RELATIVELY BOUNDED AND COMPACT PERTURBATIONS OF NTH ORDER DIFFERENTIAL OPERATORS
TERRYG.ANDERSON
Department
ofMathematical Sciences Appalachian State UniversityBoone, NC
28608, U.S.A.E-mailaddress: tga@math.appstate.edu
(Received June I0, 1996 and in revised formAugust 26, 1996)
ABSTRACT.
Aperturbationtheoryfor nth order differential operators isdeveloped.For
certainclasses
ofoperatorsL,
necessaryand sufficient conditions are obtained for aperturbing operatorB
to be relatively bounded or relatively compact withrespecttoL. Theseperturbationconditions involve explicit integral averagesof the coefficients ofB. Theproofs
involveinterpolation inequalities.KEY WORDS AND PHRASES.
Perturbationtheory, differential operators,relativelybounded, relativelycompact, integral averages, interpolation inequalities,maximal and minimaloperators, essential spectrum, Fredholm index.1991
AMS SUBJECT CLASSIFICATION CODES.
34L99, 47E05.INTRODUCTION AND MAIN RESULTS
We develop a perturbation theory for nth order differential operators.
In
the following, the differential operatorBwillberegardedas aperturbation of a(typically)higher-order differential operatorL.For
certainclasses of operatorsL,we obtainnecessaryand sufficient conditions forB
to be L-bounded orL-compact. We
employthefollowing terminologyas given inKato [5,pp.
190,194].
DEFINITION
A.B
isrelatively boundedwithrespecttoL
or simply L-bounded if D(L)_
D(B)and
B
isbounded onD(L) withrespecttothegraphnormII. II,.
ofLdefinedbyIlyll,. Ilyll
/llLyll,
y e D(L), whereD(L) denotes the domain ofL.
In
otherwords,B
isL-bounded ifD(L) D(B) and thereexistnonnegative constantsct andfl
such thaty D(L).
Thegreatestlowerbound
/0
ofallpositiveconstants/
forwhich thisinequalityholds is called the relativeboundof B
with respecttoL
orsimplytheL-boundof
B.In
general,theconstanto
willincreasewithout bound as
/
is chosencloserto/0
(sothat the infimum/0
neednotbeattained).A
sequence
{y, }
issaidtobeL-bounded if thereexists K > 0 such thatIly.ll,
<.
_>.
B
iscalled relatively compactwithrespecttoL
orsimplyL-compact
if D(L) D(B)andB
is compact on D(L) with respect to theL-norm, i.e.,B takeseveryL-bounded
sequence
into asequencewhich has a convergent
subsequence. For
example, ifL
is the identitymap,
thenL-
boundedness (L-compactness) of B is equivalent to the usual operator norm boundedness (compactness)ofB.The functionspace settingistheweightedBanach
space /.(I),
where _< p < .o, Wisa positive Lebesguemeasurable function defined on an intervalI
of the real line, and/. (I)
denotestheLebesguespaceof equivalence classes ofcomplex-valuedfunctionsywith domain
I
such thatIIMI := [], w lye" ]""
< .o. Ifw
1, we denote thisspace
byL"(1).
Thespace
ofcomplex- valued functionsywith domainI
suchthatI11.
:= =sssop [y(t)
< isdenoted byL’(1). A
localproperty is indicatedbyuseofthe subscript"loc," and
AC
is usedtoabbreviateabsolutely continuous. Thespaceof allcomplex-valued,ntimescontinuouslydifferentiable functions onI
is denoted byC"(1)" C(1)
denotes the restriction ofC"(1)
to functions withcompact support contained inI;
andC0(l
isthespace
of allcomplex-valuedfunctions onI
which areinfinitely differentiableandhave compactsupportcontained inthe interior ofI. We adoptthe definitionsof maximaland minimaloperators giveninGoldberg [4,pp.
127-128,135].
DEFINITION B.
Let be a differential expression of the formWilp
ai(t)
(D ),
whereW
is a positiveLebesgue
measurable function defined on l and eacha,
is a complex-valuedfunction onI. Then the maximal operatorL
corresponding to has domainD(L) {
yI.(I)" y-" AC(1),
/[y] e/,(I)}
and actionL[y] l[y]
,__oa,(t) (y D(L)).
Ifa, . Cl(1)
for 0 < < n anda,
0onl,then the minimal operator
L
correspondingto is definedtobe the minimal closed extension ofL
restricted tothose y D(L) whichhave compactsupportin the interiorof
L In
the Hilbertspace settingofL=(1),
mostof the smoothness requirements on the coefficientsa,
(0 < < n) are not needed, and thetheoryisdevelopedin Naimark[7,sect.17].We
considerperturbationsn-!
