I + λn 2 T + ... + T n 1 )
= λ n + Cλ (λn 1 I + λn 2 T + ... + T n 1 )
= λ n (λn 1 I + λn 2 T + ... + T n 1 )
+Cλ (λn 1 I + λn 2 T + ... + T n 1 )
(where Cλ is compact)
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Chapter 2
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Extension to power compact operators
Let T 2 L(X ) and n 2 N (n > 2) such that T n is compact. The spectral
n
mapping theorem σ(T n ) = (σ(T )) implies that σ(T )/ f0g consists at
most of isolated points. Let α 2 σ(T ) with α 6= 0. Then, for λ close to
α, (λn T n ) = (λn 1 I + λn 2 T + ... + T n 1 )(λ T ) implies
(λ
T)
1
= (λn T n ) 1 (λn 1 I + λn 2 T + ... + T n 1 )
= λ n + Cλ (λn 1 I + λn 2 T + ... + T n 1 )
= λ n (λn 1 I + λn 2 T + ... + T n 1 )
+Cλ (λn 1 I + λn 2 T + ... + T n 1 )
(where Cλ is compact) so the spectral projection
A
1
=
1
2i π
Z
C
(λ
T)
1
dλ =
1
2i π
Z
C
Cλ ( λ n
1
I + λn
2
T + ... + T n
1
)d λ
is compact.
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Chapter 2
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Extension to power compact operators
Let T 2 L(X ) and n 2 N (n > 2) such that T n is compact. The spectral
n
mapping theorem σ(T n ) = (σ(T )) implies that σ(T )/ f0g consists at
most of isolated points. Let α 2 σ(T ) with α 6= 0. Then, for λ close to
α, (λn T n ) = (λn 1 I + λn 2 T + ... + T n 1 )(λ T ) implies
(λ
T)
1
= (λn T n ) 1 (λn 1 I + λn 2 T + ... + T n 1 )
= λ n + Cλ (λn 1 I + λn 2 T + ... + T n 1 )
= λ n (λn 1 I + λn 2 T + ... + T n 1 )
+Cλ (λn 1 I + λn 2 T + ... + T n 1 )
(where Cλ is compact) so the spectral projection
A
1
=
1
2i π
Z
C
(λ
T)
1
dλ =
1
2i π
Z
C
Cλ ( λ n
1
I + λn
2
T + ... + T n
1
)d λ
is compact. Hence σ(T )/ f0g consists at most of isolated eigenvalues
with …nite algebraic multiplicities.
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Chapter 2
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Spectral mapping theorem for a resolvent
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Chapter 2
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Spectral mapping theorem for a resolvent
Let T : D (T )
X ! X be closed linear operator and λ0 2 ρ(T ).
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Chapter 2
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Spectral mapping theorem for a resolvent
Let T : D (T )
σ ( λ0
rσ ( λ 0
X ! X be closed linear operator and λ0 2 ρ(T ).
T)
T)
1
1
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n f0g = (λ0 σ(T )) 1 (so
= [dist (λ0 , σ(T ))] 1 )
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Chapter 2
19 / 57
Spectral mapping theorem for a resolvent
Let T : D (T )
σ ( λ0
rσ ( λ 0
σ p ( λ0
X ! X be closed linear operator and λ0 2 ρ(T ).
n f0g = (λ0 σ(T )) 1 (so
1 = dist ( λ , σ (T ))] 1 )
[
0
1
T)
n f0g = (λ0 σp (T )) 1
T)
T)
1
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Chapter 2
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Spectral mapping theorem for a resolvent
Let T : D (T )
σ ( λ0
rσ ( λ 0
σ p ( λ0
σap (λ0
X ! X be closed linear operator and λ0 2 ρ(T ).
n f0g = (λ0 σ(T )) 1 (so
1 = dist ( λ , σ (T ))] 1 )
[
0
1
T)
n f0g = (λ0 σp (T )) 1
T ) 1 n f0g = (λ0 σap (T )) 1
T)
T)
1
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Chapter 2
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Spectral mapping theorem for a resolvent
Let T : D (T )
σ ( λ0
rσ ( λ 0
σ p ( λ0
σap (λ0
σres (λ0
X ! X be closed linear operator and λ0 2 ρ(T ).
n f0g = (λ0 σ(T )) 1 (so
1 = dist ( λ , σ (T ))] 1 )
[
0
1
T)
n f0g = (λ0 σp (T )) 1
T ) 1 n f0g = (λ0 σap (T )) 1
T ) 1 n f0g = (λ0 σres (T )) 1
T)
T)
1
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Chapter 2
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Spectral mapping theorem for a resolvent
Let T : D (T )
X ! X be closed linear operator and λ0 2 ρ(T ).
n f0g = (λ0 σ(T )) 1 (so
1 = dist ( λ , σ (T ))] 1 )
[
0
1
σ p ( λ0 T )
n f0g = (λ0 σp (T )) 1
σap (λ0 T ) 1 n f0g = (λ0 σap (T )) 1
σres (λ0 T ) 1 n f0g = (λ0 σres (T )) 1
µ is an isolated point of σ(T ) if and only if (λ0 µ) 1 is an isolated
point of σ (λ0 T ) 1 . In this case, the residues and the orders of
1
the pole of (λ T ) 1 at µ and of λ (λ0 T ) 1
at
(λ0 µ) 1 coincide.
σ ( λ0
rσ ( λ 0
T)
T)
1
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Chapter 2
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Spectral mapping theorem for a resolvent
Let T : D (T )
X ! X be closed linear operator and λ0 2 ρ(T ).
n f0g = (λ0 σ(T )) 1 (so
1 = dist ( λ , σ (T ))] 1 )
[
0
1
σ p ( λ0 T )
n f0g = (λ0 σp (T )) 1
σap (λ0 T ) 1 n f0g = (λ0 σap (T )) 1
σres (λ0 T ) 1 n f0g = (λ0 σres (T )) 1
µ is an isolated point of σ(T ) if and only if (λ0 µ) 1 is an isolated
point of σ (λ0 T ) 1 . In this case, the residues and the orders of
1
the pole of (λ T ) 1 at µ and of λ (λ0 T ) 1
at
(λ0 µ) 1 coincide.
Application to Riesz-Schauder theory of resolvent compact operators.
σ ( λ0
rσ ( λ 0
T)
T)
1
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Chapter 2
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Fredholm operators
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Chapter 2
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Fredholm operators
A closed operator T : D (T ) X ! X is a Fredholm operator if
dim Ker (T ) < ∞ and the range R (T ) of T is closed with …nite
codimension (i.e. dim R (XT ) < ∞).
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Chapter 2
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Fredholm operators
A closed operator T : D (T ) X ! X is a Fredholm operator if
dim Ker (T ) < ∞ and the range R (T ) of T is closed with …nite
codimension (i.e. dim R (XT ) < ∞).
Let T : D (T )
domain is
X ! X be closed linear operator. Its Fredholm
ρF (T ) := fλ 2 C; λ
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T : D (T ) ! X is Fredholmg .
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Chapter 2
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Fredholm operators
A closed operator T : D (T ) X ! X is a Fredholm operator if
dim Ker (T ) < ∞ and the range R (T ) of T is closed with …nite
codimension (i.e. dim R (XT ) < ∞).
Let T : D (T )
domain is
X ! X be closed linear operator. Its Fredholm
ρF (T ) := fλ 2 C; λ
T : D (T ) ! X is Fredholmg .
