Integr. equ. oper. theory 39 (2001) 363-376
0378-620X/01/030363-14 $1.50+0.20/0
9 Birkh~iuserVerlag, Basel, 2001
ON
THE
SPECTRUMDETERMINED
PERTURBATION
I IntegralEquations
and OperatorTheory
GROWTH
ASSUMPTION
OF Co SEMIGROUPS
AND
THE
GEN-QI XU and DE-XING FENG
The spectrum determined growth property of Co semigroups in a Banach space
is studied. It is shown that if A generates a Co semigroup in a Banach space X,
which satisfies the following conditions: 1) for any G > s(A), sup{llR(A; A)II[
l~eA > G} < oo; 2) there is a Go > ca(A) such that f_+~ [IR(G0 + iT; A)x]I 2& <
~ , vx e x, and y+~ IIR(Go§ iv; A*)flledT < c~, Vf e X*, then ~a(A) = s(A).
Moreover, it is also shown that if A = Ao + B is the infinitesimal generator of a
Co semigroup in Hilbert space, where Ao is a discrete operator and B is bounded,
then ca(A) = s(A). Finally the results obtained are applied to wave equation and
thermoelastic system.
. Introduction
Let A be the infinitesimal generator of a Co semigroup T(t) on a Banaeh space X.
As usual, we define the type or growth order of the semigroup T(t) by
w(A) = ~-~
lira l~
t
'
(1.1)
and the spectral bound by
8(A)
= [ sup{Re~ I ~ e ~(A)},
l -oc,
if ~(A) # 0,
if a(A) = O,
(1.2)
where a(A) denotes the spectrum of A. If X is finitely dimensional, it is well known that
co(A) = s(A).
(1.3)
But in the infinite dimensional case, in general, the above equality (1.3) may not hold. From
the Hille-Yosida theorem we see that
s(A) < ca(A).
(1.4)
We notice that ca(A) describes the growth order of T(t). From the definition of ca(A), if
ca > ca(A), then there exists a constant M > 1 such that IIT(t)[] _< Me ~t, t > O. Therefore
the exponential stability of T(t) is equivalent to the condition that ca(A) < 0. Thus the
condition (1.3) is important because it gives a practical criterion for exponential stability of
T(t), i.e., the exponential stability of T(t) is completely determined by the spectrum of A.
So the condition (1.3) is usually called the spectrum determined growth assumption.
364
Xu, Feng
It is well known that the spectrum determined growth assumption holds for wide
classes of semigroups, for example, for analytic or compact semigroups. These classes, however, do not cover applications to hyperbolic PDEs. So it is a very important problem to
investigate which conditions will lead to the spectrum determined growth assumption.
In recent years, many authors have made effort and obtained a lot of results on the
spectrum determined growth assumption (cf. [1-16]). Among them, there are two important
results: Let A generate a Co semigroup T(t) in Hilbert space H, then
w(A) = inf {g > s(A) [ sup [1R(a + iT; A)I [ < +oo},
(1.5)
w(A)=inf{a>s(A) f_+~ IIR(~ + i~; ~)=11~ < oo, v= < ~}
0.6)
(cf. [1-4]) and
(cf. [5,6]). In general Banach spaces, an important result is that
~(A) =
sup
{~(A),
~(A)\~,(A)
(cf. [7,8]), where ~o(A) = lira t-'logll~(T(t))ll,
t~-~oo
ReA}
~(') denotes the measure of noncompa~tness
of bounded linear operator, and ~e(A) is the essential spectrum of A. A natural question
is whether or not the results obtained in Hilbert spaces can be extended to Banach spaces.
There many examples which shows that the equality (1.5) is no longer valid in arbitrary
Banach space (cf'. [9-11]), that is the boundedness of the resolvent of A is not sufficient to
characterize the bound of the spectrum of the semigroup generated by A. Therefore it would
be interesting to know which additional properties of the resolvent of A ensure the spectrum
determined growth assumption. In the remaining parts of the paper, we shall discuss this
problem. In section 2, we first establish a sufficient condition for the spectrum determined
growth property of Co semigroups in Banach spaces. Then in section 3 we shall investigate
the perturbation of Co semigroups in Hilbert space. We shall prove that if A = A0 + B is
the infinitesimal generator of a Co semigroup of operator in H, where Ao is discrete and B is
bounded, then the spectrum determined growth assumption holds. Finally we will illustrate
the usefulness of our results through several examples in controlled system.
