Theorems on Perturbations of Generators Teoremas Espectrais and Dicotomias SMA 5878 Functional Analysis II Alexandre Nolasco de Carvalho Departamento de Matem´ atica Instituto de Ciˆ encias Matem´ aticas and de Computa¸c˜ ao Universidade de S˜ ao Paulo June 06, 2018 Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
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Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
SMA 5878 Functional Analysis II
Alexandre Nolasco de Carvalho
Departamento de MatematicaInstituto de Ciencias Matematicas and de Computacao
Universidade de Sao Paulo
June 06, 2018
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Trotter Approximation Theorems
Trotter Approximation Theorems
TheoremIf A,An ∈ G (M, ω), n ∈ N, the following statements are equivalent
(a) For all x ∈ X and λ with Reλ > ω,(λ− An)−1x
n→∞−→ (λ− A)−1x .
(b) For all x ∈ X and t ≥ 0, eAntxn→∞−→ eAtx .
Besides that, the convergence in (b) is uniform for all t in boundedsubsets of R+.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Trotter Approximation Theorems
Trotter-Kato Theorem
Theorem (Trotter-Kato)
If An ∈ G (M, ω) and there exists λ0 with Reλ0 > ω such that
(a) for all x ∈ X , (λ0 − An)−1x → R(λ0)x when n→∞ and
(b) the image of R(λ0) is dense in X ,
then there exists a unique operator A ∈ G (M, ω) such thatR(λ0) = (λ0 − A)−1.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Trotter Approximation Theorems
A direct consequence of Theorems 1 and 2 is the following theorem
TheoremLet An ∈ G (M, ω),∀n ∈ N. If for some λ0∈C with Reλ0>ω,
(a) limn→∞(λ0 − An)−1x =: R(λ0)x for all x ∈ X and
(b) the image of R(λ0) is dense in X ,
then, there is a unique operator A ∈ G (M, ω) such thatR(λ0) = (λ0 − A)−1. Besides that, eAntx → eAtx for all x ∈ X ,uniformly for t in bounded subsets of R+.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Trotter Approximation Theorems
A little different consequence of the previous results is thefollowing theorem.
Theorem (Trotter)
Let An ∈ G (M, ω) and suppose that
(a) there is a dense subset D of X such that {Anx}n∈N isconvergent for all x ∈ D. Define A : D ⊂ X → X by,Ax = limn→∞ Anx for all x ∈ D,
(b) there is a λ0 with Reλ0 > ω for which (λ0 − A)D is dense inX .
Then A is closable and the closure A of A is in G (M, ω). Besides
that eAntx → eAtx for all x ∈ X , uniformly for t in boundedsubsets of R+.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Trotter Approximation Theorems
Proof: If f ∈ D, x = (λ0 − A)f and xn = (λ0 − An)f then,Anf
n→∞−→ Af and xnn→∞−→ x . Furthermore, since
‖(λ0 − An)−1‖L(X ) ≤ M(Reλ0 − ω)−1, it follows that
limn→∞
(λ0 − An)−1x = limn→∞
((λ0 − An)−1(x − xn) + f ) = f ; (1)
that is, (λ0 − An)−1 converges in the image of λ0 − A. From (b)the image of λ0 − A is dense in X and, by hypothesis,‖(λ0 − An)−1‖L(X ) is bounded, uniformly for n ∈ N. It follows that(λ0 − An)−1x converges for all x ∈ X .
