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Semigroups and Their Generators
SMA 5878 Functional Analysis II
Alexandre Nolasco de Carvalho
Departamento de MatematicaInstituto de Ciencias Matematicas and de Computacao
Universidade de Sao Paulo
May 02, 2018
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Semigroups and Their Generators
In this chapter we present some basic facts of the theory ofsemigroups of bounded linear operators which are essential for theunderstanding of the techniques of resolution of parabolic andhyperbolic semilinear PDEs.
Most of the exposition will be centered on the characterization ofthe generators of semigroups of bounded linear operatorssince, in applications of the theory, in general, we know thedifferential equation and do not know the solution operator.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
The study of semigroups of bounded linear operators is associatedto the study of linear Cauchy problems of the form
d
dtx(t) = Ax(t)
x(0) = x0
(1)
where A : D(A) ⊂ X → X is linear (in general unbounded).
The semigorup {T (t) : t ≥ 0} is the solution operator of (1), thatis, given x0 ∈ X , t 7→ T (t)x0 is the solution (in some sense) of (1).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
To beter explain this remark we consider first the case A ∈ L(X ).In this case, the semigroup t 7→ T (t) is the solution operator (inthe usual sense) of the problem
d
dtT (t) = AT (t)
T (0) = B ∈ L(X ).(2)
with B = I . This solution will be denoted by T (t) =: etA.
Let us show that there exists a unique solution for (2) and that thesemigroup properties are satisfied. This follows from the Banachcontraction principle that we state next.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
LemmaLet X be a complete metric space with metric dX : X × X → R+
and a function F :X→X such that dX (F n(x),F n(y)) ≤ k dX (x , y)for some positive integer n and k < 1 (F n is a contraction). Then,there exists a unique x ∈ X such that F (x) = x . The elementx ∈ X is called a fixed point of F .
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Now we will seek for solutions for (2) which are functions in{U(·) ∈ C ([0, τ ], L(X )) ∩ C 1((0, τ ], L(X )) : U(0) = B} thatverify (2). Let K = {U(·) ∈ C ([0, τ ],L(X )) : U(0) = B} anddefine the map F : K → K by
F (U)(t) = B +
∫ t
0AU(s)ds
and observe that a solution of (2) is a fixed point of F in K andthat is a fixed point of F is a solution of (2). Note that K is acomplete metric space with the metric induced by the norm ofC ([0, τ ],L(X )).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
We wish to show that there exists a positive integer n such that F n
is a contraction. In fact:
‖F (U)(t)− F (V )(t)‖ ≤∣∣∣∣∫ t
0‖AU(s)− AV (s)‖ds
∣∣∣∣≤ |t|‖A‖ sup
t∈[0,τ ]‖U(t)− V (t)‖
≤ τ‖A‖ supt∈[0,τ ]
‖U(t)− V (t)‖
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Suppose that, for t ∈ [0, τ ],
‖F n−1U(t)− F n−1V (t)‖ ≤ |t|n−1‖A‖n−1
(n − 1)!sup
t∈[0,τ ]‖U(t)− V (t)‖,
then
‖F n(U)(t)− F n(V )(t)‖ ≤∣∣∣∣∫ t
0‖AF n−1U(s)− AF n−1V (s)‖ds
∣∣∣∣≤ |t|
n‖A‖n
n!sup
t∈[0,τ ]‖U(t)− V (t)‖
≤ |τ |n‖A‖n
n!sup
t∈[0,τ ]‖U(t)− V (t)‖.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Noting that |τ |n‖A‖nn! → 0 as n→∞, we have that there is a
positive integer n0 such that F n0 is a contraction and it followsfrom the Banach Contraction Principle that there is a unique fixedpoint for F . It is easy to see that this fixed point is a continuouslydifferentiable function and that it satisfies (2).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Since the above reasoning holds for all τ ∈ R we obtain that thesolutions of (2) are globally defined.
Now let us verify that the semigroup property is satisfied for thesolution T (t) of (2) with B = I .
Note that U(t) = T (t + s) and V (t) = T (t)T (s) are solutions of(2) satisfying U(0) = V (0) = T (s).
