Fattorini H.O., Kerber A. The Cauchy Problem
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3.6. Isometric Propagators and Conservative Operators
157
which must coincide with A by maximality. We can then assume in any case
that A is closed. Let K = (I - A)D(A) * E. We have already observed
(Remark 3.1.12) that K is closed, so that its orthogonal complement K -`
does not reduce to (0). It is immediate that K 1 consists of all v e D(A*)
with
A*v = v. (3.6.10)
We note next that D(A)n K = (0). In fact, let u e D(A)n K 1. Then
((I - A)u, u) = 0, or (Au, u) =11u112; since Re(Au, u) < 0, u = 0 as claimed.
We define an operator A in D(A) = D(A)+ K 1 as follows:
A(u+v)=Au-v.
(3.6.11)
Clearly A is an extension of A. Taking (3.6.10) into account, we have
(A(u + v), u + v) = (Au, u)- (v, v)+2iIm(u, v)
(3.6.12)
for u + v E D(A), which proves A dissipative. This contradicts the assump-
tion that A is maximal dissipative. Accordingly (I - A)D(A) = E, thus
ending the proof.
Some additional information can be coaxed out of these arguments.
Let A be as before a closed dissipative operator (if A is not closed but
densely defined, we take its closure using Lemma 3.1.11). The operator A
can be constructed as above, and it is not difficult to show that A is
m-dissipative. In fact, take w e H and write w = (I - A)u +2v, with u E
D(A), v E K 1. Then (I - A)(u + v) = (I - A)u +2v = w, which shows
that A is m-dissipative. (It follows from Theorem 3.6.2 that A is as well
maximal dissipative.) We have then proved
3.6.3 Lemma. Let A0 be dissipative in H and assume that A0 is
either closed or densely defined. Then there exists a m-dissipative extension
of A.
Lemma 3.6.3 has the following application to the theory of the Cauchy
problem in Hilbert space: if A0 is a closed or densely defined (but otherwise
arbitrary) dissipative operator, there exists an extension A E e+ (0, 1) of A0, that is,
the Cauchy problem for
u'(t)=Au(t)
is well posed in t > 0 and the propagator satisfies IIS(t)II < 1. The requirements
on A0 are in general not difficult to meet: for instance, if A0 is a differential
operator and we take as D(A0) a space of sufficiently smooth functions in L2
(satisfying boundary conditions if we work in a portion of Euclidean space), it is
easy, using integration by parts, to establish conditions on the coefficients of A0 and
on the boundary conditions (if any) that guarantee dissipativity of A0. On the other
hand, since the elements of D(A0) are only restricted by smoothness assumptions
and (possibly) by boundary conditions, it is plain that D(A0) must be dense in L2.
However, the result refers not to Ao but to A, and the construction of A and of
158
Dissipative Operators and Applications
D(A) provided by Lemma 3.6.5 is far from explicit, since it presupposes the
identification of the subspace K or, equivalently, of the nullspace of the operator
I - A A. Because of this, the usefulness of Lemma 3.6.5 is rather limited in this
context.
3.6.4
Remark. The argument in Theorem 3.6.2 proving that a m-dissipative
operator must be maximal dissipative extends without changes to the Banach space
case. This is not the case with the converse, as the following counterexample shows.
3.6.5 Example (Lumer-Phillips [1961: 1]). Let E = Co.1 [0,11R and Au = u', where
D(A) is the set of all u E E such that Au' G E. Then (a) A is maximal dissipative,
and (b) (I-A)D(A)*E.
We note also that Lemma 3.6.3 fails in the Banach space case, since an
operator as the one in Example 3.6.5 is not m-dissipative and, being maximal
dissipative, does not admit dissipative extensions at all. We note in passing that
every dissipative operator (not necessarily closed or densely defined) in an arbitrary
Banach space E can be extended to a maximal dissipative operator by a standard
application of Zorn's lemma. This is slightly more general than Lemma 3.6.3 even in
the Hilbert space case, although the extension is even more esoteric.
Theorems 3.6.1 and 3.1.9 have interesting interpretations in Hilbert
spaces. Observe that both relations (3.6.3) and (3.6.4) coalesce into
Re(Au, u) = 0
(u E D(A)).
(3.6.13)
It follows that the operator A is conservative if and only if B = - iA is
symmetric. In fact, assume B is symmetric. Then (iBu, u) = - i(Bu, u),
where (Bu, u) is real, thus iB is conservative. On the other hand, let A be
conservative. Taking real parts in the identity (A(u + v), u + v) = (Au, u)+
(Au, v)+(Av, u)+(Av, v), we obtain
Re(Au,v)+Re(u,Av)=0 (u,veD(A)). (3.6.14)
Applying (3.6.14) to iu, v and adding the two equalities, we obtain (Au, v)+
(v, Au)=0 or
(iAu, v) = (u, iAv), (3.6.15)
which shows that B = - iA is symmetric (of course, so is iA).