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Trotter Approximation Theorems
Letlimn→∞
(λ0 − An)−1x = R(λ0)x . (2)
From (1) it follows that the image of R(λ0) contains D andtherefore is dense in X . Theorem 2 implies the existence of anoperator A′ ∈ G (M, ω) satisfying R(λ0) = (λ0 − A′)−1. Toconclude the proof we show that A = A′. If x ∈ D then,
limn→∞
(λ0 − An)−1(λ0 − A)x = (λ0 − A′)−1(λ0 − A)x . (3)
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Trotter Approximation Theorems
On the other hand, since ‖(λ0 − An)−1‖L(X ) is uniformly bounded,
(λ0 − An)−1(λ0 − A)x = (λ0 − An)−1(λ0 − An)x
+ (λ0 − An)−1(An − A)x
= x + (λ0 − An)−1(An − A)xn→∞−→ x ,
given that Anxn→∞−→ Ax for all x ∈ D. Hence,
(λ0 − A′)−1(λ0 − A)x = x , x ∈ D. (4)
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Trotter Approximation Theorems
But (4) implies that A′x = Ax for x ∈ D and therefore A′ ⊃ A.Since A′ is closed, A is closable. Next we show that A ⊃ A′. Letf ′ = A′x ′. Since (λ0 − A)D is dense in X there is a sequencexn ∈ D such that
fn = (λ0 − A′)xn = (λ0 − A)xnn→∞−→ λ0x
′ − f ′ = (λ0 − A′)x ′.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Trotter Approximation Theorems
Thus,
xn = (λ0 − A′)−1fnn→∞−→ (λ0 − A′)−1(λ0 − A′)x ′ = x ′ and (5)
Axn = λ0xn − fnn→∞−→ f ′. (6)
From (5) and (6) it follows that f ′ = Ax ′ and A ⊃ A′. HenceA = A′. The remaining statements of the theorem follow directlyfrom Theorem 3.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Spectral decomposition of semigroupsSpectral theorems for semigroups
Spectral decomposition of semigroups
When we study the stability of problems where semigroups oflinear operators are involved, one of the most fundamentalproblems is to determine the spectrum of the semigroup.
In general we known the infinitesimal generator and not theassociated semigroup of linear operators.
So, if we can compute some of the spectral properties of theinfinitesimal generator, we would like to use them to deriveproperties of the spectrum of the associated semigroup.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Spectral decomposition of semigroupsSpectral theorems for semigroups
TheoremSuppose that {T (t) : t ≥ 0} is a strongly continuous semigroupand that, for some t0 > 0, σ(T (t0)) ∩ {λ ∈ C : |λ| = eαt0} = ∅for some α ∈ R. Then, there exists P ∈ L(X ), P2 = P,PT (t) = T (t)P ∀t ≥ 0 such that, if X− = R(P) and X+ = N(P),the restrictions of T (t)
∣∣X±
are in L(X±),
σ(T (t)∣∣X−
) = σ(T (t)) ∩ {λ ∈ C : |λ| < eαt} and
σ(T (t)∣∣X+
) = σ(T (t)) ∩ {λ ∈ C : |λ| > eαt}.There are constants M ≥ 1, δ > 0 such that
‖T (t)∣∣X−‖L(X−) ≤ Me(α−δ)t , ∀t ≥ 0,
{T (t)∣∣X+
; t ≥ 0} extents to a group in L(X+) with
T (t)∣∣X+
= (T (−t)∣∣X+
)−1 for t < 0, and
‖T (t)∣∣X+‖L(X+) ≤ Me(α+δ)t , ∀t ≤ 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Spectral decomposition of semigroupsSpectral theorems for semigroups
RemarkFor α = 0, the above separation of the space X together with thedecaying properties is a particular case of exponential dichotomy.A even more special case, but clearly useful, is the case whenσ(T (t0)) ⊂ {λ ∈ C : |λ| < eαt0}, that is, P = I and X+ = {0}then,
‖T (t)‖L(X ) ≤ Me(α−δ)t , t ≥ 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Spectral decomposition of semigroupsSpectral theorems for semigroups
Proof: If C is a rectifiable closed simple curve with trace{λ ∈ C : |λ| = eαt0}, define
P =1
2πi
∫C
(λ− T (t0))−1dλ ∈ L(X ).
Then, from what we have seen before, P2 = P and P is acontinuous projection.
It is easy to see that T (t)P = PT (t) for all t ≥ 0. Hence, ifX− = R(P) and X+ = N(P) we have that T (t) takes X+ into X+
and X− em X−.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Spectral decomposition of semigroupsSpectral theorems for semigroups
Also note that σ(T (t0)∣∣X−
) is the part of σ(T (t0)) inside C and
σ(T (t0)∣∣X+
) is the part of σ(T (t0)) outside of C and that the
parts of (λ− T (t0))−1 in X+ and X− coincide with((λ− T (t0))
∣∣X+
)−1 and ((λ− T (t0))∣∣X−
)−1 respectivelly.