B W,I E bj/T
(a < < to)j=O
of the operators
T pn
D,wllp and
Z
W|/t, aP," D’
l=O
inthesettingof
L(a,
,o),where < p < andWis a positiveLebesguemeasurable function definedon(a, *,,). Definitionsand conditions forP
andP,
aregiveninthehypothesesofTheorems 1.1 and 1.2,respectively. Wegive conditions on certainaverages
oftheperturbationcoefficientsbj
(0 <j < n-1) whichare sufficient and, in some casesnecessary,
forB
tobeT-boundedorT-
compact. These resultsrely heavilyonTheorems
A
andB,
whicharespecialcases of Theorem 2.1 in Brown andHinton [3]. Thesetwotheoremsgivesufficient conditions forweighted interpolation inequalitiesof the form: there existdj
>0, r/>0,K
>0, and e > 0 such thatfor all e (0, eo)
and y in a class
D
of functions,where0 <j < n-landl < p < ,,*.
Theorem 1.1 gives integralaverageconditions on
bj
(0 < j < n-1) whicharenecessaryand sufficient for
B
tobeT-bounded orT-compact
inthecase when < p < andP
andW satisfy theconditionsin Theorem 5 inKwong
andZettl[6]. When W=-
1, these conditionsimplythat the coefficients ofT
are bounded above by the corresponding coefficients of an Euler operator.Furthermore, theperturbationconditions for
T-compactness
ofB
are sufficient for the essential spectrum and Fredholmindex tobe invariantunderperturbationsofT
by B.By
definition(Goldberg [4, pp. 162-163]),the essential spectrum ofT,
writtentr,(T),
is theset of all complex numbers
2
such that the rangeR(21
T) of21 T
isnot closed. The essentialresolventofT,writtenp,(T),
isthecomplementof thisset.By
definition(Goldberg[4,
p.102]),theFredholm indexto(T)isgiven by to(T) ct(T)
fl(T),
whereo(T)isthe dimension of the nullspaceofT
andfl(T)
isthe dimensionof/.(I)
R(T). or(T) is calledthe kernel index ofT, andfl(T)
iscalled the deficiency index ofT.In
Theorem 1.2, theresults in Theorem 1.1 for thesingle-term operatorT
areextendedtothe multi-termoperator L.An
nth orderperturbation ofL
is considered in Corollary 1.1. Sufficient conditionsaregivenforinvarianceof the essentialspectrum andFredholm indexofL
under such perturbations.Theorems 1.1 and 1.2 andCorollary 1.1 provide generalizations of results of Balslev and Gamelin[2]aspresentedinGoldberg[4, pp. 166-175]. Theirwork dealswithbounded coefficient and Euleroperatorsin theunweighted settingof
LP(a,
.,,)for < p < oo.In
Theorem 2.1, thesufficiencyconditions in Theorem 1.1 aregeneralized for operatorsT
with arbitrarily large coefficients. Again, these conditions involve integralaverages
of the perturbation coefficientsbj (0
<_ j _< n-l). Theorem 2.2 gives pointwise conditions onbj
(0 _< j _< n-l)under which the conclusions of Theorem 2.1 hold. The case inwhich p is coveredbyTheorem2.2. Also,perturbationconditions which are sufficient forL-boundednessorL-
compactnessofB
are obtained for the casep and the case in which the coefficientsofL
are arbitrarily large. These theorems rely heavily on investigations by Brown and Hinton [3]on sufficient conditions forinterpolation inequalities. Examplesof each theorem arepresented
and contrastedfor the situation in which the coefficient inT
is anexponentialfunction.The final theorem, Theorem3.1,deals exclusively with the case p 1. Sufficient,integral averageconditionsaregivenfor T-boundedness ofB.
1.
INTEGRAL AVERAGE CONDITIONS FOR EULER-LIKE OPERATORS
In
this section we consider operators whose coefficients arepowers of a fixed functions times a weightfunctionwand a bounded function.In
thesimplestcase, i.e., w(t) s(t) 1, Theorem 1.2 givesTheoremVI.8.1of[4]. For
tx 0, w(t) 1, and s(t) t,thesufficiencycondition ofpart (ii)of Theorem 1.2yields CorollaryVI.8.4 of[4]
forperturbationsof the Euler operator. Since wedonotrequire w(t) or a 0, we refertotheunperturbedoperator of Theorem 1.2 as Euler- like.
THEOREM
1.1. Let < p < andI
[a, **). Letsandwbe positive,AC,o
(I)functions
suchthat
Is’(/)l
<_No
andIs(l)w’(t)l <- Mo
w(t) a.e. onI for
someconstantsN
andMo.
Let otR,
W ws
ap,
andP
ws+"). Let T, B:l.(a,
**) --->l.(a,
**) be the maximal operators corresponding to thedifferential
expressionsz
-7 D" D
andZ
n-Ibj D ,
respectively, where eachby o(l).
For0 <j < n- andt
> O,define
D=
j=O
g,. n(t)
w()s() dT.
Thenthefollowinghold:
(i) BisT-bounded
if
andonlyif b L
I)andsup g,(t)
< (0 < j < ,-1) (1.1)aStS*.
(ii)
for
some e O,l/(2N0) ).
When(1.1)
holds, the relative boundfor B
isO.
Furthermore, the maximal operatorcorrespondingtot+ v
isTr+ T + B.
B
isT-compact if
andonlyif bj l.
(I)andlim
g,.(t)
0 (0 <j < n-l) (1.2)for
some ( O,l/(2N0)).