Condition (3.6.5) (say, for X = 1) can be written in terms of B as
follows:
(B+iI)D(B)=H,
while the same condition for - A is
(B-iI)D(B)=H.
(3.6.16)
(3.6.17)
Clearly (3.6.17) is equivalent to the vanishing of the negative deficiency
index d_ of B; likewise, (3.6.16) holds if and only if d+ = 0 and any of the
two conditions is equivalent to B being maximal symmetric (see Section 6).
We obtain in this way the following result.
3.7. Differential Equations in Banach Lattices, Nonnegative Solutions
159
3.6.6 Theorem. Let A E (2+(1,0) and let the propagator of (3.6.1) be
isometric. Then A = iB, where B is maximal symmetric. Conversely, let B be
maximal symmetric. Then either A = iB (if d+ = 0) or A = - iB (if d_ = 0)
belongs to (2+(1,0) and the propagator of (3.6.1) is isometric.
Theorem 3.6.6 is the translation of Theorem 3.6.1 to the Hilbert
space case. The corresponding version of Theorem 3.1.9 is obtained noting
that both defficiency indices of a symmetric operator are zero if and only if
the operator is self-adjoint.
3.6.7
Theorem.
The operator A belongs to 3 (0, 1) and the propaga-
tor of (3.6.1) is isometric in - oo < t < oo if and only if A = iB, where B is a
self-adjoint operator.
We note that in this case each of the isometric operators S(t) is
invertible (S(t)-' = S(- t)) thus s(t) is unitary, that is, S(t)* = S(t)-'.
These results can be applied to the operators CT of the previous
section in an obvious fashion.
3.6.8
Example. Let A1,...,Am, B satisfy the smoothness assumptions in
Section 3.5, plus condition (3.5.1l)-(3.5.12) and Re(K(x)u, u) = 0 for all
xEE Rm,uEE C',or
Q(x)=O (x E=- R-). (3.6.18)
Then i& = ido = i(do)* is self-adjoint, so that 9 E C(1,0). Condition (3.6.18)
is necessary in order that d be conservative.
On the other hand, if (3.5.1l)-(3.5.12) is violated, condition (3.6.18)
does no longer guarantee that d is conservative (the operator in Example
3.5.6 is not).
3.6.9
Example. None of the operators A, A (/3) or A (/30, /3 ) of Sections
3.2, 3.3, or 3.4 is conservative in the spaces LP (1 < p < oo) or C (see the
next section, however, for conservation of norm of nonnegative solutions
in L').
3.6.10 Example. Check Theorem 3.6.1
for the operators in Example
2.2.6.
3.7.
DIFFERENTIAL EQUATIONS IN BANACH
LATTICES. POSITIVE SOLUTIONS.
DISPERSIVE OPERATORS.
We have already observed in Section 1.3 that when the equation
au
-Au
dt
(3.7.1)
160 Dissipative Operators and Applications
is used as a model for heat or diffusion processes, the only solutions that
can have any physical meaning must be positive. Also we have seen in a
particular case that solutions of (3.7.1) that are initially positive stay positive
forever. We introduce in this section an abstract framework to deal with this
kind of situation.
Let E be a real Banach space with a partial order relation < between
its elements. Assume E is a lattice with respect to <, that is, that any two
elements of E have a least upper bound u V v and a greater lower bound
u A v. Following Birkhoff [1967: 1] (see also Schaefer [1974: 1]), we say that
E is a Banach lattice if the following relations among the linear structure of
E, its norm 11- 11, and the order relation <, are satisfied:
(I) Ifu<v,thenu+w<v+w (weE). (3.7.2)
(II) Ifu<v,then Au<Xv (X>' 0). (3.7.3)
and
(III)
If Jul < Ivl, then (lull < Ilvll,
(3.7.4)
where Iul, the modulus (or absolute value) of u, is the element of E defined
by
Given u E E, let
Jul
=uv(-u).
u+=uv0, u_=(-u)v0.