It follows from the uniqueness of solutions thatT (t + s) = T (t)T (s). Thus, {T (t) : t ∈ R} is a uniformlycontinuous group of bounded linear operators.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Clearly, we will be interested in more general situations since, inseveral applications, the operator A is not bounded.
Reciprocally, given a semigroup of bounded linear operators we canassociate it to a differential equation through the followingdefinition.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Definicoes e resultados basicos
DefinitionA semigroup of bounded linear operators in X is a family{T (t) : t ≥ 0} ⊂ L(X ) such that
(i) T (0) = IX ,
(ii) T (t + s) = T (t)T (s), for all t, s ≥ 0.
If besides that
(iii) ‖T (t)− IX‖L(X )t→0+−→ 0, we say that the semigroup is
uniformly continuous.
(iv) ‖T (t)x − x‖Xt→0+−→ 0, for each x ∈ X , we say that the
semigroup is strongly continuous.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
DefinitionIf {T (t), t ≥ 0} ⊂ L(X ) is a strongly continuous semigroup ofbounded linear operators, its infinitesimal generator is theoperator defined by A : D(A) ⊂ X → X , where
D(A) =
{x ∈ X : lim
t→0+
T (t)x − x
texists
},
Ax = limt→0+
T (t)x − x
t, ∀ x ∈ D(A).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Example
Let A ∈ L(X ) and define eAt :=∞∑n=0
Antn
n! . Then {eAt : t ∈ R} is a
uniformly continuous group of bounded linear operators withgenerator A satisfying ‖eA t‖ ≤ e |t|‖A‖.
The series∞∑n=0
Antn
n! converges absolutely, uniformly in compact
subsets of R. In fact, since ‖An‖ ≤ ‖A‖n, we have that
‖eAt‖ ≤∞∑n=0
∥∥∥∥Antn
n!
∥∥∥∥ ≤ ∞∑n=0
(|t| ‖A‖)n
n!= e |t| ‖A‖, t ∈ R and
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
∞∑n=1
∥∥∥∥ Antn−1
(n − 1)!
∥∥∥∥ ≤ ‖A‖ ∞∑n=0
(|t| ‖A‖)n
n!= ‖A‖e |t| ‖A‖, t ∈ R.
Therefored
dteAt = AeAt , t ∈ R.
Also‖eAt − I‖ ≤ |t|‖A‖e |t|‖A‖ → 0
as t → 0. It follows that {T (t) : t ∈ R} is the unique solution ofx = Ax with x(0) = I . The result now follows from the previousconsiderations.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
The following result very useful to obtain properties of regularity ofsemigroups.
Lemma (2)
Let φ be a continuous function which is right differentiable in theinterval [a, b). If D+φ is continuous in [a, b) then, φ iscontinuously differentiable in [a, b).
Proof: Exercise.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Every strongly continuous semigroup posses an exponential boundthat is given in the following theorem.
Theorem (1)
Suppose that {T (t), t ≥ 0} ⊂ L(X ) is a strongly continuoussemigroup. Then, there exists M ≥ 1 and β ∈ R such that
‖T (t)‖L(X ) ≤ Meβ t , ∀t ≥ 0.
For any ` > 0 we can choose β ≥ 1` log ‖T (`)‖L(X ) and then
choose M.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Proof: First note that there is an η > 0 such thatsupt∈[0,η] ‖T (t)‖<∞. This is a consequence of the fact that, for
each sequence {tn}n∈N em (0,∞) with tnn→∞−→ 0+, {T (tn)x}n∈N is
bounded for each x ∈ X and, from the Uniform BoundednessPrinciple, {‖T (tn)‖L(X )}n∈N is bounded.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Choose ` > 0 such that sup{‖T (t)‖L(X ), 0 ≤ t ≤ `}=M<∞ and
let β ≥ 1` log{‖T (`)‖L(X )} that is ‖T (`)‖L(X ) ≤ eβ` and then
‖T (n`+ t)‖ = ‖T (`)nT (t)‖ ≤ ‖T (`)‖n‖T (t)‖ ≤ Meβn`
≤ Me |β|`eβ(n`+t), 0 ≤ t ≤ `; n = 0, 1, 2 · · ·
and the result follows.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
The theorem that follows characterizes completely the uniformlycontinuous semigroups of bounded linear operators through itsgenerators.