2. S o m e R e s u l t s
of the Growth
Order
o f Co S e r n i g r o u p s
T h e o r e m 2.1 Let T(t) be the Co semigroup generated by a closed densely defined
linear operator A on a complex Banach space X. If the following conditions are satisfied:
1) for all cr > s(A),
sup{[[R(A;A)l I [ReA __ or} < 0%
(2.1)
2) there is a cr0 > w(A) such that
f ~ 5 [IR(~o + iT; A)xl[ZdT<
oo,
Vx e X,
(2.2)
Xu, Feng
365
/_~
Then
w(A)
=
IIR(o0 + iv; A*)flI2& <
V/E X*.
(2.3)
s(A).
P r o o f We only need to prove that in the case s(A) > - o c , for any r > 0, there
must be M~ > I such that
I[T(t)ll < M~e ('(A)+~)t, t > o.
Set A1 :
A - (s(A) + e), it is easily seen that the infinitesimal generator of T~(t) =
e-(~(A)+~)tT(t) is A~ and s(Ax) = - e . Therefore it is enough to prove that the semigroup
T~(t) is uniformly bounded. For this purpose, we first estimate the norm of the resolvent of
A~. Since R(A; Aa) = R(A + s(A) + e; A), from hypothesis 1), we know that
sup{HR(A; ttl)11 I R e A
> ~
> -~} < oo.
(2.4)
Denote
M:
sup
KeA>_-c/2
IIR(~;AJIh
then for A = cr + iT with ReA >_ -e/2, we have
[IR(~ + i~; &)~ll S (x + MI~;
-
(2.5)
@IIR(4 + iv; A~)4,
where a~ = a0 + s(A) + r Thus for any cr > w(A1), x E X, f e X* and t > 0, we have
(T~(t)x, f) = ~
1
r~+icr
J~-i~ e~t(R(A; A j x , f)dA
r~
~+iT r~+:~ eA~
1 ~--~+oo
lim [ T (R(A; AJx, f) ~-~, + J~-i
2~ri
1 /~+ioo eat
2~i ~=-,~ --~-<R(A;
T <R(A; A1)2x' f)dA
]
AJ2x, f)dA,
where we have used the property of Laplace transform: ]lR(cr + iT;&)x[I --+ 0 (Iq ~ ~).
eat
For any fixed t _> 1, the function -/-(R(A;A1)2x, f> is analytical in the zone {A E
r -~e _< ReA _< or}. It follows from (2.4) and (2.5) that
sup
-89
[<R(A; A1)x, f>l -< (1 § [a § elM)[ls
§ iT; &)xll lifll -+ 0 (~- --~ oo).
Hence, from the Cauchy integral theory of complex functions, it follows that
1 f+ioo eat
<Tl(t)x, f> = ~ i ~-ioo -~--(R(A; AJ2x, f)dA.
(2.6)
It follows from (2.2) (2.3) and (2.5) that the above integral is absolutely convergent. Then
there is a constant #(x, f ) only dependent on x and f such that
p(x, f)
[(Tl(t)x,f)[ < - - ,
t
Vx E X, f E X*, t_> 1.
366
Therefore
Xu, Feng
by the uniform
boundedness
principle, there exists a constant
M'
such that
IlTl(t)ll < M',Vt __ 1. Setting M1 = max{M', sup IlTx(t)ll}, we obtain
0_<t<l
IlT~(t)H _< M1, vt > o,
and the proof is then finished.
|
R e m a r k In Theorem 2.1 we only consider the case of s(A) > - c o , if s(A) = -oo,
in general, the condition 1) of Theorem 2.1 does not hold, but we can define
coR(A) = inf{w e/R; sup IIR(A;A)[I < c~}
Re~,>>_w
instead of s(A) in "Theorem 2.1, and the condition 2) is still satisfied, then co(A) = c~R(A).
