Much more on semigroups is available in Yosida [32]. This is just an introduction to the
subject.
Definition 24.9.1A strongly continuous semigroup defined on H,a Banachspace is a function S : [0,∞) → H which satisfies the following for all x_{0}∈ H.
S(t) ∈ ℒ (H, H ),S (t+ s) = S (t)S (s),
t → S (t)x0 is continuous, lim S (t)x0 = x0
t→0+
Sometimes such a semigroup is said to be C_{0}. It is said to have the linear operator A as itsgenerator if
{ }
D (A) ≡ x : lim S-(h)x−-x exists
h→0 h
and for x ∈ D
(A )
, A is defined by
lim S-(h)x−-x ≡ Ax
h→0 h
The assertion that t → S
(t)
x_{0} is continuous and that S
(t)
∈ℒ
(H,H )
is not
sufficient to say there is a bound on
||S (t)||
for all t. Also the assertion that for each
x_{0},
tl→im0+ S(t)x0 = x0
is not the same as saying that S
(t)
→ I in ℒ
(H,H )
. It is a much weaker assertion. The next
theorem gives information on the growth of
||S(t)||
. It turns out it has exponential
growth.
Lemma 24.9.2Let M ≡ sup
{||S(t)|| : t ∈ [0,T]}
. Then M < ∞.
Proof: If this is not true, then there exists t_{n}∈
[0,T]
such that
||S (tn)||
≥ n. That is the
operators S
(tn)
are not uniformly bounded. By the uniform boundedness principle, Theorem
23.1.8, there exists x ∈ H such that
||S(tn)x||
is not bounded. However, this is impossible
because it is given that t → S
(t)
x is continuous on
[0,T ]
and so t →
||S(t)x||
must achieve
its maximum on this compact set.
Now here is the main result for growth of
||S(t)||
.
Theorem 24.9.3For M described in Lemma 24.9.2, there exists α suchthat
αt
||S (t)|| ≤ M e .
In fact, α can be chosen such that M^{1∕T} = e^{α}.
Proof:Let t be arbitrary. Then t = mT + r
(t)
where 0 ≤ r
(t)
< T. Then by the
semigroup property
||S(t)|| = ||S (mT + r(t))||
m m+1
= ||S(r(t))S (T) || ≤ M
Now mT ≤ t ≤ mT + r
(t)
≤
(m + 1)
T and so
m ≤ t-≤ m + 1
T
Therefore,
( )t
||S (t)|| ≤ M (t∕T)+1 = M M 1∕T .
Let M^{1∕T}≡ e^{α} and then
||S (t)|| ≤ M eαt
This proves the theorem.
Definition 24.9.4Let S
(t)
be a continuous semigroup as described above. It iscalled a contractionsemigroup if for all t ≥ 0
||S (t)|| ≤ 1.
It is called a bounded semigroup if there exists M such that for all t ≥ 0,
||S(t)|| ≤ M
Note that for S
(t)
an arbitrary continuous semigroup satisfying
||S (t)|| ≤ M eαt,
It follows that the semigroup,
T (t) = e−αtS (t)
is a bounded semigroup which satisfies
||T (t)|| ≤ M.
Proposition 24.9.5Given a continuous semigroup S
(t)
, its generator A exists and is aclosed densely defined operator. Furthermore, for
||S(t)|| ≤ M eαt
and λ > α, λI −A is onto and
(λI − A)
^{−1}maps H onto D
(A )
and is in ℒ
(H,H )
. Also forthese values of λ,
∫
−1 ∞ −λt
(λI − A) x = 0 e S(t)xdt.
For λ > α, the following estimate holds.
|||| −1|||| --M---
||(λI − A ) || ≤ |λ− α |
Proof: First note D
(A )
≠∅. In fact 0 ∈ D
(A)
. It follows from Theorem 24.9.3 that for all
λ large enough, one can define a Laplace transform,
∫
R (λ)x ≡ ∞e− λtS (t)xdt ∈ H.
0
Here the integral is the ordinary improper Riemann integral. I claim each of these is in
D
(A)
.
∫ ∞ ∫ ∞
S(h)-0-e−-λtS-(t)xdt−--0--e−-λtS-(t)xdt-
h
Using the semigroup property and changing the variables in the first of the above integrals,
this equals
( ∫ ∞ ∫ ∞ )
= 1- eλh e− λtS (t)xdt − e−λtS(t)xdt
h h 0
( ∫ ∞ ∫ h )
= 1- (eλh − 1) e−λtS(t)xdt− eλh e−λtS(t)xdt
h 0 0
The limit as h → 0exists and equals
λR (λ )x− x
Thus R
(λ)
x ∈ D
(A)
as claimed and
AR (λ)x = λR (λ)x − x
Hence
x = (λI − A)R (λ)x. (24.37)
(24.37)
Since x is arbitrary, this shows that for λ large enough, λI − A is onto.