We call u+ (u_) the positive (negative) part of u. It follows from (I) (taking
w = - u - v) that u < v implies - v < - u; we obtain easily from this that
uvv=-((-u)A(-v)), uAv=-((-u)v(-v)) (3.7.5)
and it is a direct consequence of (I) that
(u V v)+w= (u+w)v(v+w),
(uAv)+w=(u+w)A(v+w)
(wEE) (3.7.6)
and of (II) that
X(uVv)=(Xu)V(Xv),
(x>- 0). (3.7.7)
X(u A v) _ (Xu)A(Xv)
It follows from (3.7.5) and (3.7.6) that u - u+ = u - u V 0 = u +(- u)AO =
OA u=-(Ov(-u))=-u_, thus
u = u+ - u_.
(3.7.8)
On the other hand, u + l u l= u + u V (- u) = 2u V O= 2u, so that
Jul =u++u_.
(3.7.9)
Other useful relations, immediate consequences of the three postulates and
3.7. Differential Equations in Banach Lattices, Nonnegative Solutions
161
of the previously shown equalities, are
lul=O if and onlyifu=0, (3.7.10)
IXul =1Xi Jul,
(3.7.11)
and the "triangle inequality"
lu+vl<lul+ivl
(3.7.12)
valid for arbitrary u, v E E and X E R. Note also that (3.7.4) implies
iiiul11=llull.
(3.7.13)
We have u=v+(u-v)<v++lu-vl, so that u+<v++lu-vi.
Interchanging u and v and combining the two inequalities, we obtain
lu+-v+I<lu-vl,
which implies, in view of (III),
Ilu+ - v+II < Ilu - vll,
(3.7.14)
so that the map u - u+ is uniformly continuous. Naturally, the same is true
of u -. u-; it may be shown by the same elementary means that (u, v) -
u A v and (u, v) -* u V v are uniformly continuous from E X E into E. As a
consequence of these remarks, we see that the set
E+={u;u>, 0}=(u;u_=0)
of nonnegative5 elements of E is closed.
All the real spaces hitherto used in our treatment of differential
operators are Banach lattices under the natural order relations: in the space
C(K) = C(K)R we say that u < v if u(x) < v(x) for x E K. The same
definition applies to the spaces C0(K) and the various subspaces used. In
the
spaces LP(I, µ) = LP(2; OR (1 < p < oo), u < v if u(x) < v(x)
.t-almost everywhere in Q.
A duality map 0: E -* E* is called proper if it satisfies the following
two conditions:
(IV)
If u, v > 0, then (B(u), v) >_ 0. (3.7.15)
(V)
(9(u+), u)=IIu+112
(uEE).
(3.7.16)
A cursory examination of the duality maps in question establishes the
following result.
3.7.1
Lemma. (a) Let K be a closed subset of IW' and E = CO(K),
C(K) (if K is compact), or a closed subspace thereof. Then any duality map
0: E -* E * is proper. (b) Let 0 be a measurable subset of QB m, E = L P (2).
Then if 1 < p < oo, the only duality map 0: E - E* is proper; if p = 1, a
5 In this context, "nonnegative" does not mean "not negative."
162
Dissipative Operators and Applications
duality map 0 is proper if and only if
0(u)(x) = 0 a.e. where u(x) = 0.
An operator B : E -p E is said to be positive if and only if
Bu >, 0 whenever u >, 0, (3.7.17)
and an operator A : D(A) - E will be called dispersive (with respect to a
proper duality map 0) if and only if
(0(u+ ), Au) < 0 (u E D(A)). (3.7.18)
Our goal is to characterize those operators A E (2+(1,0) for which the
propagator of
u'(t) = Au(t) (3.7.19)
is positive for all t. This will be done in the following Theorem 3.7.3. We
begin by establishing an auxiliary result of considerable interest in itself.
3.7.2 Lemma.
Let A E+(w). Then the propagator of (3.7.19) is
positive for all t > 0 if and only if R(X) is positive for all A > w.
Proof We use (2.1.10) and (2.1.27), valid for real as well as for
complex spaces by virtue of Remark 3.1.13:
R(X)u= f e-'"S(t)udt (X>),
(3.7.20)
0
S(t)u=nlimo(n t) R( n t)"u
(t>0). (3.7.21)
The result follows then from the fact that the set (u; u > 0) is closed.
3.7.3 Theorem. Let A E ?+(0,1) and assume the propagator of
(3.7.19) is positive for all t (i.e., solutions with positive initial value remain
positive forever). Then A is dispersive with respect to any proper duality map
0. Conversely, assume that A is dispersive with respect to some proper duality
map 0 and
(XI - A)D(A) = E (3.7.22)
for some A > 0. Then A E (2+(0,1) and the propagator of (3.7.19) is positive for
all t>, 0.
Proof.
