24.9 General Theory Of Continuous Semigroups
Much more on semigroups is available in Yosida . This is just an introduction to the
Definition 24.9.1 A strongly continuous semigroup defined on H,a Banach
space is a function S : [0,∞) → H which satisfies the following for all x0 ∈ H.
Sometimes such a semigroup is said to be C0. It is said to have the linear operator A as its
and for x ∈ D
, A is defined by
The assertion that t → S
is continuous and that S
sufficient to say there is a bound on
Also the assertion that for each
is not the same as saying that S
It is a much weaker assertion. The next
theorem gives information on the growth of
It turns out it has exponential
Lemma 24.9.2 Let M ≡ sup
. Then M < ∞.
Proof: If this is not true, then there exists tn ∈
That is the
are not uniformly bounded. By the uniform boundedness principle, Theorem
, there exists x ∈ H
is not bounded. However, this is impossible
because it is given that
t → S
is continuous on
its maximum on this compact set.
Now here is the main result for growth of
Theorem 24.9.3 For M described in Lemma 24.9.2, there exists α such
In fact, α can be chosen such that M1∕T = eα.
Proof: Let t be arbitrary. Then t = mT + r
. Then by the
Now mT ≤ t ≤ mT + r
Let M1∕T ≡ eα and then
This proves the theorem.
Definition 24.9.4 Let S
be a continuous semigroup as described above. It is
called a contraction semigroup if for all t ≥
It is called a bounded semigroup if there exists M such that for all t ≥ 0,
Note that for S
an arbitrary continuous semigroup satisfying
It follows that the semigroup,
is a bounded semigroup which satisfies
Proposition 24.9.5 Given a continuous semigroup S
, its generator A exists and is a
closed densely defined operator. Furthermore, for
and λ > α, λI −A is onto and
−1 maps H onto D
and is in ℒ
. Also for
these values of λ,
For λ > α, the following estimate holds.
Proof: First note D
In fact 0 ∈ D
. It follows from Theorem
that for all
large enough, one can define a Laplace transform,
Here the integral is the ordinary improper Riemann integral. I claim each of these is in
Using the semigroup property and changing the variables in the first of the above integrals,
The limit as h → 0 exists and equals
x ∈ D
as claimed and
Since x is arbitrary, this shows that for λ large enough, λI − A is onto.
Why is D
dense? It was shown above that
and thererfore λR
x ∈ D
Then for λ > α
Now since S
x − x →
it follows that for h
is large enough. Thus D
is dense as claimed.
Let x ∈ D
Then for y∗∈ H′,
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
Thus since y∗
is arbitrary, for x ∈ D
Why is A closed? Suppose xn → x and xn ∈ D
Axn → z.
From what was just
and so, passing to the limit this yields
which shows Ax = z and x ∈ D
Because of 24.37 it follows R
and this also proves the last estimate. Also from
. This proves the proposition.
The linear mapping
is called the resolvent.
The above proof contains an argument which implies the following corollary.
Corollary 24.9.6 Let S
be a continuous semigroup and let A be its generator. Then
0 < a < b and x ∈ D
and also for t > 0 you can take the derivative from the left,
Proof:Letting y∗∈ H′,
The difference quotients are bounded because they converge to Ax. Therefore, from the
dominated convergence theorem,
is arbitrary, this proves the first part. Now from what was just shown, if t >
0 and h
is small enough,
which converges to S
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 24.9.7 Suppose A is a densely defined linear operator which has the
property that for all λ > 0,
which means that λI − A : D
→ H is one to one and onto with continuous inverse.
Suppose also that for all n ∈ ℕ,
Then there exists a continuous semigroup, S
which has A as its generator and satisfies
≤ M and A is closed. In fact letting
it follows limλ→∞Sλ
x uniformly on finite intervals. Conversely, if A is
the generator of S
, a bounded continuous semigroup having
≤ M, then
for all λ >
0 and 24.38 holds.
Proof: Consider the operator
which makes sense on all of H, not just on D
. Also this last expression equals
on all of H because λI −A is given to be onto. Denote this as Aλ to save notation. Thus on