Now the spectral radius of T (t0)∣∣X−
is strictly smaller than eαt0 ,
that is,r(T (t0)
∣∣X−
) < e(α−δ)t0 ,
for some δ > 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Spectral decomposition of semigroupsSpectral theorems for semigroups
If t > 0, for each m ∈ N there are n = n(m) ∈ N and τ ∈ [0, t0)such that mt = nt0 + τ . It is clear that n(m)
m→∞−→ ∞ and
r(T (t)∣∣X−
) = limm→∞
‖T (mt)∣∣X−‖
1m
L(X−)
= limn→∞
‖T (nt0 + τ)∣∣X−‖
tnt0+τ
L(X−)
≤ limn→∞
‖T (nt0)∣∣X−‖
tnt0+τ
L(X−)‖T (τ)∣∣X−‖
tnt0+τ
L(X−)
= r(T (t0)∣∣X−
)t/t0 < e(α−δ)t
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Spectral decomposition of semigroupsSpectral theorems for semigroups
Also, there is an integer N ≥ 1 such that Nt0 ≥ t, consequently
T (Nt0 − t)(T (t0)
∣∣X+
)−Nis the inverse of T (t)
∣∣X+
, that is, T (−t)∣∣X+
and arguing as abovewe can show that
r(T (t)∣∣X+
) < e(α+δ)t , t < 0.
It is easy to see that (considering the components in both spaces)
σ(T (t)) = σ(T (t)∣∣X+
) ∪ σ(T (t)∣∣X−
), t > 0,
and the statements about the spectral radius prove the estimateson the spectrum.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Spectral decomposition of semigroupsSpectral theorems for semigroups
The estimates for the norms are simple. For example, sincer(T (t0)
∣∣X−
) < e(α−δ)t0 ,
‖T (nt0)∣∣X−‖1/nL(X−) < e(α−δ)t0
when n is large, hence
‖T (nt0)∣∣X−‖L(X−) ≤ M0e
n(α−δ)t0
for all n ≥ 0 and some M0 ≥ 1.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Spectral decomposition of semigroupsSpectral theorems for semigroups
So, for n = 0, 1, 2, · · · and 0 ≤ τ < t0,
‖T (nt0 + τ)∣∣X−‖L(X−) ≤ M0e
n(α−δ)t0‖T (τ)∣∣X−‖L(X−)
≤ Me(α−δ)(nt0+τ)
where M = M0 sup0≤τ≤t0
e−(α−δ)τ‖T (τ)∣∣X−‖L(X−).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Spectral decomposition of semigroupsSpectral theorems for semigroups
Spectral theorems for semigroups
It follows from the Spectral Mapping Theorem thatσ(f (A)) = f (σe(A)) when A is a closed operator with non-emptyresolvent and f ∈ U∞(A), this does not hold, in general, if A isunbounded and f /∈ U∞(A).
Since C 3 λ 7→ eλt ∈ C does not belong to U∞(A) for Aunbounded, in general, we cannot say that σ(eAt) = eσe(A)t .
Next we study the relations between the spectrum of a semigroupand the spectrum of its generator.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Spectral decomposition of semigroupsSpectral theorems for semigroups
LemmaLet {eAt : t ≥ 0} be a strongly continuous semigroup. If
Bλ(t)x =
∫ t
0eλ(t−s)eAsxds (7)
then,(λ− A)Bλ(t)x = eλtx − eAtx , ∀x ∈ X (8)
andBλ(t)(λ− A)x = eλtx − eAtx , ∀x ∈ D(A). (9)
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Spectral decomposition of semigroupsSpectral theorems for semigroups
Proof: For all λ and t fixed, Bλ(t) defined by (7) is an operator inL(X ). Besides that, for all x ∈ X we have
eAh − I
hBλ(t)x =
eλh−1
h
∫ t
heλ(t−s)eAsxds +
eλh
h
∫ t+h
teλ(t−s)eAsxds
− 1
h
∫ h
0eλ(t−s)eAsxds
h→0+
−→ λBλ(t)x + eAtx − eλtx .
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Theorems on Perturbations of GeneratorsTeoremas Espectrais and Dicotomias
Spectral decomposition of semigroupsSpectral theorems for semigroups