When (1.2)holds,r
andTr+
have thesameessentialspectrum and
I
ep,(T) to(M-T) r(M-T/o),
wherep,(T)
is the essential resolventof T
andto(T)istheFredholmindexof
T.Thefollowingtheorem ispartof Theorem 2.1 in
Brown
and Hinton[3]. It
givessufficient conditionsforweighted interpolation inequalities.THEOREMA.
Let <p <.0, l=[a, ..),and 0 < j < n-1. LetiV, W, andPbe positive .nu,zJ.,zd,J,.sio.j.ab .h,. :(:),.
j’"" " L
II-q’, ,’-’’ ,,(,5
+
1"for
p 1, W-j, P-
arelocally
essentially boundedonL Suppose
thereexistsP q
e > 0andapositivecontinuous
function f f(t)
onI
such that:=
.,,,,,-,,,
and
for
alle (0,eo),
wherellP-’il..,,.,/,,.
Lef
JtJ
p=l
<p<.,,
withsimilar
definitions for T.(W).
Then thereexistsK
> 0 such thatfor
all e (0,e o)
andyeD,
where
D {
y"y(n-,, AC,(1), I WlyIP
< *,,, andI Ply’’’I
<"}"
PROOF OF THEOREM
1.1.(i) Sufficiency.
Suppose
(1.1)holds for some d/ 0,1 ). We
will show that TheoremA appliestothe choicesf
s, NIbj[ ’
e//,
andWandP
as inTheorem1.1. Basicestimatesare obtainedfrom thefollowinglemma in[3, pp.
575-576].
LEMMAA. Let
sandwbe as inTheorem 1.1. Thenfor fixed
I, 0 <e
< 1/N and< < t+es(t), we have that
(1-eNo)s(t)
<_ s(r) <_O+eNo)s(t)
andexp/-o )
wCt)< wCz)<exp(-o )
w(t).Thisimpliesthatbothpositiveandnegativepowersof s(z) and w(z) areessentially constant for
< <
+
e s(t)and fixedt.By
l.emmaAand the definitions ofP
andW,T.(P) [:/.,o) w(.)-qp s(.)-+,)
d’t"<- Ct w(t)-’
s(t)-+’) (1.3)and similarly
Tt. e(W
< Cw(t)-’
s(t) (1.4)for all e
I
ande
(0,d/),
whereC,
andC2
areindependentof ande. UsingLemma A
again, weobtainfora constantCef(t) e
s(t)e
s(t) w s<
C3
w(t) s(t)(a+)g.(t)
for all I, e (0, g;). Hence, by (I. 1),there is aconstantC
>
0such that[,+cf(,
N < C w(t) s(t)<=*j) (1.5) .f(t)for all e 1,t e (0,
6).
ThusS(e)
<sup,,, { s(t)<"-J) C w(t)-’
s(t)-a+")C__e
w(t)S(t)("m)p}
sothat
S,(e)
_<CC,,
0 <e
<6. (1.6)
Similarly,
S(e)
<CC:,
0 <e
<6.
(1.7)Hence, byTheoremA,there isaconstant
K
such that for all y eD
D(T),Use
oftheelementary inequality(a + bP)
’i < a+
b (a,b > 0) givesfor all y e D(T), 0 <j < n-l, where
K, K ’.
Restrict e < 1. Then therightside can be bounded aboveindependentlyofj, and thetriangle inequality givesfor all y e D(T). Since p > 1, itfollows that
B
is T-bounded with relativebound 0. The resultTr/
T+ B
followsbyan argument given onpp. 169-170inGoldberg[4].
Necessity.
Suppose B
isT-bounded.Let
bea function inC’(R)
suchthat on[0, 1]
andsupport(0) t-2,
21.
Fix6
0,l/(2No) ). For
eachr>
a,definet > a. (1.9)
,(t) 6s(r)
Then
Cr
on [r,r+6s(r)]
andsupport(0,) [r-26s(r), r+26s(r)]. We proceed
by an inductionargument. First consider j 0 in(1.1). Fix r > a.Note
thatB 0r Wt,, b0
on[r,r
+ t
s(r)], so thatgo.(r)
s-r)
WSapconstantC independentofrsuch that
Now,
applyingLemma
A,there is ago. (r)
< w(r)s(r)C +apI
r+s(r)WIB rl <
w(r)s(r)C /aplIB ,11’
(11 ,11" + lit rl[ p)
(1.10)<
w(r) s(r)’+ap
for some constant
K
independentof r, where the lastinequality followsfrom thehypothesisthatB
is T-bounded.Using the compact support of
,, Lemma
A, a change of variable, and the fact thatC0
(R), we have for someconstant C0,< Cw(r)s(r)a"
Ii I (u)l" ,(r)du
<
C
w(r) s(r)aE+ (1.11)for someconstant
CI
independentofr. Similarly, for some C0,lIT IrlIP I: w IT rlP fP n)lP
Jr-2cs(r) W<
Cow(r)
s(r)’a+", dtC w(r) s(r)(a+")’
p<")(u) 6
s(r)dt"s(r)"
< C w(r) s(r)a+’ (1.12)
for some constant
C2
independent of r.Use
of (1.11) and (1.12) in(1.10)
yieldsgo.,(r)
<K(C + C),
r [a, **). Therefore,(1.1)holdsforj=0andalld; 0,l/(2N0)).