ρF (T ) is open
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Chapter 2
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Fredholm operators
A closed operator T : D (T ) X ! X is a Fredholm operator if
dim Ker (T ) < ∞ and the range R (T ) of T is closed with …nite
codimension (i.e. dim R (XT ) < ∞).
Let T : D (T )
domain is
X ! X be closed linear operator. Its Fredholm
ρF (T ) := fλ 2 C; λ
T : D (T ) ! X is Fredholmg .
ρF (T ) is open
ρ (T )
ρF (T ).
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Chapter 2
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Fredholm operators
A closed operator T : D (T ) X ! X is a Fredholm operator if
dim Ker (T ) < ∞ and the range R (T ) of T is closed with …nite
codimension (i.e. dim R (XT ) < ∞).
Let T : D (T )
domain is
X ! X be closed linear operator. Its Fredholm
ρF (T ) := fλ 2 C; λ
T : D (T ) ! X is Fredholmg .
ρF (T ) is open
ρ (T )
ρF (T ).
If λ0 be an isolated eigenvalue of T with …nite algebraic multiplicity
then λ0 T : D (T ) ! X is Fredholm, (see Kato Chapter IV).
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Chapter 2
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Essential spectrum
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Chapter 2
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Essential spectrum
σess (T ) := CnρF (T ).
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Chapter 2
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The Calkin algebra
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Chapter 2
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The Calkin algebra
Let K(X )
L(X ) be the closed ideal of compact operators.
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Chapter 2
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The Calkin algebra
Let K(X )
algebra
L(X ) be the closed ideal of compact operators. The Calkin
C(X ) :=
L(X )
K(X )
b := T + K(X ))
is endowed with the quotient norm (for T
b
T
C(X )
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=
inf
K 2K(X )
kT + K k = dist (T , K(X )).
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Chapter 2
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On bounded Fredholm operators
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Chapter 2
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On bounded Fredholm operators
T 2 L(X ) is Fredholm if and only if there exists S 2 L(X ) such that
I ST and I TS are …nite rank operators (see Gohberg, Goldberg,
Kaashoek, p. 190).
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Chapter 2
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On bounded Fredholm operators
T 2 L(X ) is Fredholm if and only if there exists S 2 L(X ) such that
I ST and I TS are …nite rank operators (see Gohberg, Goldberg,
Kaashoek, p. 190).
b ) and σess (T ) = σ(T
b ).
ρ (T ) = ρ (T
F
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Chapter 2
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Essential norm
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Chapter 2
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Essential norm
The essential norm of T 2 L(X )
b
kT kess := T
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C(X )
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.
Chapter 2
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Essential norm
The essential norm of T 2 L(X )
b
kT kess := T
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C(X )
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.
Chapter 2
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Essential norm
The essential norm of T 2 L(X )
In particular, kT kess
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b
kT kess := T
C(X )
.
kT k .
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Chapter 2
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Essential norm
The essential norm of T 2 L(X )
In particular, kT kess
k.kess
b
kT kess := T
C(X )
.
kT k .
is submultiplicative, i.e.
kT1 T2 kess
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kT1 kess kT2 kess (Ti 2 L(X ), i = 1, 2).
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Chapter 2
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Essential radius
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Chapter 2
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Essential radius
The essential radius of T 2 L(X )
ress (T )
:
n
o
b ) = sup jλj ; λ 2 σ(T
b)
= rσ (T
= sup fjλj ; λ 2 σess (T )g .
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Chapter 2
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Essential radius
The essential radius of T 2 L(X )
ress (T )
:
n
o
b ) = sup jλj ; λ 2 σ(T
b)
= rσ (T
= sup fjλj ; λ 2 σess (T )g .
b ) = limn !∞
ress (T ) = rσ (T
1
n
limn !∞ kT n kess .
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b
T
n
1
n
C(X )
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= limn !∞
cn
T
1
n
C(X )
=
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25 / 57
Alternative de…nition of essential radius
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Chapter 2
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Alternative de…nition of essential radius
The unbounded component of ρF (T ) consists of resolvent set and at
most of isolated eigenvalues with …nite algebraic multiplicities, (see
Gohberg, Goldberg, Kaashoek, p. 204).
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Alternative de…nition of essential radius
The unbounded component of ρF (T ) consists of resolvent set and at
most of isolated eigenvalues with …nite algebraic multiplicities, (see
Gohberg, Goldberg, Kaashoek, p. 204).
The essential radius of T 2 L(X ) is given by
inf fr > 0; λ 2 σ(T ), jλj > r ) λ 2 σdiscr (T ) g
where σdiscr (T ) refers to the isolated eigenvalues of T with …nite
algebraic multiplicities.
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Chapter 2
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Alternative de…nition of essential radius
The unbounded component of ρF (T ) consists of resolvent set and at
most of isolated eigenvalues with …nite algebraic multiplicities, (see
Gohberg, Goldberg, Kaashoek, p. 204).
The essential radius of T 2 L(X ) is given by
inf fr > 0; λ 2 σ(T ), jλj > r ) λ 2 σdiscr (T ) g
where σdiscr (T ) refers to the isolated eigenvalues of T with …nite
algebraic multiplicities.
For any ε > 0
σ(T ) \ fjλj > ress (T ) + εg
consists at most of …nitely many eigenvalues with …nite algebraic
multiplicities.
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Chapter 2
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Semigroups and generators
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Chapter 2
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Semigroups and generators
A complex Banach space X and a C0 -semigroup (S (t ))t >0 , i.e.
S (t ) 2 L(X ), S (0) = I , S (t )S (s ) = S (t + s ) and 8x 2 X ,
t > 0 ! S (t )x 2 X is continuous.
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Chapter 2
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Semigroups and generators
A complex Banach space X and a C0 -semigroup (S (t ))t >0 , i.e.
S (t ) 2 L(X ), S (0) = I , S (t )S (s ) = S (t + s ) and 8x 2 X ,
t > 0 ! S (t )x 2 X is continuous. Its generator
T : x 2 D (T )
n
with D (T ) = x; limt !0
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S (t )x x
t
S (t )x
t !0
t
o
exists in X .
X ! lim
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x
2X
Chapter 2
27 / 57
Semigroups and generators
A complex Banach space X and a C0 -semigroup (S (t ))t >0 , i.e.
S (t ) 2 L(X ), S (0) = I , S (t )S (s ) = S (t + s ) and 8x 2 X ,
t > 0 ! S (t )x 2 X is continuous. Its generator
T : x 2 D (T )
n
with D (T ) = x; limt !0
S (t )x x
t
S (t )x
t !0
t
o
exists in X .
X ! lim
x
2X
T is closed and densely de…ned.
D (T ) is invariant under S (t ) and S (t )Tx = TS (t )x 8x 2 D (T ).
8x 2 D (T ), f : t > 0 ! S (t )x is C 1 and f 0 (t ) = Tf (t ), f (0) = x.
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Chapter 2
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Semigroups and generators
A complex Banach space X and a C0 -semigroup (S (t ))t >0 , i.e.
S (t ) 2 L(X ), S (0) = I , S (t )S (s ) = S (t + s ) and 8x 2 X ,
t > 0 ! S (t )x 2 X is continuous. Its generator
T : x 2 D (T )
n
with D (T ) = x; limt !0
S (t )x x
t
S (t )x
t !0
t
o
exists in X .
X ! lim
x
2X
T is closed and densely de…ned.
D (T ) is invariant under S (t ) and S (t )Tx = TS (t )x 8x 2 D (T ).