TheoremGiven a strongly continous semigroup {T (t), t ≥ 0} ⊂ L(X ), thefollowing statements are equivalent:
(a) The semigroup is uniformly continuous,
(b) The infinitesimal generator is defined in all of X ,
(c) For some A in L(X ), T (t) = et A.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Proof: If T (t) = et A for some A ∈ L(X ) the other statementswere proved in Example 1.
If the infinitesimal generator of {T (t) : t ≥ 0} is globally defined
then,{∥∥∥T (t)x−x
t
∥∥∥X
}0≤t≤1
is bounded for each x and from the
Uniform Boundedness Principle
{∥∥∥T (t)−It
∥∥∥L(X )
}0≤t≤1
is bounded
and therefore T (t)→ I in L(X ) as t → 0+.
It is enough to prove that, if T (t)t→0+−→ I in L(X ), there is
A ∈ L(X ) with T (t) = eAt .
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Supposing that T (t)→ I in L(X ) as t → 0+, there is a δ > 0such that ‖T (t)− I‖L(X ) ≤ 1/2, 0 ≤ t ≤ δ. So, for t > 0,
‖T (t + h)− T (t)‖L(X ) = ‖(T (h)− I )T (t)‖L(X ) → 0,
‖T (t)− T (t − h)‖L(X ) = ‖(T (h)− I )T (t − h)‖L(X ) → 0
as h→ 0+, since ‖T (t)‖L(X ) is bounded in [0, δ].
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Therefore t → T (t) : R+ → L(X ) is continuous and the integral∫ t
0T (s)ds is well defined. Besides that,
∥∥∥∥1
δ
∫ δ
0T (s)ds − I
∥∥∥∥L(X )
≤ 1/2
and therefore
(∫ δ
0T (s)ds
)−1∈ L(X ). Define
A = (T (δ)− I )
(∫ δ
0T (s)ds
)−1.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
For each h > 0,
h−1(T (h)−I )∫ δ
0T (s)ds = h−1
{∫ δ+h
hT (s)ds −
∫ δ
0T (s)ds
}= h−1
∫ δ+h
δT (s)ds − h−1
∫ h
0T (s)ds
h→0+−→ T (δ)− I .
Hence h−1(T (h)− I )h→0+−→ A and h−1(T (t + h)− T (t)) =
T (t)T (h)−Ih = T (h)−I
h T (t)h→0+−→ T (t)A = AT (t).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Thus t → T (t) has a right derivative
d+
dtT (t) = T (t)A = AT (t)
which is continuous for t ≥ 0.
It follows from Lemma 2 that t 7→ T (t) is continuouslydifferentiable and, from the uniqueness of solutions for the problemx = Ax with x(0) = I , consequently, T (t) = eAt , t ≥ 0.
In view of this theorem the theory of semigroups concentrates inthe study of strongly continuous semigroups and their generators.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Our next result collects some important facts about stronglycontinuous semigroups of bounded linear operators that will befrequently used in the rest of this chapter.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
TheoremLet {T (t)} be a strongly continuous semigroup. Then,
1. For each x ∈ X , t → T (t)x is continuous for t ≥ 0.
2. t → ‖T (t)‖L(X ) is lower semicontinuous and thereforemeasurable.
3. If A is the generator of T (t), then, A is densely defined andclosed. For x ∈ D(A), t 7→ T (t)x is continuouslydifferentiable and
d
dtT (t)x = AT (t)x = T (t)Ax , t > 0.
4. ∩m≥1D(Am) is dense in X .
5. For Reλ > β (β from Theorem 1), λ ∈ ρ(A) and
(λ− A)−1x =
∫ ∞0
e−λtT (t)xdt, ∀x ∈ X
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Proof: 1. The continuity of t 7→ T (t)x is a consequence ofTheorem 1 and from the fact that, t > 0 and x ∈ X ,
‖T (t + h)x − T (t)x‖X = ‖(T (h)− I )T (t)x‖Xh→0+−→ 0,
‖T (t)x − T (t − h)x‖X ≤ ‖T (t − h)‖L(X )‖T (h)x − x‖Xh→0+−→ 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
2. Let us show that {t ≥ 0 : ‖T (t)‖L(X ) > b} is open in [0,∞) foreach b which implies the statement.
But ‖T (t0)‖L(X ) > b implies that there exists x ∈ X with‖x‖X = 1 such that ‖T (t0)x‖ > b.