Nextfix k _< n-1.
Suppose
(1.1)holds for 0 <j _< k- andsome d; 0,1/(2N o) ).
k-I
Let a
be the maximaloperatorwith actiongiven bya 17 bj D By
the sufficiency1--0
argument above,
A
isT-bounded. Thus sinceB
isT-bounded, Minkowski’sinequality impliesthatA
n-!
+ B
isT-bounded. Notethat (A+
B)yW/,
, bj yO),
y D(T). With and,
definedl=k
asabove(see (1.9)),define
h(t)
O(t) .,
t>a.Then h
C’(R)
and h(k)-=
on[0,1].For
eachr > a,define(1.13)
hr(t
s(r) h(u), > a, (1.14)hk)(t)
where u t-r
ds(r)"
Thenh:)(t) h()(u),
support(h
[r-2ds(r),
r+ 26s(r)].
Thusb
on[r, r+ 6
s(r)].(A
+ B)h,=
W
pfor r < <
r+s(r),
and(1.15)
By Lemma
A,weobtain for aconstantC,g. (r) r)
ws(+*ms(r---
W S(Ot+k)pC [r+;(r)
w(r) s(r)(a+*m+’
WI(A +
B)h,]"
< Cw(r) s(r)<a/*’’/’
II(a + m h, ll’
<
C
w(r) s(r)’+’)’+’
llh,’ + liT hrll
p),
(I. 16)where the lastinequality follows fromthe relative boundedness of
A + B
withrespectto T.By
calculations like those usedinderiving (1.11)and(1.12),we obtain forr >_ a,
Ilh, il"
<_c,
w(r) s(r) (1.17)and
where
C
and C areconstantsindependentofr. Thus(1.6)
impliesthat(1.1)holds forj kandanyd (0, 1/(2N0)).
This establishesnecessityof(1.1).(ii) Sufficiency.
Suppose
(1.2) holds for some6
0,1/(2No) ). We
will use an argument similartothat inGoldberg[4, pp. 171-172]. For
eachpositive integerN >
a, defineB#
{yon[a,N],
on D(T) by
By We
show thatB
N convergestoB
in thespace
ofon (N, .o).
boundedoperatorson D(T) withthe
T-norm.
First note thatT
isclosed.To
see this, letf, -- f
and
Tf. --.
g in/.,(a,
.,,). Let Jbe acompactsubinterval of[a, *,,) and restrict the functionsf, f,,
and gtoJ.
Define Tj/.(J)
---)/.(J)
tobe the maximaloperator correspondingtoz
on J.Clearly,
f.
--4f
in/,(J)
andf. D(T).
SinceTf, (Tf.)l,, Tf.
g inL(J). By
Theorems VI.3.1 and IV.1.7 inGoldberg [4],
T
isclosed. Therefore,f D(T)
and Tjf g.Thus,
f
D(T) andTf
g. HenceT
isdosed.Therefore D(T) is complete under the
T-norm.
D(T) D(B). For y D(T),
From
(i),B
is T-bounded. So(1.19)
By
the argument used inprovingsufficiency in(i),TheoremA
appliestothe interval I [N, .o) withthesame choices for the weights,f,
ande
0.By (1.3)
and(1.4),for 0 < e <S(e)
< C supIv.**){w(t)-’s(t)
-/)e
s(t)[,/.,O)]bl, } (1.20)
and the same estimate holds for
S(e)
up to a multiplicative constant.By Lemma
A, forO<e<d/,
[,+.,,,, -< c
w(t)
s(t),a/,,, g,.n(t) IN,
oo). (1.21)es(t) e
Hence
S(e)
<_ C supg,.n(t)
(1.22)E
with asimilar estimate for
S:(e),
0 <e
<6,
whereC
isa constantindependentofN
and e. It follows from TheoremA
thatfor ally e D(T),I’ b, Y"’I" <- - I W IY[
p+ I? P ’(’)
where
Cj
isindependentofyandN (but dependsone).Use
of(1.23)in(1.19)
givesIIy- Ily]l >1
_<Z
,=oc,
,,,.-,supgj.,(t)
(1.24)for all y D(T)such that y 0.
By
(1.2),thetermontherightsideapproaches0 asN
---> ,,*.Therefore,
Bu
--->B
in thespaceofbounded operators onD(T)withtheT-norm.Next,
we show that eachBu
isT-compact. Let {ft}
be a T-boundedsequence,
sayIIf, L -< ’
forall I. Wewillshow that{fJ’},
0 <j < n-1, is uniformly bounded on [a, N].Partition
I [a,N]
byJ, [t,,t,+],
< < k, witht
a,t,+ t, + es(t,),
ande e (0,
d)
chosen such thatN tk+ tk + es(t). From
theproof ofTheorem 2.1 inBrown
and Hinton[3],withJ,,
f
If,,,,<,)l
_<[s<,,>]-" z "
(w>e s(t,) I,, wlf, l" + [es(t,)]
’"-’)pT ,(P) e s(t,) I,,
Use
of(1.3)and(1.4) yieldsfor someC (dependingone),w(t,) s(t,) {I,, wlzr + I,,
for E
J,.