8x 2 D (T ), f : t > 0 ! S (t )x is C 1 and f 0 (t ) = Tf (t ), f (0) = x.
see e.g. B. Davies.
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Chapter 2
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On subadditive functions
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Chapter 2
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On subadditive functions
Let
p : R+ ! [ ∞, +∞[
be subadditive (i.e. p (t + s )
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p (t ) + p (s )) and locally bounded above.
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Chapter 2
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On subadditive functions
Let
p : R+ ! [ ∞, +∞[
be subadditive (i.e. p (t + s )
Then
p (t ) + p (s )) and locally bounded above.
p (t )
p (t )
= inf
.
t !+∞ t
t >0 t
lim
see e.g. B. Davies.
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Chapter 2
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On the type of a semigroup
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Chapter 2
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On the type of a semigroup
t = 0 ! p (t ) := ln(kS (t )k) 2 [ ∞, +∞[
is subadditive and locally bounded above.
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Chapter 2
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On the type of a semigroup
t = 0 ! p (t ) := ln(kS (t )k) 2 [ ∞, +∞[
is subadditive and locally bounded above. So
ω := inf
t >0
ln(kS (t )k)
ln(kS (t )k)
= lim
2 [ ∞, +∞[ .
t !+∞
t
t
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Chapter 2
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On the type of a semigroup
t = 0 ! p (t ) := ln(kS (t )k) 2 [ ∞, +∞[
is subadditive and locally bounded above. So
ω := inf
t >0
ln(kS (t )k)
ln(kS (t )k)
= lim
2 [ ∞, +∞[ .
t !+∞
t
t
In particular (S (t ))t >0 is exponentially bounded, i.e. 8α > ω 9Mα > 1
such that kS (t )k Mα e αt 8t = 0. (ω is called the type or growth bound
of (S (t ))t >0 ).
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Chapter 2
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On the type of a semigroup
t = 0 ! p (t ) := ln(kS (t )k) 2 [ ∞, +∞[
is subadditive and locally bounded above. So
ω := inf
t >0
ln(kS (t )k)
ln(kS (t )k)
= lim
2 [ ∞, +∞[ .
t !+∞
t
t
In particular (S (t ))t >0 is exponentially bounded, i.e. 8α > ω 9Mα > 1
such that kS (t )k Mα e αt 8t = 0. (ω is called the type or growth bound
of (S (t ))t >0 ). In addition, for any t > 0
rσ (S (t )) =
=
1
1
lim kS (t )n k n = lim kS (nt )k n
n !+∞
lim exp
n !+∞
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n !+∞
1
1
ln kS (nt )k = lim exp t ln kS (nt )k = e ωt .
n !+∞
n
nt
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Chapter 2
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Resolvent of the generator
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Chapter 2
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Resolvent of the generator
fRe λ > ω g
ρ(T ) and
(λ
T)
1
=
Z +∞
e
λt
S (t )dt
0
(Re λ > ω )
where the integral converges in operator norm.
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Chapter 2
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Resolvent of the generator
fRe λ > ω g
ρ(T ) and
(λ
T)
1
=
Z +∞
e
λt
S (t )dt
0
(Re λ > ω )
where the integral converges in operator norm. Thus σ(T )
and the spectral bound of T
s (T ) : sup fRe λ; λ 2 σ(T )g
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fRe λ
ωg
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30 / 57
ω
Resolvent of the generator
fRe λ > ω g
ρ(T ) and
(λ
T)
1
=
Z +∞
e
λt
S (t )dt
0
(Re λ > ω )
where the integral converges in operator norm. Thus σ(T )
and the spectral bound of T
s (T ) : sup fRe λ; λ 2 σ(T )g
fRe λ
ωg
ω
For any α > ω 9Mα > 1 and
(λ
T)
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1 n
Mα
(Re λ > α) 8n 2 N.
(Re λ α)n
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Chapter 2
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Hille-Yosida theorem
Theorem
Let T : D (T )
X ! X be a closed densely de…ned linear operator.
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Chapter 2
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Hille-Yosida theorem
Theorem
Let T : D (T )
(λ
X ! X be a closed densely de…ned linear operator. If
T)
1 n
Mα
(Re λ > α) 8n 2 N
(Re λ α)n
then there exists a C0 -semigroup (S (t ))t >0 with generator T satisfying
kS (t )k Mα e αt 8t = 0.
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Chapter 2
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Dual semigroup
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Chapter 2
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Dual semigroup
Let X be complex Banach space. Let (S (t ))t >0 be a C0 -semigroup with
generator T .
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Dual semigroup
Let X be complex Banach space. Let (S (t ))t >0 be a C0 -semigroup with
generator T . If X is re‡exive then (S 0 (t ))t >0 is a C0 -semigroup with
generator T 0 .
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Chapter 2
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Dual semigroup
Let X be complex Banach space. Let (S (t ))t >0 be a C0 -semigroup with
generator T . If X is re‡exive then (S 0 (t ))t >0 is a C0 -semigroup with
generator T 0 . In particular
ω = ω 0 and s (T ) = s (T 0 ).
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Chapter 2
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An identity for semigroups
Theorem
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T .
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Chapter 2
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An identity for semigroups
Theorem
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T .Then, for any λ 2 C,
e λt x
S (t )x
= (λ
=
Z t
0
T)
Z t
e λ (t
s)
0
e λ (t
s)
S (s )(λ
S (s )xds (x 2 X )
T )xds (x 2 D (T )).
(See, for e.g. Engel-Nagel, Chapter IV).
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Chapter 2
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Partial spectral mapping theorem for semigroups
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Partial spectral mapping theorem for semigroups
Theorem
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T .Then:
(i) e t σ(T ) σ(S (t ))n f0g .
(ii) e t σp (T ) = σp (S (t ))n f0g .
(iii) e t σres (T ) = σres (S (t ))n f0g .
(iv) mg (λ, T ) mg (e λt , S (t ))
(v) ma (λ, T ) ma (e λt , S (t ))
(v) k (λ, T ) k (e λt , S (t )).
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Chapter 2
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Partial spectral mapping theorem for semigroups
Theorem
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T .Then:
(i) e t σ(T ) σ(S (t ))n f0g .
(ii) e t σp (T ) = σp (S (t ))n f0g .
(iii) e t σres (T ) = σres (S (t ))n f0g .
(iv) mg (λ, T ) mg (e λt , S (t ))
(v) ma (λ, T ) ma (e λt , S (t ))
(v) k (λ, T ) k (e λt , S (t )).
(See, for e.g. Engel-Nagel, Chapter IV).
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Partial spectral mapping theorem for semigroups
Theorem
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T .Then:
(i) e t σ(T ) σ(S (t ))n f0g .
(ii) e t σp (T ) = σp (S (t ))n f0g .
(iii) e t σres (T ) = σres (S (t ))n f0g .
(iv) mg (λ, T ) mg (e λt , S (t ))
(v) ma (λ, T ) ma (e λt , S (t ))
(v) k (λ, T ) k (e λt , S (t )).
(See, for e.g. Engel-Nagel, Chapter IV). The possible failure of the
spectral mapping theorem stems from the approximate point
spectrum.
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Eigenspaces for semigroups, Part 1
Theorem
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T .
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Eigenspaces for semigroups, Part 1
Theorem
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T .Then:
(i) Ker (µ T ) = \t >0 Ker (e µt S (t )).
(ii) Ker (e µt S (t )) = linn 2Z Ker (µ + 2i tπn T ) 8t > 0.