Assume that S(t) is positive for t > 0, and let 0 be a proper
(but otherwise arbitrary) duality map. If u E E, u, u , 0; then S(t) u 0,
and, since 0 is proper,
(0(u+),S(t)u_)>1 0.
3.7. Differential Equations in Banach Lattices, Nonnegative Solutions
163
Accordingly,
(B(u+), u) =11u+112%IIS(t)u+II llu+ll %(B(u+),S(t)u+)
(e (u+
S(t) u+
(e (u+ ), SW U_
=(9(u+),S(t)u).
Taking u E D(A), we obtain
(9(u+),Au)= lim h-((9(u+),S(h)u)-(9(u+),u))<O,
h
0+
which proves that A is dispersive. In order to show the converse, let A > 0 be
such that (3.7.22) is satisfied and let v >_ 0, u the solution of
Au - Au = v. (3.7.23)
Then, if A is dispersive with respect to the proper duality map 9, we have
AIIu_112=All(-u)+112=A(9((-u)+),-u)
A(9((-u)+),-u)-(9((-u)+),A(-u))
=u)+),v) <0.
Clearly this implies that u_ = 0, so that u >, 0 and therefore
AIIuI12 = A11u+ 112
= A(9(u+ ), u) < A(9(u+ ), u) - (9(u+ ), Au)
= (9(u+), v) <IIu+II Ilvll = lull hull,
whence we obtain
(lull <
hull
for the solution u of (3.7.22) under the assumption that v > 0. Applying the
result to v = 0, we see that Al - A is one-to-one. Accordingly, R(A) =
(Al - A)-' exists and the preceding arguments show that R (X) is a positive
operator and
IIR(A)ull < XIIu11
(3.7.24)
if u > 0. To prove that this inequality actually holds for any u E E, we only
have to observe that
IR(A)ul = (R(A)u)++ (R(A)u)-
=R(A)u++R(A)u_
= R(A)(u++ u_ )
= R(A)lul
164 Dissipative Operators and Applications
so that, in view of (3.7.24) and of the fact that R(X) is positive,
IIR(A)ull = IIR(A)Iul II < A-'II Jul II = A-'IIuiI
We have thus shown that
IIR(A)II < x
(A > 0). (3.7.25)
We apply then Theorem 3.1.1 (as modified by Remark 3.1.13) and Lemma
3.7.2 to deduce that A E (2+(1, 0) and that S(t) is positive for all t. This ends
the proof.
We obtain as a byproduct of Theorem 3.7.3 that every dispersive operator
that satisfies (3.7.22) must be dissipative. It is apparently unknown whether the
implication holds in absence of (3.7.22). However, we have:
3.7.4
Example (Phillips [1962: 1]).
(a) Let 9 be a proper duality map such that:
for every u there exist scalars a, ,O > 0 (depending on u) with
0(u)=ao(u+)-00(u_).
(3.7.26)
Then if A is dispersive with respect to 0, it is also dissipative with respect to 0. (b)
All the proper duality maps appearing in Lemma 3.7.1 (b) satisfy the requirements
in (a).
We shall use most of the time Theorem 3.7.3 "at half power": we
start with an operator A already known to belong to the class (2+ and prove
that A is in addition dispersive, so that S(t) is positive for all t.
As a first application of the results in this section we reexamine the
ordinary differential operators considered in Sections 2, 3, and 4 of this
chapter, or rather their restrictions to the corresponding spaces of real-
valued functions. To simplify the notation, these restrictions will be denoted
by the same symbols. Consider the case E = Co(- oo, oo). If u e D(A) and
µ E O(u+), then p. is a positive measure and every point x in the support of
µ is a maximum of u+, hence of u; we have then u'(x) = 0, u"(x) < 0. Thus
(IA, Au> = f (au"+ bu'+ cu) dµ < 0
(3.7.27)
if the assumptions of Lemma 3.2.1 are satisfied, so that the corresponding A
is dispersive. We deal next with the case E = LP(- oo, oo); for purposes of
identification, we call AP the operator defined in part (b) of Theorem 3.2.7.
Direct verification of dispersivity is difficult in this case, since integration by
parts is prevented by lack of sufficient information on the behavior of
elements of D(A) at infinity. We follow then a more roundabout way,
examining directly the solution operators S.
of u'(t) = AP u (t) and S( - )
of u'(t) = Au(t) (A the operator in Co in Theorem 3.2.7(a)). It was observed
in the course of the proof of Theorem 3.2.7 that if A > 0, v is a function, say,
in 6D, and if u is the solution of Au - APu = v in D(AP) C LP, then u
3.7. Differential Equations in Banach Lattices, Nonnegative Solutions
165
actually belongs to D(A) c Co; it follows that
R(X; Ap)v = R(X; A)v.