Sincewand s arepositive,continuous functions on [a, ) and t,J,
c: [a, N], we havefor someC dependingone,(1.25) for [a,N], 0 <j < n-l. Since
{f}isT-bounded, {f(’)},
0 <j < n-l, isuniformlybounded on[a,N].
Next
we show{f(s)},
0 <j < n-1, isequicontinuouson [a, N].Let
r/ > 0 be given.For
t,s e [a, N],If,<"(: t, -< LFI IS: w ’’"
byHolder’s inequality, where
+
1. Sincew andsarepositive, continuous functions onP q
[a, oo),
W
ws"’
isboundedabove andbelowon [a,N]. Hence
fort, s e [a, N],If,<’>(t) -f,<"(:’)t
<c lt-
" (1.26)
where theconstantC dependsonW. Forthe case 0 <j _< n 2, theargumentused toobtain (1.25) appliesto j
+
< n andyieldsIf,/"(t
_<c IV, L,
[a, N]. Thisimpliesthat,sinceWisboundedon[a,N],witha newC,
ils,<,+’>ll
L;(a.N) <c ILf, L., o
<j < n- 2 (1.27)Forthecase j n 1,
(")
Ils,<’+"ll,,.<o.,, Ils, L.<o.,, <- wiry,
<c rs,
<c IIs, L
(1.28)since W/P is bounded on[a, N]. Thus,inanycase,(1.28)holds for 0 <j < n 1.
So (1.26)
impliesthatIf/"(’) -f,("(s)! -< c lt- ,1" IIs;,ll,, <-
Mit- 1",
(1.29)where
M c s.p{ IiZL
1>1},
since{fl}
is T-bounded. Since p > 1, llq >O.
Therefore,
{f(J)}
isequicontinuousandboundedon[a, N], 0 < j < n 1.By
theArzela-Ascoli Theorem,{f}
has a subsequence{f, 0}
whichconvergesuniformly on [a, N],and{f0}
has asubsequence
{fl}
whichconverges
uniformly on [a, N].Hence {f,,}
and{f,}
convergeuniformly on [a, N]. Repeatingthisprocedure,asubsequence
{g,}
of{ft}
is obtained such that for 0 <j < n 1,{gJ)}
convergesuniformly on[a, Nq.By
definitionofB,,
(1.30)
It
followsthat{B g,} converges in/_.(a, .0)
as--->
**. ThusBn
isT-compact
foreachN,and soB
is
T-compact,
beingthe uniform limitofT-compact
operators.Necessity.
Suppose B
isT-compact.
First we showthat(1.2)holds for j 0. We proceed byacontradiction argument.Suppose
that forany d; 0,l/(2N0) ),
there exists e > 0 and a sequence{r}7__i
ofpositivenumbers such that r--> and> e, > 1. (1.31)
Fix
t
O,1/(2No) ).
Let{,}
be the functions definedby (1.9). Asbefore,Bi,
W,,
b0, on[r, r+ 6s(r)].
(1.32)Itfollows from(1.31)andLemma
A
that[ I’
e <_
r,+,<r,
W
l,,
<s(r)
wSw(r,) s(r,)
’+a"whereC is aconstantindependentof
I.
Foreach r > a, defineI[Ir(t)
w’r’l/’t
s,(t), _
[a, 0.). (1.34)Then
and(1.33)implies that
(1.36)
subsequence. Relabel indices so that
{Br } converges
inL(a,
**)to someYo. We
show thatYo
0 a.e. in [a, *,,). LetJo
be a finite subinterval ofIn,
-0). Sincer
---> as --)*,, andsupport(Vr,) [r
2ts(rt), r +
2ts(r)],
wehaveI//r,
m 0 onJo
andBr
0 onJo
for<
IlYo BI//,,Ii.
SinceBrr,
-’>Yo
as --> and thetermon the left sideisindependentofl,Ilyoll ,,o, o.
This holdsfor anarbitraryfinite subinterval
Jo
ofIn,
-0),andsoYo
0 a.e. inIn,
-0). Therefore,Blp’,, -->
0 inL(a,
*,,)as -->.,,.
This contradicts(1.36).
Thus(1.2)
holdsforj 0.To
establish (1.2) for _<j_< n- 1, we use an induction argument. Fix k _< n- 1.Suppose
(1.2)holdsfor 0 _< j _< k and somet
0,1/(2No) ). Suppose
(1.2)doesnotholdfor j k. Then there exists e > 0 and a
sequence {rt}
ofpositivenumbers such thatr --
asg,.(r)
>_ eo, >_ 1. (1.37)As in the proof of necessity in (i), let A be the maximal operator with action defined by
k-i
A ,, X bj D .
ThenA
isT-compact
bythesufficiencyargument in(ii). SinceB
isT-
compact,B
isT-bounded. Therefore, the estimate preceding(1.16)
yields, withh,
as in(1.14),w(r) s(r)
"+)/’For
each r > a, define pr(t)w(r),/, s(r)++j/" h,(t),
> a. (1.39)Then
g. ,(rt)
<C (A + B),r, ll’
(1.40)and
I,,, w<r, ,<r, :"/"’ (liar, + limb’, II)" .41)
By
(1.17)and(1.18),{p,, }
isT-bounded. SinceA
andB
arebothT-compact, A + B
isT-compact.