The proof is same as above.
C o r o l l a r y 2.2 Let X be a Hilbert space. If the condition (2.1) is satisfied, then
w(A) = s(A).
P r o o f When A is the infinitesimal generator of a Co semigroup in H, for ~ > a~(A)
and for any x , y e H, we have He-~'T(.)xl], ]le-~'T*(.)y]] e L2(Z%). Since H has Fourier
type 2, there always hold
][R(a + i.; A)x[] 6 L2(~) and [JR(or - i.; A*)y[[ E L2(h~),
Vx, y C H.
So the desired assertion follows.
II
R e m a r k In the proof of Corollary 2.2 we have used the notion of Fourier type.
Recall that a Banach space X has Fourier type p for some p with 1 _< p < 2, if the
Fourier transform can be extended to a bounded operator from the Lebesgue-Bochner space
/ 7 ( ~ , X) to LP'(1R, X ) ( 1 + ~, = 1), i.e., the vector-valued Hausdorff-Young theorem holds
for exponent p in X. Every Banach space has Fourier type 1 but only a Hilbert space has
Fourier type 2 (of. [18]).
The conclusion of Corollary 2.2 was first given by Gearhart (1978) for contraction
semigroup, and obtained independently by Prfiss (1984) and Huang (1985) (see [14]).
T h e o r e m 2.3 Let T(t) be a Co semigroup on a complex Banach space X and A
be its infinitesimM generator. Then the following assertions are equivalent:
1) for any c~ > s(A),
sup{ [[R(A;A)I[ [ ReA 2 ~} < co;
(2.7)
1
2) for any cr > s(A), and 0 < c < ~(a
- s(A)), the family Y of analytic functions
defined by
.F = {fe~(z) = (R((z + iT + z; A)x, f) [ Vx E X, f C X*, [[x[I = II/ll = 1; ~- ~ ~ , Izl _< c},
(2.8)
is normal.
Proof
Assume
that the condition i) holds. For any (r > s(A), taking ~ such that
0 < c < 1-(os(A)), we have
2
-
sup [[R(cr+iT + z;A)I [ <_ M <
r~t, Izl<_e
oe.
Xu, Feng
367
Thus Vx e X and f E X* with []x[[ =
Ilfll
= 1,
[fi,(z)[ = ](Tt(c~+ iv + z;A)x,f)[ <_ M,
Vv e 1R.
According to the Montel's theorem [19], the family .7 is normal.
Now let condition 2) holds. If there is some ~ > 8(A) such that
sup IIR(o- + iv; A)II = ~ ,
"rE.~
then according to uniform boundedness principle, there exists an x E X with ][x[] -- 1 such
that
sup II.~(~ + iv; A)xll = co.
(2.9)
refit
Thus for each v E JT~, there is a functional fr E X* with [[re][ = 1 such that
(R(~ + iv; A)x, f~) = IIR(~ + iv; A)xll.
Set
.71 = {(R(a + iT + Z; A)x, f~) [ r E ~ , Iz] < e}.
Obviously, .71 C .7, so 3rl is normal. It follows from (2.9) that there exists a sequence
{Tn In > I} C ~ such that % --+ oo as n --+ oo and
l i m IIR(~ + ivy; A)xll = oo.
Denote
fn(z) = (R(cr + iv~ § z; A)x, f ~ ) E .71, n >_ 1.
Then there exists a subsequence {f~k} such that fn~ --+ oo as k -+ oo. The limit is uniformly
on tzl _< e. Notice that each analytic function fn can be extended to horizontal set Z~:
Z~={zEC
Izl<e}U{zE~
Rez~0,
[Imz[_~e},
(2.10)
hence f~k (z) -+ co uniformly on Z6 as k --+ oo. But it is impossible, because for cr + Rez >
w(A), there holds
M
I A ( z ) l = [(R(~ + iv~ + z ; A ) x , f ~ ) l <_ NR(~ + iv~ + z;A)N <_ Rez + a - w ( A ) "
Therefore we have
sup([[R(A;A)[[ [ ReA _~ or} < oc.