(3.7.28)
We note next that the preceding inequality extends to the powers of R(X),
since (3.7.28) holds for all A > 0 and powers of the resolvent can be
expressed by means of derivatives with respect to A (see Section 3). It
follows then from (3.7.21) that
SS(t)v = S(t)v
for all t>0and vE61.
Let now u E LP, u, 0; choose a sequence of nonnegative functions
in 6withlly - ullp - 0. Then
S(t)v being positive in (- oo, oo). Since some subsequence must
converge almost everywhere, it follows that SS(t)u(x), 0 almost every-
where, so that Sp (t) is a positive operator for all t , 0 and all p , 1.
An entirely similar analysis can be carried out in the cases covered
by Theorems 3.3.9 and 3.4.3. Details are left to the reader.
3.7.5
Theorem. Let A be any of the operators considered in Theo-
rems 3.2.7, 3.3.6, or 3.4.3. Then A is dispersive with respect to any proper
duality map; equivalently, the solution operator of u' = Au is positive for all
t , 0.
We close this section with some remarks on conservation of norms of
nonnegative solutions.
It was observed in Section 1.3
that, while the
diffusion equation does not possess an isometric propagator in any LP space
(or in C), the L' norm of a solution with nonnegative initial value remains
constant in time. This property admits an abstract formulation.
3.7.6
Theorem. Let A E (2+(0,1), and let
be the solution opera-
tor of (3.7.19). Assume N C E, N fl D(A2) is dense in N,
S(t)N C N (t, 0) (3.7.29)
and
(9(u), Au) = 0,
u E N fl D(A2), (3.7.30)
where 6: N fl D(A2) * E* is a duality map. Then,
IIS(t)ull =11ull
(t, 0)
(3.7.31)
for u E N. Conversely, assume that N is a set satisfying (3.7.29), and (3.7.31)
for u E N. Then there exists a duality map 0: N fl D(A) - E* such that
(3.7.30) holds for u E N fl D(A).
The proof of Theorem 3.7.6 can be read off that of Theorem 3.6.1
(where N = E) and will be left to the reader.
Theorem 3.7.6 can of course be applied to the differential operators
considered in this chapter. We limit ourselves to the finite interval case.
166
Dissipative Operators and Applications
3.7.7 Corollary. Let A(/30, /3I) be one of the operators in Lemma
3.4.2. Assume that
a(x)>0, a"(x)-b'(x)+c(x)=0 (0<x<1) (3.7.32)
and that the boundary conditions at 0 and at l are of type (I) with
y0a(0)-a'(0)+b(0)=y,a(l)-a'(1)+b(l)=0
(3.7.33)
Then A(/3o, /3I) is m-dissipative in L1(0,1), dispersive with respect to any
proper duality map, and (3.7.31) holds for any u >, 0.
The proof applies Theorem 3.7.6 with N = (u; u >- 0). It is just as
simple, however, to give a proof based on an argument similar to (1.3.17): it
suffices to observe that
dtf'u(t,x)dx=0
for every solution u(t, x) = u(t)(x) of u'(t) = A(P0,01) u(t), which follows
from (3.7.32) and (3.7.33), integrating by parts.
We note that when Ao is written in variational form (3.2.35) the two
conditions in Corollary 3.7.7 are:
a(x)>0, b(x)=c(x) (0<x<1) (3.7.34)
y0a(0)=b(0),
y,a(1)=b(l) (3.7.35)
3.7.8 Example. Check Theorem 3.7.3 for the operators in Examples 2.2.4,
2.2.5, 2.2.6, and 2.2.7.
3.8. MISCELLANEOUS COMMENTS
A result of Lyapunov [1892:
11 can be stated in a slightly generalized
version as follows: the eigenvalues of the matrix A have negative real parts
if and only if a positive definite matrix Y can be found satisfying
Re(YAu,Yu)<-K(u,u) (uEC"`)
(3.8.1)
for some K > 0 (see Daleckii-Krein [1970: 1]). This property of the eigenval-
ues of A was recognized much before to be equivalent to
Ilexp(tA)II - 0ast- +oo;
(3.8.2)
algebraic criteria were known in particular cases to physicists and engineers
like Maxwell working on regulator theory and were given generality by
Routh and Hurwitz during the last years of the nineteenth century. Thus,
Lyapunov's result can be considered as a forerunner of the theory of
dissipative operators presented in this chapter.
The Hilbert space theory was initiated and developed by Phillips
[1959:
1] although dissipativity and closely associated ideas were used