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Eigenspaces for semigroups, Part 2
Theorem
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T and let t > 0 be …xed.
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Eigenspaces for semigroups, Part 2
Theorem
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T and let t > 0 be …xed. Let e µt be a pole of S (t ) of
order k and let Q be the corresponding residue.
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Eigenspaces for semigroups, Part 2
Theorem
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T and let t > 0 be …xed. Let e µt be a pole of S (t ) of
order k and let Q be the corresponding residue. Then
(i) For every n 2 Z, µ + 2i tπn is ( at most) a pole of (λ T ) 1 of order
at most k and residue Pn .
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Eigenspaces for semigroups, Part 2
Theorem
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T and let t > 0 be …xed. Let e µt be a pole of S (t ) of
order k and let Q be the corresponding residue. Then
(i) For every n 2 Z, µ + 2i tπn is ( at most) a pole of (λ T ) 1 of order
at most k and residue Pn .
(ii) QX = linn 2Z Pn X .
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Eigenspaces for semigroups, Part 2
Theorem
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T and let t > 0 be …xed. Let e µt be a pole of S (t ) of
order k and let Q be the corresponding residue. Then
(i) For every n 2 Z, µ + 2i tπn is ( at most) a pole of (λ T ) 1 of order
at most k and residue Pn .
(ii) QX = linn 2Z Pn X .
See (G. Greiner, Proposition 1.10, Math. Z.. 177 (1981) 401-423; or
Engel-Nagel, p. 283).
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Corollary
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T and let t > 0 be …xed.
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Corollary
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T and let t > 0 be …xed. Let α 6= 0 be an isolated
eigenvalue of S (t ) with …nite algebraic multiplicity and with residue Q.
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Corollary
Let X be a complex Banach space and (S (t ))t >0 be a C0 -semigroup on X
with generator T and let t > 0 be …xed. Let α 6= 0 be an isolated
eigenvalue of S (t ) with …nite algebraic multiplicity and with residue Q.
Then Q = ∑nj=1 Pj where the Pj are the residues of (λ T ) 1 at
fλ1 , ..., λn g, the (…nite and nonempty) set of eigenvalues of T such that
e λi t = α.
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Essential type of a semigroup
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Essential type of a semigroup
The fact that k.kess is submultiplicative implies that
t = 0 ! pess (t ) := ln(kS (t )kess ) 2 [ ∞, +∞[
is subadditive.
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Essential type of a semigroup
The fact that k.kess is submultiplicative implies that
t = 0 ! pess (t ) := ln(kS (t )kess ) 2 [ ∞, +∞[
is subadditive. It is also locally bounded above so
ln(kS (t )kess )
ln(kS (t )kess )
= lim
2 [ ∞, ω ] .
t
!+
∞
t >0
t
t
ω ess := inf
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Essential type of a semigroup
The fact that k.kess is submultiplicative implies that
t = 0 ! pess (t ) := ln(kS (t )kess ) 2 [ ∞, +∞[
is subadditive. It is also locally bounded above so
ln(kS (t )kess )
ln(kS (t )kess )
= lim
2 [ ∞, ω ] .
t
!+
∞
t >0
t
t
ω ess := inf
In particular 8α > ω ess 9Mα > 1 such that
kS (t )kess
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Mα e αt 8t = 0.
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Essential type of a semigroup
The fact that k.kess is submultiplicative implies that
t = 0 ! pess (t ) := ln(kS (t )kess ) 2 [ ∞, +∞[
is subadditive. It is also locally bounded above so
ln(kS (t )kess )
ln(kS (t )kess )
= lim
2 [ ∞, ω ] .
t
!+
∞
t >0
t
t
ω ess := inf
In particular 8α > ω ess 9Mα > 1 such that
kS (t )kess
Mα e αt 8t = 0.
ω ess is called the essential type (or essential growth bound) of (S (t ))t >0 .
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For any t > 0
ress (S (t )) =
1
1
n
n
= lim kS (nt )kess
lim kS (t )n kess
n !+∞
n !+∞
1
= lim exp ln (kS (nt )kess )
n !+∞
n
1
= lim exp t ln (kS (nt )kess ) = e ωess t .
n !+∞
nt
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Essentially compact semigroups
De…nition
A C0 -semigroup (S (t ))t >0 on a complex Banach space X is said to be
essentially compact if its essential type is less than its type (i.e.
ω ess < ω ).
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Theorem
Let X be a complex Banach space and (S (t ))t >0 be an essentially
compact C0 -semigroup on X with generator T .
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Theorem
Let X be a complex Banach space and (S (t ))t >0 be an essentially
compact C0 -semigroup on X with generator T . Then:
(i) σ(T ) \ fRe λ > ω ess g consists of a nonempty set of isolated
eigenvalues with …nite algebraic multiplicities.
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Theorem
Let X be a complex Banach space and (S (t ))t >0 be an essentially
compact C0 -semigroup on X with generator T . Then:
(i) σ(T ) \ fRe λ > ω ess g consists of a nonempty set of isolated
eigenvalues with …nite algebraic multiplicities.
(ii) For any ω 0 such that ω ess < ω 0 < ω, σ(T ) \ fRe λ > ω 0 g consists
of a …nite set (depending on ω 0 ) fλ1 , ..., λm g of eigenvalues of T .
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Theorem
Let X be a complex Banach space and (S (t ))t >0 be an essentially
compact C0 -semigroup on X with generator T . Then:
(i) σ(T ) \ fRe λ > ω ess g consists of a nonempty set of isolated
eigenvalues with …nite algebraic multiplicities.
(ii) For any ω 0 such that ω ess < ω 0 < ω, σ(T ) \ fRe λ > ω 0 g consists
of a …nite set (depending on ω 0 ) fλ1 , ..., λm g of eigenvalues of T .
(iii) Let Pj be the residues of (λ T ) 1 at λj and let
P : = ∑m
j =1 Pj .
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Theorem
Let X be a complex Banach space and (S (t ))t >0 be an essentially
compact C0 -semigroup on X with generator T . Then:
(i) σ(T ) \ fRe λ > ω ess g consists of a nonempty set of isolated
eigenvalues with …nite algebraic multiplicities.
(ii) For any ω 0 such that ω ess < ω 0 < ω, σ(T ) \ fRe λ > ω 0 g consists
of a …nite set (depending on ω 0 ) fλ1 , ..., λm g of eigenvalues of T .
(iii) Let Pj be the residues of (λ T ) 1 at λj and let
P : = ∑m
j =1 Pj . Then the projector P reduces (S (t ))t >0 and
m
S (t ) =
∑ e λ t e tD Pj + O (e (ω
j
j
0
ε )t
)
j =1
(for some ε > 0) where Dj := (T λj )Pj are nilpotent bounded operators
k
(Dj j = 0 where kj is the order of the pole λj ).
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Proof
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Proof
Let ω 0 be such that ω ess < ω 0 < ω. Let t > 0 be …xed.
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Proof
Let ω 0 be such that ω ess < ω 0 < ω. Let t > 0 be …xed. Then
0
e ωess t < e ω t < e ωt and
n
o
0
σ (S (t )) \ µ; jµj > e ω t
consists of a …nite (and nonempty ) set of eigenvalues with …nite algebraic
multiplicities fµ1 , ..., µn g while
o
n
o n
0
0
σ(S (t )) \ µ; jµj < e ω t
µ; jµj < e (ω ε)t
o
n
l
for some ε > 0. For each j (1 j n) let λ1j , ..., λjj be the (…nite and
nonempty ) set of eigenvalues λ of T such that e λt = µj .