It follows from 1. that ‖T (t)x‖ > b for all t sufficiently close to t0,so ‖T (t)‖L(X ) > b for t in a neighborhood of t0 and the resultfollows.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
3. Let x ∈ X and for ε > 0, xε = 1ε
∫ ε
0T (t)x dt then, xε → x as
ε→ 0+ and, for h > 0,
h−1(T (h)xε − xε) =1
εh
{∫ ε+h
εT (t)x dt −
∫ h
0T (t)x dt
}h→0+−→ 1
ε(T (ε)x − x).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
So xε ∈ D(A). It will be an immediate consequence of 5. that A isclosed since (λ− A)−1 ∈ L(X ).
If x ∈ D(A) it is clear that
d+
dtT (t)x = lim
h→0+
1
h{T (t + h)x − T (t)x} = AT (t)x = T (t)Ax
is continuous and any function with right derivative continuous iscontinuously differentiable.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
4. Let φ : R→ R be a function in C∞(R) and φ(t) = 0 in aneighborhood of t = 0 and also for t sufficiently large, let x ∈ X
and f =
∫ ∞0φ(t)T (t)x dt.
It follows easily from
h−1(T (h)f − f ) = h−1∫ ∞h
(φ(t − h)− φ(t))T (t)x dt
that f ∈ D(A) and that Af = −∫ ∞0φ′(t)T (t)x dt.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Since −φ′ satisfies the same conditions that φ,
Amf = (−1)m∫ ∞0φ(m)(t)T (t)x dt
for all m ≥ 1 e f ∈ ∩m≥1D(Am).
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
To show that such set of points is dense in X , choose φ above
satisfying also
∫ ∞0φ(t)dt = 1 then, if
fn =
∫ ∞0
nφ(nt)T (t)xdt =
∫ ∞0φ(s)T (s/n)xds, n = 1, 2, 3, · · · ,
we have that fn ∈ ∩m≥1D(Am) and fn → x as n→∞.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
5. Define R(λ) ∈ L(X ) by
R(λ)x =
∫ ∞0
e−λtT (t)xdt.
Note that ‖R(λ)‖≤ MReλ−β for all λ ∈ C such that Reλ>β and
‖T (t)‖L(X ) ≤ Meβt .
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Let x ∈ X and h > 0
h−1(T (h)− I )R(λ)x = R(λ)T (h)x − x
h
= h−1[∫ ∞
he−λt+λhT (t)x −
∫ ∞0
e−λtT (t)x
]= h−1
[−∫ h
0eλ(h−t)T (t)x +
∫ ∞0
(eλh − 1)e−λtT (t)x
]h→0+−→ −x + λR(λ)x .
(3)
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Therefore R(λ)x ∈ D(A) and (λ− A)R(λ)x = x , and λ− A isonto. Also, if x ∈ D(A) then, integrating by parts,R(λ)Ax = λR(λ)x − x = AR(λ)x .
It follows that (λ−A)R(λ)x = x = R(λ)(λ−A)x for all x ∈ D(A)and λ−A is also injective. Hence (λ−A) is a bijection from D(A)onto X with bounded inverse R(λ) and the proof is complete.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
TheoremLet {T (t), t ≥ 0} and {S(t), t ≥ 0} strongly continuoussemigroups with generators A and B respectively. If A = B thenT (t) = S(t), t ≥ 0.
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II
Semigroups and Their Generators
Proof: Let x ∈ D(A) = D(B). From Theorem 3 it follows easilythat the function s 7→ T (t − s)S(s)x is differentiable and that
d
dsT (t − s)S(s)x = −AT (t − s)S(s)x + T (t − s)BS(s)x
= −T (t − s)AS(s)x + T (t − s)BS(s)x = 0.
Therefore s 7→ T (t − s)S(s)x is constant and in particular itsvalues at s = 0 and s = t are the same, that is, T (t)x = S(t)x .
This holds for all x ∈ D(A) and since D(A) is dense in X andS(t), T (t) are bounded, T (t)x = S(t)x for all x ∈ X .
Alexandre Nolasco de Carvalho ICMC - USP SMA 5878 Functional Analysis II