The proof is then complete.
3. T h e P e r t u r b a t i o n
|
o f Co S e m i g r o u p s
in H i l b e r t S p a c e s
In this section we always assume that H is a complex Hitbert space. Denote by
L ( H ) the Banach space of all bounded linear operators on H. Let A0 be a closed dense
defined linear operator in H. Assume that A0 satisfies following hypotheses:
368
Xu, Feng
(H1) o-(A0) consists of isolated eigenvalues: o-(A0) = Crp(Ao) = (#~,; n _> t}, and
each eigenvalue of A0 is semi-simple, which means that its geometrical multiplicity is equal
to the algebraic multiplicity;
(H2) Denote by P~ the spectral projector corresponding to eigenvalue #~. For
each x E H, x = ~ P~x, and there exist positive constants M and m such that
~ll~Jl _< ( ~ llp~xll2) ~ _< MIIxll, vx e
~.
r~:l
For example, a normal operator with compact resolvent in H has these properties.
For A ~ ~,r > O, denote by P(A,r) the disc with radius r and center at A, i.e.,
T h e o r e m 3.1 Let A0 be a closed dense defined linear operator satisfying the
hypotheses (H1) and (H2) and B r s
If A -- VAoV -1 + t3, where V, V -I c L:(H),
then or(A) C U C(p~,r) with r -- ~-IIV-lSvII.
P r o o f For y r H, consider the inhomogeneous linear equation
(A - Ao - V - ~ B V ) f = g,
where f = V-ix, g = V-ly. If A ~ ~(A0), then
f - R(A; m o ) Y - l B Y f = R(A; Ao)g.
It is easily seen that if ][R(A; Ao)V-1BVH < 1, then
f = [I - R(A; Ao)V-~BV]-IR(A; Ao)g,
which means that A C p(A).
We now estimate the norm I]R(A; Ao)V-1BV[[. From the hypothesis (H2) we have
f = ~ P~f, 9 = ~ P~9, and for f e ~9(Ao),
n=l
n=l
Aof = ~ ~P,d,
it follows
co
R(A; Ao)V-1BVf = ~ ,
n=l
P~V-1BVf
A
- -
#n
Therefore
oo
m2 E P~V-1BVf 2
P~V-1BVf 2
n=l
P~V-1BVf 2
n=l
Since
P~V-~Bvf 2 ~=~IIP~V-~zVfll2
Xu, Feng
369
it yields that
~a
P,~V-XBVf 2 < (M~2 IIV-~BVIll 2
=
-f---~
- ~-J~ d
So
_
][R(A; Ao)V-1BVfH
~
~
(M) IlV-~BVfll
< ,,m dist(A, cr(Ao))"
(~) Nv-,.wll
-~llv-lsvll,
Thus A E p(A) when ~ dist(a,~(Ao)) < 1. Set r =
dist(A, a(Ao)) > r, this implies that A E p(A). Therefore
if A r U P ( f ~ , r ) , then
a(A) C U P(#~,r).
n_>l
The proof is finished.
|
R e m a r k Theorem 3.1 is an extension in Hilbert space of Bauer and Fike's result
in [2o].
Let A0 be an operator satisfying hypotheses (.//1) and (H2), and F be a subset of
a(Ao). Denote k = M and
PF= ~2 P~,
then
liP41 <_ k, I[QFII -< k,
and for any A E
QF=_r-PF,
p(Ao), it holds that
k
II-a( A; AoQF)II < dist(A, a(Ao) \ F)'
I]R(A; Ao)QPBQFIf
Now set A = Ao + B and r =
<_
k~llBII
dist(A, ~(Ao) \ F )
2k2[[B[[. Denote
F~ = F(A, r) n
a(Ao)
Proposition 3.2 Let A0 be an operator satisfying hypotheses (HI) and (H2), and
B be bounded. The operators P~, Q~ are defined as above. Then the operator AI - AoQ~ Q,~BQ~ has the bounded inverse on Q~H = T4(Q~), where T4(S) denote the range of S .