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Proof
Let ω 0 be such that ω ess < ω 0 < ω. Let t > 0 be …xed. Then
0
e ωess t < e ω t < e ωt and
n
o
0
σ (S (t )) \ µ; jµj > e ω t
consists of a …nite (and nonempty ) set of eigenvalues with …nite algebraic
multiplicities fµ1 , ..., µn g while
o
n
o n
0
0
σ(S (t )) \ µ; jµj < e ω t
µ; jµj < e (ω ε)t
o
n
l
for some ε > 0. For each j (1 j n) let λ1j , ..., λjj be the (…nite and
nonempty ) set of eigenvalues λ of T such that e λt = µj . Then the residue
of the pole µj of the resolvent of S (t ) is given by
lj
Qj =
∑ Pjk
k =1
where
Pjk
is the residue of the
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λkj
of the resolvent of T .
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Let Q = ∑nj=1 Qj be the spectral projection corresponding to the
n
o
0
eigenvalues fµ1 , ..., µn g of S (t ) in µ; jµj > e ω t .
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Let Q = ∑nj=1 Qj be the spectral projection corresponding to the
n
o
0
eigenvalues fµ1 , ..., µn g of S (t ) in µ; jµj > e ω t . One sees that
l
Q = ∑nj=1 ∑kj =1 Pjk is nothing but the spectral projection corresponding to
the eigenvalues of T in fRe λ > ω 0 g .
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Let Q = ∑nj=1 Qj be the spectral projection corresponding to the
n
o
0
eigenvalues fµ1 , ..., µn g of S (t ) in µ; jµj > e ω t . One sees that
l
Q = ∑nj=1 ∑kj =1 Pjk is nothing but the spectral projection corresponding to
the eigenvalues of T in fRe λ > ω 0 g .
We decompose S (t ) as S (t )Q + S (t )(I Q ).
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Let Q = ∑nj=1 Qj be the spectral projection corresponding to the
n
o
0
eigenvalues fµ1 , ..., µn g of S (t ) in µ; jµj > e ω t . One sees that
l
Q = ∑nj=1 ∑kj =1 Pjk is nothing but the spectral projection corresponding to
the eigenvalues of T in fRe λ > ω 0 g .
We decompose S (t ) as S (t )Q + S (t )(I Q ). Wenknow that
o
σ(S (t )jIm Q ) = fµ1 , ..., µn g while σ(S (t )jKerQ )
the type of S (t )jKerQ is
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ω0
µ; jµj < e (ω
0
ε )t
so
ε.
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Let Q = ∑nj=1 Qj be the spectral projection corresponding to the
n
o
0
eigenvalues fµ1 , ..., µn g of S (t ) in µ; jµj > e ω t . One sees that
l
Q = ∑nj=1 ∑kj =1 Pjk is nothing but the spectral projection corresponding to
the eigenvalues of T in fRe λ > ω 0 g .
We decompose S (t ) as S (t )Q + S (t )(I Q ). Wenknow that
o
σ(S (t )jIm Q ) = fµ1 , ..., µn g while σ(S (t )jKerQ )
the type of S (t )jKerQ is
bounded operator
m
T ( ∑ Pj ) =
j =1
m
ω0
0
ε )t
so
ε. Finally, S (t )jIm Q is generated by the
m
∑ TPj = ∑ [λj Pj + (T
j =1
µ; jµj < e (ω
j =1
m
λj )Pj ] =
∑ [λj Pj + Dj ]
j =1
λj t e tD j P .
so S (t )jIm Q = ∑m
j
j =1 e
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The role of positivity
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The role of positivity
For the sake of simplicity, we restrict ourselves to Lebesgue spaces
X = Lp (Ω, A, µ)
(1
p
+∞)
where (Ω, A, µ) is a measure space (i.e. Ω is a set, A is a σ-algebra of
subsets of Ω and µ is a σ-…nite measure on A).
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The role of positivity
For the sake of simplicity, we restrict ourselves to Lebesgue spaces
X = Lp (Ω, A, µ)
(1
p
+∞)
where (Ω, A, µ) is a measure space (i.e. Ω is a set, A is a σ-algebra of
subsets of Ω and µ is a σ-…nite measure on A).
Lp+ (Ω, A, µ) the (closed) positive cone of nonnegative functions,
i.e. f 2 Lp+ (Ω, A, µ) if and only if f (x ) > 0 for almost all x 2 Ω.
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The role of positivity
For the sake of simplicity, we restrict ourselves to Lebesgue spaces
X = Lp (Ω, A, µ)
(1
p
+∞)
where (Ω, A, µ) is a measure space (i.e. Ω is a set, A is a σ-algebra of
subsets of Ω and µ is a σ-…nite measure on A).
Lp+ (Ω, A, µ) the (closed) positive cone of nonnegative functions,
i.e. f 2 Lp+ (Ω, A, µ) if and only if f (x ) > 0 for almost all x 2 Ω.
This induces a partial order
f
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g ,g
f 2 Lp+ (Ω, A, µ) .
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A canonical decomposition
f = f+
f ,
8f 2 Lp (Ω, A, µ)
where
f+ = sup ff , 0g , f = sup f f , 0g .
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A canonical decomposition
f = f+
f ,
8f 2 Lp (Ω, A, µ)
where
f+ = sup ff , 0g , f = sup f f , 0g .
In particular
jf j = f+ + f , kf k = kjf jk
(where jf j (x ) := jf (x )j),
k f k p = k f+ k p + k f k p
and
Lp (Ω, A, µ) = Lp+ (Ω, A, µ)
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Lp+ (Ω, A, µ) .
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Positive operators
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Positive operators
An operator G 2 L(X ) is positive if
Gf 2 Lp+ (Ω, A, µ) 8f 2 Lp+ (Ω, A, µ). We write G > 0.
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Positive operators
An operator G 2 L(X ) is positive if
Gf 2 Lp+ (Ω, A, µ) 8f 2 Lp+ (Ω, A, µ). We write G > 0.
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Positive operators
An operator G 2 L(X ) is positive if
Gf 2 Lp+ (Ω, A, µ) 8f 2 Lp+ (Ω, A, µ). We write G > 0. Then G is
"nondecreasing", i.e.8f
h, G (h f ) > 0 i.e.
Gf
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Gh, 8f
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h.
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Positive operators
An operator G 2 L(X ) is positive if
Gf 2 Lp+ (Ω, A, µ) 8f 2 Lp+ (Ω, A, µ). We write G > 0. Then G is
"nondecreasing", i.e.8f
h, G (h f ) > 0 i.e.
Gh, 8f
Gf
h.
If G 2 L(X ) is positive then
jGf j = jGf+
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Gf j
Gf+ + Gf = G (jf j) .
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Positive operators
An operator G 2 L(X ) is positive if
Gf 2 Lp+ (Ω, A, µ) 8f 2 Lp+ (Ω, A, µ). We write G > 0. Then G is
"nondecreasing", i.e.8f
h, G (h f ) > 0 i.e.
Gh, 8f
Gf
h.
If G 2 L(X ) is positive then
jGf j = jGf+
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Gf j
Gf+ + Gf = G (jf j) .
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Positive operators
An operator G 2 L(X ) is positive if
Gf 2 Lp+ (Ω, A, µ) 8f 2 Lp+ (Ω, A, µ). We write G > 0. Then G is
"nondecreasing", i.e.8f
h, G (h f ) > 0 i.e.