Therefore,
{(A + B)p,,}
contains a convergent subsequence,say
(after relabeling indices) (A+ B)p,,
-->zo in/.(a,
**). Weshow thatzo
0 a.e. on[a, **).Let Jo
c [a, **)bea finiteinterval. Since
support(p,)
[r 2d;s(r),
r+
2d;s(r)], p,, =-
0 onJo
and hence(A
+ B)p,,
0 onJ0
for all sufficiently large.For
suchl,IlZol[t4,o., Ijo Wlzo(t)
(A+ B)Pr, (t)[’
dt <[Izo
(A+B),11’ o
(l -->Thus
j,,
WIzo]
0 foranyfinite subintervalJo
of[a, ,). Therefore,Zo
0 a.e. on [a, oo)and (A+ B)p,,
0 inL(a,
.,,).Hence
(1.40) implies thatgk.(r)
---> 0 as contradicting (1.37). Therefore,(1.2) holds for k. This establishes necessity of(1.2) forT-
compactness ofB. Thus Theorem 1.1isproved. 1Note
that Theorem 1.1 deals withperturbations of asingle-term operator T.In
thenext theorem, we extend Theorem 1.1toa multi-termoperator L.THEOREM
1.2. Let p, s, w, W, P, B, and gj.s be as in Theorem 1.1.L" Ifw(a,
**) -->Ifw(a,
**)be the maximal operator correspondingtoLet
lip
W,p a, P, D’,
wherem, ai (0 < < n) L"(a, **)and
P,
wst/p. Then the following hold:an
(i)
B
isL-boundedif
andonlyif bj o(a,
**)andsup
gj.s(t)
< (0 <j < n 1) (1.42)at<*
for
somed (0, l/(2N0)).
When (1.42) holds, the relative boundfor B
is O.Furthermore,the maximal operatorcorrespondingto
+
vis/v L +
B.(ii)
B
isL-compact if
andonlyif bj o(a,
.0)andlim
gj.s(t)
0 (0 < j < n- 1) (1.43)for
some(0, l/(2No)).
When(1.43)
holds,L
andLt/
have the same essentialspectrum and /%
p,(L) r(M-
L)r(M- Lt/,).
To
prove
Theorem 1.2, we will use thefollowinglemmas.LEMMA
1.1. SupposeA, C, andDarelinearoperators such thatDis C-boundedwith relative bound less than 1.(i)
If A
isC-bounded, thenA
is(C+
D)-bounded. Furthermore,if
the relativeboundof A
withrespecttoCisO,then the relative bound
of
Awith respecttoC +D
isO.(ii)
If
AisC-compact,
thenAis (C+
D)-compact.PROOF.
For (i), we have D(C)_
D(D), D(C) c: D(A),IlDyll-< K, IlYtl
/ ellcMI
(y D(C)) forsome
K >
0ande
(0, 1), andIIal -< K= IlYtl
/,llcti
(y D(C)) forsome
K=, d
> 0. Therefore, D(C+
D) D(C) c:_ D(A). Fix y D(C). ThenIIAyll-< K INI + all(c +
O)y-o>tl-< c Ilyll + &lKc + o)ytl + &lloyll
_<
(m:= + z<’,)Ilyli + Zll(c + o)Yll + ,llcyll.
Noting that
Ilcyll-< IIc + D)I + IIZl -< IC + oYll + K, IlYll
+ecl, ,
obtainIlcyll-<
1-II(c + D) +
1-IlYl. Hence IIAyll K IlYll +
1-I[(c + D)MI,
whereK
isindendent
of y. Therefore,A is(C + D)-boundedand thestatementconcerningrelative bounds followseasily.For (ii),
suppose {Yn}
is (C+ D)-bounded,
i.e.,y,
D(C+
D) and]lYnll + [[(C + D)yn[
<_K
for some constantK
independent of n. ThenYn
D(C) andIlcyll <-II<c + D)y, + Dynll
<K + K, Ily, + e]Cy,
by the C-boundednessofD.
Since 0 < e < andilY, II-< K
w haIICy,
<K (1
1-e+ K,)
Therefore,{Yn}
is C-bounded.Since
a
isC-compact, {ayn}
containsaconvergent subsequence. Since{Yn}
was an arbitrary (C+
D)-bounded sequence,A
is (C+ D)-compact. 1LEM1VIA1.2. LetB,
L
andT
be the operatorsinTheorems1.1and1.2. Then:(i)
B
isL-boundedif
andonlyif B
isT-bounded. Further, the relative boundfor B
with respect toL
is0if
andonlyif
the relativeboundfor B
with respecttoT
isO.
(ii)
B
isL-compact if
andonlyif B
isT-compact.
PROOF.