Proof It is easily seen that AoQ~ = Q,~Ao, a(AoQ~) = a(Ao) \ FA, This shows that
dist(A, a(AoQ~)) > r and
IIR(~; AoQ~)Q~BQ~,][ <_1/2.
Thus the operator AI -
AoQ~ - Q~BQ~ is revertible and
[l(AI
-
A o @ , - Q:,BQ),)-I[[ <_
1
kJ[sl---/
370
Xu, Feng
For a sufficient small r > 0, define an operator H:~(z) as follows:
H~(z) = ((~ + z ) I - AoQ:, - Q~BQ~) < ,
[zl <_ e.
We can choose ~ such that IIH~(z)ll 6k/~ for all ]zl ~ ~.
P r o p o s i t i o n 3.3 Let A0, B, P~, QA be the operators as proposition 3.2. Define the
mapping A),(.): P(0,e) - + / : ( H ) by
Aa(z) = AoP~ + P~BPa + P~BQ~Ha(z)QaBPa.
Then for any z E F(0, e), the operator A~(z) is bounded on P:,H and AA(z) is
respect to z E r(0,e). Moreover, A + z E p(A) if and only if (~ + z) - A~(z)
reverse on PAH.
P r o o f Obviously, A~(z) is bounded on P~,H and A~(z) is analytic
to z C F(0, e). Now we prove the second assertion of Proposition. Consider
equation
(A+z)f-Aof-Bf=g,
VgcH,
analytic with
has bounded
with respect
the resolvent
which is equivalent to
(A + z)P:~f - AoP~f - P:~BP~f - P~,B@,f = P~g,
(3.1)
(~ + z ) Q ~ f - AoQ:~f - Q~BQ:,f - Q ~ B P ~ f = Q~g.
(3.2)
and
From (3.2) it follows that
q ~ f = H),(z)Q~BP~,f + H:,(z)Q:~g.
(3.3)
[(), + z) - A~(z)]P~f = P~g + P~BQ~H~(z)Q~g.
(3.4)
Hence we have
If (A + z) - A~(z) has bounded inverse on P~H, then
P~f = [(~ + z) - A~(z)]-l[P~g + P~BQaH~(z)Q~g],
and
f = P~f + Q:~f = R(A + z; A)g
= [(A + z) - A:,(z)]-l[P~g + P~BQ~H~(z)Q~g] + H~(z)Q~g
+H:~(z)Q:~B[(A + z) - A~,(z)]-l[P~,g + P~BQ~H~(z)@,g].
The proof is then complete.
|
T h e o r e m 3.4 Let T(t) be a Co semigroup on complex separable Hilbert space H
and A be its infinitesimal generator. Assume A = A0 + B, where A0 is an operator satisfying
(H1) and (H2), and B is a bounded linear operator in H. Furthermore, assume that the
following conditions are satisfied:
Xu, Feng
371
1)
for a sufficient large constant M > 0, the set
{A e ~; IAI _> M} Aa(A0)
consists of isolated eigenvalues of A0 with finite multiplicity;
2) for any ~ > s(A),
sup I1(~ + iT + z) - A ~ + ~ ( z ) l l ~<~) < oo,
]~I>M
where re(r) = dim(TZ(P~+i~)), and both P~ a n d / ~ are defined as before.
Then w(A) = s(A).