Gh, 8f
Gf
h.
If G 2 L(X ) is positive then
jGf j = jGf+
Gf j
Gf+ + Gf = G (jf j) .
If G > 0 then
kG k =
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sup
kf k 1, f 2L p+
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kGf k .
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Positive operators
An operator G 2 L(X ) is positive if
Gf 2 Lp+ (Ω, A, µ) 8f 2 Lp+ (Ω, A, µ). We write G > 0. Then G is
"nondecreasing", i.e.8f
h, G (h f ) > 0 i.e.
Gh, 8f
Gf
h.
If G 2 L(X ) is positive then
jGf j = jGf+
Gf j
Gf+ + Gf = G (jf j) .
If G > 0 then
kG k =
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sup
kf k 1, f 2L p+
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kGf k .
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Positive operators
An operator G 2 L(X ) is positive if
Gf 2 Lp+ (Ω, A, µ) 8f 2 Lp+ (Ω, A, µ). We write G > 0. Then G is
"nondecreasing", i.e.8f
h, G (h f ) > 0 i.e.
Gh, 8f
Gf
h.
If G 2 L(X ) is positive then
jGf j = jGf+
Gf j
Gf+ + Gf = G (jf j) .
If G > 0 then
kG k =
If 0
G1
sup
kf k 1, f 2L p+
kGf k .
G2 with Gi 2 L(Lp ) (i = 1, 2) then kG1 k
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kG2 k.
Chapter 2
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Positive operators
An operator G 2 L(X ) is positive if
Gf 2 Lp+ (Ω, A, µ) 8f 2 Lp+ (Ω, A, µ). We write G > 0. Then G is
"nondecreasing", i.e.8f
h, G (h f ) > 0 i.e.
Gh, 8f
Gf
h.
If G 2 L(X ) is positive then
jGf j = jGf+
Gf j
Gf+ + Gf = G (jf j) .
If G > 0 then
kG k =
If 0
G1
sup
kf k 1, f 2L p+
kGf k .
G2 with Gi 2 L(Lp ) (i = 1, 2) then kG1 k
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kG2 k.
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Positive operators
An operator G 2 L(X ) is positive if
Gf 2 Lp+ (Ω, A, µ) 8f 2 Lp+ (Ω, A, µ). We write G > 0. Then G is
"nondecreasing", i.e.8f
h, G (h f ) > 0 i.e.
Gh, 8f
Gf
h.
If G 2 L(X ) is positive then
jGf j = jGf+
Gf j
Gf+ + Gf = G (jf j) .
If G > 0 then
kG k =
sup
kf k 1, f 2L p+
kGf k .
If 0 G1 G2 with Gi 2 L(Lp ) (i = 1, 2) then kG1 k kG2 k.
0
G 2 L(Lp ) is positive if and only if its dual operator G 0 2 L(Lp ) is
positive.
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Positive operators
An operator G 2 L(X ) is positive if
Gf 2 Lp+ (Ω, A, µ) 8f 2 Lp+ (Ω, A, µ). We write G > 0. Then G is
"nondecreasing", i.e.8f
h, G (h f ) > 0 i.e.
Gh, 8f
Gf
h.
If G 2 L(X ) is positive then
jGf j = jGf+
Gf j
Gf+ + Gf = G (jf j) .
If G > 0 then
kG k =
sup
kf k 1, f 2L p+
kGf k .
If 0 G1 G2 with Gi 2 L(Lp ) (i = 1, 2) then kG1 k kG2 k.
0
G 2 L(Lp ) is positive if and only if its dual operator G 0 2 L(Lp ) is
positive.
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Chapter 2
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Positive operators
An operator G 2 L(X ) is positive if
Gf 2 Lp+ (Ω, A, µ) 8f 2 Lp+ (Ω, A, µ). We write G > 0. Then G is
"nondecreasing", i.e.8f
h, G (h f ) > 0 i.e.
Gh, 8f
Gf
h.
If G 2 L(X ) is positive then
jGf j = jGf+
Gf j
Gf+ + Gf = G (jf j) .
If G > 0 then
kG k =
sup
kf k 1, f 2L p+
kGf k .
If 0 G1 G2 with Gi 2 L(Lp ) (i = 1, 2) then kG1 k kG2 k.
0
G 2 L(Lp ) is positive if and only if its dual operator G 0 2 L(Lp ) is
positive.
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De…nition
A C0 -semigroup (S (t ))t >0 on X is positive if 8t > 0, S (t ) is a positive
operator.
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Resolvent characterization
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Resolvent characterization
A C0 -semigroup (S (t ))t >0 with generator T is positive if and only if the
resolvent (λ T ) 1 is positive for λ real and large enough.
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Resolvent characterization
A C0 -semigroup (S (t ))t >0 with generator T is positive if and only if the
resolvent (λ T ) 1 is positive for λ real and large enough. This follows
from
Z
(λ
T)
1
+∞
f =
e
λt
S (t )fdt
(λ > ω )
0
and the exponential formula
S (t )f = lim (I
n !+∞
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t
T)
n
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n
f.
Chapter 2
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A spectral property of positive operators
rσ (G ) 2 σ(G ), 8G > 0.
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Chapter 2
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A spectral property of positive operators
rσ (G ) 2 σ(G ), 8G > 0.
This basic result is linked to
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Chapter 2
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A spectral property of positive operators
rσ (G ) 2 σ(G ), 8G > 0.
This basic result is linked to
Pringsheim’s theorem: If f (x ) = ∑0∞ an x n is the sum of a power series
with nonnegative coe¢ cients an with radius of convergence r then
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A spectral property of positive operators
rσ (G ) 2 σ(G ), 8G > 0.
This basic result is linked to
Pringsheim’s theorem: If f (x ) = ∑0∞ an x n is the sum of a power series
with nonnegative coe¢ cients an with radius of convergence r then r is a
singular point of f (x ).
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Spectral bound of generators
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Chapter 2
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Spectral bound of generators
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . Then:
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Chapter 2
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Spectral bound of generators
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . Then:
The type of (S (t ))t >0 coincides with the spectral bound s (T ) of
T . (see e.g. Engel-Nagel).
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Chapter 2
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Spectral bound of generators
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . Then:
The type of (S (t ))t >0 coincides with the spectral bound s (T ) of
T . (see e.g. Engel-Nagel).
If s (T ) > ∞ then s (T ) 2 σ (T ).
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Chapter 2
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Spectral bound of generators
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . Then:
The type of (S (t ))t >0 coincides with the spectral bound s (T ) of
T . (see e.g. Engel-Nagel).
If s (T ) > ∞ then s (T ) 2 σ (T ).
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Chapter 2
50 / 57
Spectral bound of generators
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . Then:
The type of (S (t ))t >0 coincides with the spectral bound s (T ) of
T . (see e.g. Engel-Nagel).
If s (T ) > ∞ then s (T ) 2 σ (T ).
Indeed
(λ
so (λ
1
T)
f
Z +∞
e
Re λt
0
T)
1
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(Re λ
T)
S (t ) jf j dt
1
(8 Re λ > s (T ))
(8 Re λ > s (T )) .
CIMPA School Muizemberg
Chapter 2
50 / 57
Spectral bound of generators
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . Then:
The type of (S (t ))t >0 coincides with the spectral bound s (T ) of
T . (see e.g. Engel-Nagel).
If s (T ) > ∞ then s (T ) 2 σ (T ).