( ∫ ) ∫ ( )
∗ t t ∗ S(h)x-−-x
y 0 S(s)Axds = 0 y S(s)hli→m0+ h ds
The difference quotient is given to have a limit and so the difference quotients are bounded.
Therefore, one can use the dominated convergence theorem to take the limit outside the
integral and write the above equals
∫ ( )
t ∗ S-(h)x−-x
hli→m0+ 0 y S(s) h ds
( (∫ t+h ∫ t ) )
= lim y∗ 1- S (s)xds− S(s)xds
h→0+ h h 0
(∫ t+h ∫ h )
= lim y∗ S (s)xds − S(s)x
h→0+ t 0
= y∗(S(t)x− x).
Thus since y^{∗} is arbitrary, for x ∈ D
(A)
∫
t
S(t)x = x + 0 S(s)Axds
Why is A closed? Suppose x_{n}→ x and x_{n}∈ D
(A )
while Ax_{n}→ z. From what was just
shown
∫ t
S (t)xn = xn + S (s)Axnds
0
and so, passing to the limit this yields
∫
S (t)x = x + tS (s)zds
0
which implies
∫
S(t)x−-x- 1 t
tli→m0+ h = hli→m0+ t 0 S (s)zds = z
||∫ ||
|||| ∞ −λt ||||
||R (λ)x|| ≤ || 0 e S (t)xdt||
∫
≤ ∞ e−λtM eαtdt||x|| ≤--M---||x||
0 |λ− α|
so R
(λ)
=
(λI − A)
^{−1}∈ℒ
(H, H)
and this also proves the last estimate. Also from 24.37,
R
(λ)
maps H onto D
(A)
. This proves the proposition.
The linear mapping
(λI − A)
^{−1} is called the resolvent.
The above proof contains an argument which implies the following corollary.
Corollary 24.9.6Let S
(t)
be a continuous semigroup and let A be its generator. Thenfor 0 < a < b and x ∈ D
(A)
∫ b
S(b)x− S (a) x = S (t)Axdt
a
and also for t > 0 you can take the derivative from the left,
lim S(t)x−-S-(t−-h)x-= S(t)Ax
h→0+ h
Proof:Letting y^{∗}∈ H^{′},
( ∫ b ) ∫ b ( )
y∗ S(t)Axdt = y∗ S(t) lim S-(h)x−-x dt
a a h→0 h
The difference quotients are bounded because they converge to Ax. Therefore, from the
dominated convergence theorem,
(∫ b ) ∫ b ( )
y∗ S (t)Axdt = lim y∗ S (t) S(h)x-−-x dt
a h→0 a( h )
∫ b S(h)x − x
= hli→m0 y∗ S (t) ----h----dt
a
( )
∗ -1∫ b+h 1∫ b
= lhim→0 y h a+h S (t)xdt− h a S (t)xdt
( ∫ ∫ )
∗ -1 b+h 1- a+h
= lhim→0 y h b S (t)xdt− h a S (t)xdt
∗
= y (S (b)x − S(a)x)
Since y^{∗} is arbitrary, this proves the first part. Now from what was just shown, if t > 0 and h
is small enough,
∫ t
S-(t)x-−-S(t−-h)x-= 1- S (s)Axds
h h t−h
which converges to S
(t)
Ax as h → 0 + . This proves the corollary.
Given a closed densely defined operator, when is it the generator of a bounded
semigroup? This is answered in the following theorem which is called the Hille Yosida
theorem.
Theorem 24.9.7Suppose A is a densely defined linear operator which has theproperty that for all λ > 0,
(λI − A )−1 ∈ ℒ (H, H )
which means that λI − A : D
(A )
→ H is one to one and onto with continuous inverse.Suppose also that for all n ∈ ℕ,
||||( −1)n|||| M
|| (λI − A ) || ≤ λn. (24.38)
(24.38)
Then there exists a continuous semigroup, S
(t)
which has A as its generator and satisfies
||S(t)||
≤ M and A is closed. In fact letting
( )
Sλ (t) ≡ exp − λ + λ2(λI − A )−1
it follows lim_{λ→∞}S_{λ}
(t)
x = S
(t)
x uniformly on finite intervals. Conversely, if A isthe generator of S