Consider the differentialexpression l-’rW
Ipp,
tl,D’. Its
I--0
coefficientssatisfytheperturbationconditions(1.1)sincefor e
I
and0 < < n 1,p, <
(constant).
s(t) la, w s"+’) s(t)
--,
a, (0
< < n-1)
eL"(1). Hence
byTheorem1.1(i),L- T
an
is T-boundedwithrelative bound 0. Application of
L
emma 1.1 (withA D /-_}L- T
and C T) yields that
L-
TisT + L- T
L-bounded with relativebound 0.
(i)
Suppose
B isL-bounded. ThenB
is(-,/L-bounded
sincean
L"(1). Anotherapplicationof
Lemma
1.1 (withA =B,C ml
L, andD T- __1
L) shows thatB isT- bounded.Next, suppose BisT-bounded.
By
Lemma 1.1 (with A B, C T, andD
is The statementaboutzerorelative
bounds also follows from
Lemma
1.1.(ii) Thispartisprovedin asimilar mannerusing
Lemma
1.1(ii).I
PROOF OF THEOREM
1.2.(i) Sufficiency.
Suppose
(1.42)holdsfor 0 _< j _< n landsomet$ 0,1/(2N0)). By
Theorem 1.1(i),
B
is T-bounded with relative bound 0.Hence Lemma
1.3 impliesthatB
isL-
bounded with relative bound 0. The resultD(//v)
D(L) followsbythe sameargumentused in showingthatD(T/v)
D(T)intheproofof Theorem 1.1.Necessity.
Suppose
B isL-bounded. ThenB
is T-bounded by Lemma 1.2. Henceby Theorem1.1,bj
(0 _< j _< n 1) satisfy(1.42)forsome t$ 0,1/(2N0)).
(ii) Sufficiency. S.lppose(1.43)holdsfor 0 j P- andsome
/0, No )
ThenbyTheorem1.1,
B
isT-compact
andhenceL-compact
byLemma
1.2. The invarianceofthe essential spectrum and FredholmindexofLunderperturbations by Bfollow as in theproofof Theorem 1.1.Necessity.
Suppose B
isL-compact.
ThenB
isT-compact
byLemma
1.2.By
Theorem 1.1, there exists t$ 0,1/(2N0)
suchthatbj
(0 _< j _< n 1) satisfy(l.43). 1REMARK.
Theorems 1.1 and 1.2applytooperatorsT
andL
with coefficientseventuallybounded abovebythecorrespondingcoefficientsofanEuleroperator. Tosee this,notethat thehypothesiss’(t)l -< No
a.e. onI
impliesthat there exists apositiveconstantC
such that s(t) < C for all sufficiently large. Now, by definition ofP,
and W and the hypothesis thata,
(0 < < n)L"(1),
we havela,(t)l
,(t) s(t)’
<C, t’
(1.44)for all sufficientlylarge,where
C,
areconstantsindependentof and 0 < < n.EXAMPLE
1.1. Let n 2, p 2, w 1, tz 0, andsbeanypositive,AC,o([a,**))
functionsuch that
Is’(t)]
_<No
forI
[a, **). ThenW
andP,(t)
s(t)2’ for i=0, 1, 2.Consider
Ly a2(t)
s(t)y" + a,(t)
s(t)y" + ao(t
y (1.45)and
By b(t) y" + bo(t)
y, (1.46)where --, ao,
a,
a L** (I)and b0,b /.
(I). Thena2
f’/’"’" Ib(’r
d’rgj.
(t) s-
s(’r)2j (j=O, ). (1.47)By
Theorem 1.2,B
isL-bounded if andonlyif supg. (t)
< (j 0, andL-compact
if andtel
onlyif lim
g.,(t)
0 if=0, 1) forsomed (0, l/(2N0)).
Nextwe prove acorollaryof Theorem 1.2 in which an nth order perturbation
B
ofLis considered. The perturbation is such that the coefficients of thehighest-ordertermsinL
andL
+B
obeythesamehypotheses. Before statingthecorollary,weprovealemmaconcerningthedomains of the single-term operatorT
and multi-term operator L.LEMMA
1.3. LetTandL
beas inTheorems 1.1and l.2. Then D(L) D(T).PROOF.
First consider the case in which a-=
1.By
Theorem 1.1 withv
Wtp a,P,PD’,B
is T-bounded andL T/
T+B. Thus D(T) D(B),and soD(L)
D(T + B)
D(T).For
general a such that an, 1/aL’(1),
we may replaceT
bya, T
withoutaffecting T-boundednessofB.It
followsthat D(L) D(a T) D(T).I
COROLLARY
1.1.Let
p,s, w,W,P,,
andL
be asinTheorem1.2.Let B: lYw(a,
**)--> IYw(a,
0.) be the maximal operatorcorrespondingtoWIIp
where bn,
L’(I), bj I.(I) (0
<j < n),an +
b(1.48)
and
:rn. s(t--S
w(r)sfr)’a+’)’ dr 0 (0 < j < n-l) (1.49)for
some(0, l/(2N0)). Let R" l.(a,
**) -->l.(a,
**) be the maximal operatorcorresponding to
+
v. Then D(L) D(R),r,(L) eye(R),
and/%
pc(L) r(M-
L)r(M-
R).PROOF. In
view ofTheorem 1.2, it suffices toprove
the corollary for the operator RL + b. P2/’ D n. As
inTheorem 1.1, letT" L(a,
,,*) --->L(a,
.,,)be the maximal operator corresponding tor P
lipD n.