Indeed
(λ
T)
1
f
Z +∞
0
e
Re λt
S (t ) jf j dt
(8 Re λ > s (T ))
so (λ T ) 1
(Re λ T ) 1 (8 Re λ > s (T )) . By
assumption there exists a sequence ( βn )n σ(T ) such that
Re λn ! s (T ).
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Chapter 2
50 / 57
Spectral bound of generators
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . Then:
The type of (S (t ))t >0 coincides with the spectral bound s (T ) of
T . (see e.g. Engel-Nagel).
If s (T ) > ∞ then s (T ) 2 σ (T ).
Indeed
(λ
T)
1
f
Z +∞
0
e
Re λt
S (t ) jf j dt
(8 Re λ > s (T ))
so (λ T ) 1
(Re λ T ) 1 (8 Re λ > s (T )) . By
assumption there exists a sequence ( βn )n σ(T ) such that
Re λn ! s (T ). We build a sequence (λn )n with Re λn > s (T ) (so
(λn )n ρ(T )), Im λn = Im βn and Re λn ! s (T ).
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Chapter 2
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Spectral bound of generators
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . Then:
The type of (S (t ))t >0 coincides with the spectral bound s (T ) of
T . (see e.g. Engel-Nagel).
If s (T ) > ∞ then s (T ) 2 σ (T ).
Indeed
(λ
T)
1
f
Z +∞
0
e
Re λt
S (t ) jf j dt
(8 Re λ > s (T ))
so (λ T ) 1
(Re λ T ) 1 (8 Re λ > s (T )) . By
assumption there exists a sequence ( βn )n σ(T ) such that
Re λn ! s (T ). We build a sequence (λn )n with Re λn > s (T ) (so
(λn )n ρ(T )), Im λn = Im βn and Re λn ! s (T ). Then
jλn βn j ! 0 and (λn T ) 1 ! +∞.
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Chapter 2
50 / 57
Spectral bound of generators
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . Then:
The type of (S (t ))t >0 coincides with the spectral bound s (T ) of
T . (see e.g. Engel-Nagel).
If s (T ) > ∞ then s (T ) 2 σ (T ).
Indeed
(λ
T)
1
f
Z +∞
0
e
Re λt
S (t ) jf j dt
(8 Re λ > s (T ))
so (λ T ) 1
(Re λ T ) 1 (8 Re λ > s (T )) . By
assumption there exists a sequence ( βn )n σ(T ) such that
Re λn ! s (T ). We build a sequence (λn )n with Re λn > s (T ) (so
(λn )n ρ(T )), Im λn = Im βn and Re λn ! s (T ). Then
jλn βn j ! 0 and (λn T ) 1 ! +∞. Thus
(Re λn T ) 1 ! +∞ and consequently s (T ) 2 σ(T ).
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Chapter 2
50 / 57
Spectral bound of generators
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . Then:
The type of (S (t ))t >0 coincides with the spectral bound s (T ) of
T . (see e.g. Engel-Nagel).
If s (T ) > ∞ then s (T ) 2 σ (T ).
Indeed
(λ
T)
1
f
Z +∞
0
e
Re λt
S (t ) jf j dt
(8 Re λ > s (T ))
so (λ T ) 1
(Re λ T ) 1 (8 Re λ > s (T )) . By
assumption there exists a sequence ( βn )n σ(T ) such that
Re λn ! s (T ). We build a sequence (λn )n with Re λn > s (T ) (so
(λn )n ρ(T )), Im λn = Im βn and Re λn ! s (T ). Then
jλn βn j ! 0 and (λn T ) 1 ! +∞. Thus
(Re λn T ) 1 ! +∞ and consequently s (T ) 2 σ(T ).
rσ (µ T ) 1 = (µ s (T )) 1 , 8µ > s (T ).
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Chapter 2
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Irreducible operators
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Chapter 2
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Irreducible operators
Let G 2 L(Lp ) be positive.
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Chapter 2
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Irreducible operators
Let G 2 L(Lp ) be positive.
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Chapter 2
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Irreducible operators
Let G 2 L(Lp ) be positive. We that G is irreducible if
0
8f 2 Lp+ (Ω), f 6= 0 and 8g 2 Lp+ (Ω), g 6= 0 there exists n 2 N
such that
hG n f , g iLp ,Lp 0 > 0.
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Irreducible operators
Let G 2 L(Lp ) be positive. We that G is irreducible if
0
8f 2 Lp+ (Ω), f 6= 0 and 8g 2 Lp+ (Ω), g 6= 0 there exists n 2 N
such that
hG n f , g iLp ,Lp 0 > 0.
For p < +∞, this is equivalent to saying that there is no (non trivial)
closed subspace Lp (Ω0 ) invariant by G .
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Irreducible operators
Let G 2 L(Lp ) be positive. We that G is irreducible if
0
8f 2 Lp+ (Ω), f 6= 0 and 8g 2 Lp+ (Ω), g 6= 0 there exists n 2 N
such that
hG n f , g iLp ,Lp 0 > 0.
For p < +∞, this is equivalent to saying that there is no (non trivial)
closed subspace Lp (Ω0 ) invariant by G .
For instance, if Gf > 0 a.e. 8f 2 Lp+ (Ω), f 6= 0 (we say that G is
positivity-improving) then G is irreducible.
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Chapter 2
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Irreducible operators
Let G 2 L(Lp ) be positive. We that G is irreducible if
0
8f 2 Lp+ (Ω), f 6= 0 and 8g 2 Lp+ (Ω), g 6= 0 there exists n 2 N
such that
hG n f , g iLp ,Lp 0 > 0.
For p < +∞, this is equivalent to saying that there is no (non trivial)
closed subspace Lp (Ω0 ) invariant by G .
For instance, if Gf > 0 a.e. 8f 2 Lp+ (Ω), f 6= 0 (we say that G is
positivity-improving) then G is irreducible.
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Chapter 2
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Irreducible operators
Let G 2 L(Lp ) be positive. We that G is irreducible if
0
8f 2 Lp+ (Ω), f 6= 0 and 8g 2 Lp+ (Ω), g 6= 0 there exists n 2 N
such that
hG n f , g iLp ,Lp 0 > 0.
For p < +∞, this is equivalent to saying that there is no (non trivial)
closed subspace Lp (Ω0 ) invariant by G .
For instance, if Gf > 0 a.e. 8f 2 Lp+ (Ω), f 6= 0 (we say that G is
positivity-improving) then G is irreducible.
A positive C0 -semigroup (S (t ))t >0 is said to be irreducible if
0
8f 2 Lp+ (Ω), f 6= 0 and 8g 2 Lp+ (Ω), g 6= 0 there exists t > 0 such
that
hS (t )f , g iLp ,Lp 0 > 0.
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Chapter 2
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Irreducible operators
Let G 2 L(Lp ) be positive. We that G is irreducible if
0
8f 2 Lp+ (Ω), f 6= 0 and 8g 2 Lp+ (Ω), g 6= 0 there exists n 2 N
such that
hG n f , g iLp ,Lp 0 > 0.
For p < +∞, this is equivalent to saying that there is no (non trivial)
closed subspace Lp (Ω0 ) invariant by G .
For instance, if Gf > 0 a.e. 8f 2 Lp+ (Ω), f 6= 0 (we say that G is
positivity-improving) then G is irreducible.