P r o o f For any f i x e d ~ > s(A), denote A = ~ + i z , take r such that 0 < r <
89
then A + z 6 p(A) as ]z] < r for all r C 9{. From Proposition 3.3 it follows that
(A + z) e p(Ax(z)), and [[(A + z) - Ax(z)H is uniformly bounded on the disc ]z I < r for all
r C ~9. Particularly, if 1rt > M + ltBIt and dim(P~H) = re(T) < 0% then Vx C PxH,
A(x,z)
[(~ -}- Z) -- / ~ A ( Z ) ] - i x = d e t [ ( ~ -~- z) - a A ( z ) ] '
where det[S] is the determinant of S and f~(x, z) is a vector valued analytic function on
Izl < c. It follows from [21] that there is a positive constant u such that
I1[(~ + z) -- A ~ ( z ) ] < l l <
II(A + z) - A ~ ( z ) l l < r ) -1
Idet[(A + z) - A~(z)] I '
and u is independent of (A + z) - A~(z), depending only on the norm of H. Thus it follows
that
llA(x, z)ll _< -II(A + z) - A~(~)llm(~)-~llxll,
and
0 < ]det[(A + z) - A~(z)][ _< ~I](A + z) - A~(z)l] "~(~).
By the hypothesis 2) and Montel's theorem (see [19]), it follows that the family of analytic
functions
9v2 = {det[(A + z) - A~(z)] [Irl > M + I[BII, Izl < s}
is normal. Noticing that for each r C 2~, det[(A + z) - A~(z)] can be extended analytically
to the domain Z~ defined by (2.10), we have 0 r ~ . Therefore
sup II[(A + z) - a ~ ( ~ ) ] < l l
< oo.
I-I>~
From Proposition 3.3 and its proof, it follows that
sup [[R(A + z;A)l. [ < oo,
rE~
which means that co(A) = s(A) according to Corollary 2.2.
|
372
Xu, Feng
Remark
In the proof of Theorem
sup
lJ( +
3.4 we see that it always holds that
+ z) -
<
oo.
I<_>M
If for sufficiently large lwl _> M, dim(_P~H) = re(T) < constant, the condition 2) in Theorem
3.4 is satisfied naturally. In practice, many problems such as in [23-27] do satisfy the
conditions in Theorem 3.4. Renardy [22] proved that if A0 is a normal operator and for
sufficiently large Irl _> M,m(7) <_ K hold uniform]y~ K is a constant, then w(A) = s(A).
Obviously it is a special case of Theorem 3.4, So our result is a better extension.
Finally, as an extension of Theorem 3.4, we give a more useful result in practice.
T h e o r e m 3.5 Let H be a separable Hiibert space and A = A0 + B be the infinitesimal generator of a Co semigroup in H. Assume that B is bounded and A0 satisfies
the following conditions:
1) a(A0) consists of isolated eigenvalues with finite algebraic multiplicity;
2) all eigenvalues of A0 with sufficiently large module are semi-simple;
3) the sequence of the generalized eigenvectors of A0 forms a Riesz basis in H;
4) and for any r > 0, there exists an integer k such that, for any given A E (~, it
holds that
dim(
E
P•H) < k,
~ez(/o)nr(;~,r)
where Ps is the spectral projector corresponding to eigenvalue g, and F(A,r) defined as
before.
Then the spectrum determined growth assumption is valid, i.e., w(A) = s(A).
P r o o f Let or(A) = {A.;n _> 1} with [A,~] _< [An+l[,Vn >_ 1, and P,~ be the spectral
projector corresponding to A~, then from the hypotheses it follows that ~ P= = I and there
is an integer N such that A0 is expressed as
Ao =
+ X:(A P
n=N+l
+
n=l
where D~, 1 < n < N, is a bounded linear operator with the property that there is an integer
kn such that D ~ ~ -- 0. Denote by A01 the scalar part of Ao, i.e.,
oo
n=l
and set O ----Ao - A0> Obviously, O is a bounded operator, and A m satisfiesall conditions
of Theorem 3.4 with re(T) _< k. Applying Theorem 3.4 to A = A0z + C + B, the assertion
of Theorem
3.5 follows immediately.
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4. E x a m p l e s
In the present section, we list two examples to explain the application of the results
obtained previously.
Xu, Feng
373
The first example is concerned in the wave equation with linear damping and tangential force feedback control:
{ w~(~,t) = w ~ ( ~ , t )
w ( o , t ) = o,t > o
- b(~)W~(~,t) - c(~)W~(x,t), 0 < 9 < 1,t > O,
(4.1)
w ~ ( 1 , t ) = - k W ~ ( 1 , t ) , k > o , t > o,
where b(z), c(x) are nonnegative bounded measurable functions.
