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4 The Abstract Cauchy Problem
T(t) is obviously a Co semigroup of contractions on X. Its infinitesimal
generator A is given by D( A) = {f E Y': f' E X} and Af = f' for f E D( A).
Denoting the norm in X by II II, we consider the space Y of elements
g E Y' for which g' E X. We equip Y with the norm Ilgll y = IIgll + Ilg'll
for g E Y. The norm II II y is stronger than II II, Y is closed in the norm
II II y and it is easy to see that the semigroup T(t) defined by (5.4) leaves Y
invariant and is a Co semigroup in Y. Thus Y is A-admissible.
bzstrttttsesnoss
Theorem 5.5. Let T( t) be a Co semigroup on X and let A be its infinitesimal
generator.
A subspace Y of X is A -admissible if and only if
(i) Y is an invariant subspace of R(A: A) for all A > w.
(ii) A, the part of A in Y, is the infinitesimal generator of a Co semigroup
on Y.
Moreover, if Y is A-admissible then A is the infinitesimal generator of the
restriction of T( t) to Y.
PROOF. Assume that Y is A-admissible. Since T(t)Y c Y for t 0 and
since II II y is stronger than II II and the restriction of T(t) to Y is a Co
semigroup in Y it follows from (5.1) that there is an w such that for A> w
R(A: A)Y c Y. Let Al be the infinitesimal generator of the restriction of
T( t) to Y. From the definition of the infinitesimal generator it follows
readily that D(AI) c D(A) () Yand that for x E D(AI)' Alx = Ax and so
A :J AI. On the other hand if x E D(A) then Ax E Yand the equality
T(t)x - x
= {T(s )Ax ds
(5.5)
holds in Y. Dividing (5.5) by t > 0 and letting t ! 0 it follows that x E D( A I)
and so D(A 1) :J DC'I). Thus A = Al and A is the infinitesimal generator
of a Co semigroup on Y, namely, the restriction of T(t) to Y.
Conversely assume that (i) and (ii) are satisfied and denote by S(t) the Co
semigroup generated by A on Y. From the assumption (i) and Lemma 5.2 it
follows that R(A: A)x = R(A: A)x for every x E Yand therefore also
(5.6)
for all n large enough and x E Y. Passing to the limit as n -+ 00 it follows
from the exponential formula (Theorem 1.8.3) that the left-hand side of
(5.6) converges in Y, and hence also in X, to S(t)x while the right-hand side
converges in X to T(t)x. Therefore S(t)x = T(t)x for every x E Y which
implies both that Y is an invariant subspace for T( t) and that T( t) is a Co
semigroup on Y.
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Scmigroups of Linear Operators
pre
Corollary 5.6. Y is A -admissible if and only if
(i) For
large A, Y is an invariant subspace of RCA. : A).
(ii) There exist constants M and /3 such that
IIR(A:
(iii) For A >
Arllt·:s; M(A.
/3, R( A: A) Y is
- /3)-n,
n
A > {j,
=
1,2,....
(5.7)
dense in Y.
Condition (i) is the same as in Theorem 5.5. From (i) and Lemma
5.2 it follows that R(A: A)x = R(A: A)x for x E Y and A> w. Therefore
we can replace A by A in (5.7) and in Condition (iii). Condition (iii) is then
equivalent to the fact that D(A) = R(A: A)Y = R(A: A)Y is dense in Y
and from Theorem 1.5.3 it follows that A generates a Co semigroup on Y if
and only if (ii) and (iii) are satisfied. From Theorem 5.5 it then follows that
Y is A-admissible if and only if (i)-(iii) are satisfied.
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PROOF.
Remark 5.7. If in Corollary 5.6 Y is reflexive the Condition (iii) follows
from (i) and (ii). Indeed, for A, p. E peA) we have the resolvent identity
R(A: A) - R(p.: A)
=
(p. - A)R(A: A)R(p.: A)
which implies directly that D = R(A: A)Y is independent of A E peA).
From (5.7) with n = I it follows that for x E Y, AR(A: A)x is bounded in
Y as A -> 00. The reflexivity of Y then implies that there is a sequence
An -> 00 such A" R( An: A)x converges weakly in Y to some y E Y. Since A
is the infinitesimal generator of a Co semigroup on X, AR(A: A)x -> x
strongly in X as A -> 00 (Lemma 1.3.2) so y = x. Since for large values
A", A"R(A,,: A)x E D we conclude that the weak closure of D in Y is all of
Y. But the weak and strong closures of a linear subspace of a Banach space
are the same and so D is dense in Y.
We conclude this section with a useful criterion for a subspace Y of X to
be A-admissible.
Theorem 5.S. Let Y be the closure of Y in the norm of X. Let S be all
isomorphism of Y onto Y. Y is A -admissible if alld only if A I = SA S - I is the
infinitesimal generator of a Cu semigroup on
Y. In this case we have in Y
Tsetse
where TI (t) is the semigroup generated by A I'
PROOF.
Let
A be
the part of A in Y. From the definition of A I we have
D(A I )
=
{x E
=
{x E
Y:
Y:
S-I X E D(A),AS-Ix E Y}
S-I X E D(A)}
It follows that D( A I) is dense in
Y if
=
SD(A).
and only if D( A) is dense in Y.