A positive C0 -semigroup (S (t ))t >0 is said to be irreducible if
0
8f 2 Lp+ (Ω), f 6= 0 and 8g 2 Lp+ (Ω), g 6= 0 there exists t > 0 such
that
hS (t )f , g iLp ,Lp 0 > 0.
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Chapter 2
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Irreducible operators
Let G 2 L(Lp ) be positive. We that G is irreducible if
0
8f 2 Lp+ (Ω), f 6= 0 and 8g 2 Lp+ (Ω), g 6= 0 there exists n 2 N
such that
hG n f , g iLp ,Lp 0 > 0.
For p < +∞, this is equivalent to saying that there is no (non trivial)
closed subspace Lp (Ω0 ) invariant by G .
For instance, if Gf > 0 a.e. 8f 2 Lp+ (Ω), f 6= 0 (we say that G is
positivity-improving) then G is irreducible.
A positive C0 -semigroup (S (t ))t >0 is said to be irreducible if
0
8f 2 Lp+ (Ω), f 6= 0 and 8g 2 Lp+ (Ω), g 6= 0 there exists t > 0 such
that
hS (t )f , g iLp ,Lp 0 > 0.
For p < +∞, this is equivalent to saying that there is no (non trivial)
closed subspace Lp (Ω0 ) invariant by all S (t ).
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A positive C0 -semigroup (S (t ))t >0 with generator T is irreducible if and
only if (λ T ) 1 is positivity-improving for some λ > s (T ).
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A positive C0 -semigroup (S (t ))t >0 with generator T is irreducible if and
only if (λ T ) 1 is positivity-improving for some λ > s (T ). This
follows easily from
h(λ
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T)
1
f , gi =
Z +∞
e
λt
0
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hS (t )f , g idt.
Chapter 2
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Irreducible power compact operators
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Irreducible power compact operators
Theorem
Let G 2 L(X ) be compact and irreducible. Then rσ (G ) > 0.
See B. de Pagter, Math Z, 192 (1986) 149-153.
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Irreducible power compact operators
Theorem
Let G 2 L(X ) be compact and irreducible. Then rσ (G ) > 0.
See B. de Pagter, Math Z, 192 (1986) 149-153.
Corollary
If some power of G 2 L(X ) is compact and irreducible. Then
rσ (G ) > 0.
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Irreducible power compact operators
Theorem
Let G 2 L(X ) be compact and irreducible. Then rσ (G ) > 0.
See B. de Pagter, Math Z, 192 (1986) 149-153.
Corollary
If some power of G 2 L(X ) is compact and irreducible. Then
rσ (G ) > 0.
It follows from: rσ (G n ) = rσ (G )n .
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Peripheral spectral theory
The following result can be found in R. Nagel (Ed), Chapter CIII.
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Peripheral spectral theory
The following result can be found in R. Nagel (Ed), Chapter CIII.
Theorem
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T.
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Peripheral spectral theory
The following result can be found in R. Nagel (Ed), Chapter CIII.
Theorem
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . If s (T ) is a pole of the (λ T ) 1 then the boundary spectrum
σb (T ) := σ(T ) \ (s (T ) + iR)
consists of poles of the resolvent and is cyclic
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Peripheral spectral theory
The following result can be found in R. Nagel (Ed), Chapter CIII.
Theorem
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . If s (T ) is a pole of the (λ T ) 1 then the boundary spectrum
σb (T ) := σ(T ) \ (s (T ) + iR)
consists of poles of the resolvent and is cyclic in the sense that there exists
α > 0 such that
σb (T ) := s (T ) + i αZ.
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Corollary
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . We assume that (S (t ))t >0 is essentially compact (i.e. ω ess < ω).
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Corollary
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . We assume that (S (t ))t >0 is essentially compact (i.e. ω ess < ω).
Then
σb (T ) = fs (T )g
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Corollary
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . We assume that (S (t ))t >0 is essentially compact (i.e. ω ess < ω).
Then
σb (T ) = fs (T )g
i.e. s (T ) is the leading eigenvalue and is strictly dominant (i.e.
9ε > 0; Re λ s (T ) ε 8λ 2 σ(T ), λ 6= s (T )).
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Corollary
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . We assume that (S (t ))t >0 is essentially compact (i.e. ω ess < ω).
Then
σb (T ) = fs (T )g
i.e. s (T ) is the leading eigenvalue and is strictly dominant (i.e.
9ε > 0; Re λ s (T ) ε 8λ 2 σ(T ), λ 6= s (T )).
Proof: According to the theorem above, σb (T ) is either unbounded or
reduces to fs (T )g .
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Corollary
Let (S (t ))t >0 be a positive C0 -semigroup on Lp (Ω, A, µ) with generator
T . We assume that (S (t ))t >0 is essentially compact (i.e. ω ess < ω).
Then
σb (T ) = fs (T )g
i.e. s (T ) is the leading eigenvalue and is strictly dominant (i.e.
9ε > 0; Re λ s (T ) ε 8λ 2 σ(T ), λ 6= s (T )).
Proof: According to the theorem above, σb (T ) is either unbounded or
reduces to fs (T )g . The fact that ω ess < ω implies that σb (T ) is …nite.
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Asymptotic structure of essentially compact irreducible
semigroups
Theorem
Let (S (t ))t >0 be an irreducible C0 -semigroup on Lp (Ω, A, µ) with
generator T .
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Asymptotic structure of essentially compact irreducible
semigroups
Theorem
Let (S (t ))t >0 be an irreducible C0 -semigroup on Lp (Ω, A, µ) with
generator T . We assume that (S (t ))t >0 is essentially compact (i.e.
ω ess < ω).
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Asymptotic structure of essentially compact irreducible
semigroups
Theorem
Let (S (t ))t >0 be an irreducible C0 -semigroup on Lp (Ω, A, µ) with
generator T . We assume that (S (t ))t >0 is essentially compact (i.e.
ω ess < ω). Then s (T ) is the leading eigenvalue, is strictly dominant and
is algebraically simple.
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Asymptotic structure of essentially compact irreducible
semigroups
Theorem
Let (S (t ))t >0 be an irreducible C0 -semigroup on Lp (Ω, A, µ) with
generator T . We assume that (S (t ))t >0 is essentially compact (i.e.
ω ess < ω). Then s (T ) is the leading eigenvalue, is strictly dominant and
is algebraically simple. In particular there exists ε > 0 such that
S ( t ) f = e s (T )t
Z
f (x )v (x )µ(dx ) u + O (e (s (T )
ε )t
)
where u is the (strictly positive almost everywhere) eigenfunction of T
associated to s (T ) and v is the (strictly positive almost everywhere)
eigenfunction
of T 0 associated to s (T 0 ) = s (T ) with the normalization
R
u (x )v (x )µ(dx ) = 1.
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The irreducibility assumption implies the spectral projection P associated
with s (T ) is one-dimensional
Pf =
Z
f (x )v (x )µ(dx ) u
where u is a strictly positive almost everywhere eigenfunction of T
associated to s (T ) and v is the strictly positive
almost everywhere dual
R
eigenfunction of T 0 with the normalization u (x )v (x )µ(dx ) = 1.
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The irreducibility assumption implies the spectral projection P associated
with s (T ) is one-dimensional
Pf =
Z
f (x )v (x )µ(dx ) u
where u is a strictly positive almost everywhere eigenfunction of T
associated to s (T ) and v is the strictly positive
almost everywhere dual
R
eigenfunction of T 0 with the normalization u (x )v (x )µ(dx ) = 1. See
Nagel (Ed), Prop 3.5, p. 310.
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