Set Vol(0, 1) = { f E Ht(0, 1), f(0) = 0}, where Hk(0, 1) is the usual Sobolev space
of order k. 7 / = V~(O, 1) x L2(O, 1) is the product Hilbert space with the norm
It(f,n)lff = fo ~ [If~(~)l ~-+ Ig(x)?]d~,
for f ~ yo~,g ~ L~(0, a).
Define operators Ak and B in 7 / b y
Ak(f , g) = (g, fx~),
D(Ak) = {(f,g) e 7/;f e H 2 n Vol,g e V01,fx(1) = " k g ( 1 ) } ,
and
S(f,g) = ( 0 , - b ( x ) f ~ ( x ) - c(x)g(x)),
V(/,g) e 7/.
Obviously B is bounded. Set A = Ak + B. Then the equation (4.1) can be written as the
abstract evolution equation in 7 / :
d u ( t ) = AU(t)
with
U(t) = (W(x,t), Wt(x,t)), for t > 0.
It is easily seen that A is the infinitesimal generator of a Co semigroup.
It is shown in [28] that for all k > 1, the operator A~ has compact resolvent and
the spectrum of Ak is
1+ k
(2~ + 1)~
o},
and every eigenvalue of Ak is algebraically simple. The sequence of eigenvectors of Ak is a
Riesz basis in 7/.
Thus it follows that the operator Ak satisfies the all conditions of theorem 3.4, so
we have w(d) = s(A).
Another example is concerned in the linear thermoelastic system with damping:
l utt(z,t) + uO~(x,t) = u~(x,t) - b(x)u~(x,t) - c(x)ut(x,t), 0 < z < 1,t > 0
0,(x, t) + - ~ t ( x , t) - ko~(~, t) = o, o < x < 1, t > o,
(4.2)
u(0, t) = u(1, t) = 0, t > 0,
0(o, t) = o(1, t) : o, t > 0,
where u > 0, k > 0 are constants, u usually much less than 1, and b(z), c(x) are nonnegative
bounded measurable functions.
374
Xu, Feng
Set H~(0, 1) = {f 9 Hi(0, 1), f(0) = 0, f(1) = 0}, 7 / = H~(0, 1) x L2(0, 1) x L2(0, 1)
is the product Hilbert space equipped with the norm
II(f,g,h)]12=
2
[lf~(z)12 + [g(x)12 + lh(x)I~]dx,
( f , g , h ) 9 7/.
Define operators Ao and B in 7-/by
A o ( f , 9, h) = (g, fxx - zJhz, -L,g~ + khan)
~(A0) = {(f,g, h) ~ ~ ; f e H~nH~,g e ~ , h c H~nH~,},
and
B ( f , 9, h) = (0, - b ( x ) f x ( x ) - c(x)g(x), 0),
V(f, g, h) G 7/.
Then 13 is bounded, and A = A0 + 13 is the infinitesimal generator of a Co semigroup.
It is pointed out in [29] that the set of generalized eigenfunctions of A0 forms a
Riesz basis in 7-t, in addition, all eigenvalues with sufficiently large module are algebraically
simple. Furthermore, Ao ~ exists, and the eigenvalues of Ao consist of a real sequence {cry}
and a sequence of conjugate pairs {AN,~-~} with the following asymptotic properties
{a~ = - k ( ~ ) 2 + -~
~2 + o(~),
A~
//2
- ~ + i ~ + o(~),
for positive integer n large enough.
From [29] we know that the operator Ao satisfies all conditions of Theorem 3.5, so
we can assert that co(A) = s(A).
Acknowledgment
This research was supported by the National Key Project of
China and the Natural Science Foundation of Shanxi.
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Institute of Systems Science
Academy of Mathematics and Systems Control
Chinese Academy of Sciences, Beijing 100080
Emaih [email protected]
Fax: (8610)62587343
AMS S u b j e c t Classification" 47D06, 47D03
Submitted: November 17, 1999
Revised: May 9, 2000