ThenR L +
b T.By
Lemma 1.3,D(L) D(T) and D(R) D(T).
Hence
D(L) D(R).For
any scalarA
andy D(R) D(L),
(M- R)y .y- Ly-
bnP’.‘"
/%y_Ly+ an b {A.y_Ly+ ,=oZa’P"’Y<"
n-,Zy Axy + Say
PERTURBATIONS OF nTH ORDER DIFFERENTIAL OPERATORS 63
where
At
andS;t
are the maximal operators associated with(1 + a/(AJ-
and- a , D A
WI/p
bn p
b I, respectively.An
applicationof Theorem 1.2(with L, B, andan an
L/
replaced byA;t, St,
andAJ-
R, respectively) yields thatSx
isAa-compact,
cr,(Ax) O’,(M- R),
and0
p,(Aa) = K’(Ax) g’(:l/- R).
(1.50)By
definition ofAx,
RI_L=( an a, + bn I A. Let
han +bn a,
Then h,1/h
L’(1)
andR(R/- L) {hg"
gR(A)}.
TheresultthatR(A)
isclosedifandonlyifR(M L)
isclosedfollowsfrom thenextlemma.LEMMA
1.4. LetM
beaclosedsubspaceof lfw(a,
**)andN
hM{hg"
gM},
whereh, 11 h
L"(a,
oo). ThenNisclosed.PROOF. Suppose
hgN
withgn
eM
and hg-- z.
Since 1/hL"(a,
00),’gn
-’> z/h. Since Misclosed, z/h M. Therefore,z h.(z/h)
N.So
Nisclosed. 1Since
tr,(Ax) o’,()I/- R), p,(Ax) p,(itl- R),
i.e.,{/z" R(/Z/ A)is closed} {/2" R(/z/ (R/ R))is closed}.
Therefore,
R(A:t
closed =:,R(,qJ R)
closed. Itfollows thatp,(L) p,(R);
and so o’,(L)=
It remains to show that it e
p,(L)
=#(AJ- L) ()d- R). Let
it ep,(L).
Then
R(A/- L)is
closed andL(a,
*,,)R(it/- L)
M, whereM N(I- I.:).
SinceL*y
ity hasat most nL(a,
oo)solutions,M
isfinite-dimensional.Let gt=
an+b
Then, L"(1)
andA (;t/- L). Any f /.(a,*,,)
cana
be written as
f (:hr L)g +
m, whereg e D(L)andm M. Thuswith
vf l.(a,
**),(/1I- L)g R(A),
andgan vM. Now,
sinceR(ft/- L)
closed(a)
losd,<a,
**)(a) * N
wh S--Vt {"
mL’*(a, *,,), dimN dimM.
By
definition, the deficiencyindexofA isfl(Ax) dim[/.(a,-0) R(Ax)
dimN dimM dim[/.(a,
oo)R(t/- L)] fl(:t/- L).
Since
A;t gt(M L)
and Ig,: 0 (because L’(a, **)),N(Ax) N(Zt- L).
Therefore,ct(Aa) ct(M L).
Thustc(At K’(M- L).
SinceR(Ax)is
closed, 0p,(ax). Hence
by<.5o),
(a) (Z- ).
Threor,:(Z- .) :(Z- ).
REMARK.
Notethat(1.49) and(1.43)are identical conditions on the lower-order perturbation coefficientsbj,
0 <j < n 1. Theorem 1.2 is a result for lower-order perturbations ofL
Wi/pa,P, D’,
whereto,a,
(0 < < n 1) EL’(a,**).
Corollaryl.lappliesto nthorderperturbationsofLoftheformR
a+ bn) P" D + 2 (a, P,l’" + b,)O’
I=0
where b satisfies(1.48)and
a,
+b,,
L’(a, **) (inanalogytothe conditionsona
intheoperator L).
2.
CONDITIONS FOR OPERATORS WITH LARGE COEFFICIENTS
Recall that Theorem1.1appliestooperatorssuch that
P(t)
]’
< Ct W(t)JforsomeconstantCandall sufficientlylarge. Thefollowingtheorem generalizes thesufficiency conditions inTheorem 1.1for operators
T
witharbitrarilylargecoefficients.THEOREM
2.1. Let < p < and I [a, **). LetP
andW
be nondecreasing, positive continuousfunctions
onI
such that W-qlp,
p-q/i,L(1),
where--+--
1. Let T,P q
B L
(I) --> Ifw(I) be the maximal operatorscorrespondingto 7 pIIp- D
and
n-I
j=O
respectively, where each
b
Eo(l). For
0 <j < n and>
O,define
(0
If
thereexists5 >
0 such thatsup/z.(t)
< (0 <j < n- 1), (2.1)tel
then
B
isT-boundedwith relativeboundO.(ii)
If
thereexists > 0such thatlim
/j.(t)
0 (0 <j < n 1), (2.2)thenBis
T-compact.
PROOF.
(i)Suppose
(2.1)holds for some > 0. Wewillshow that TheoremA appliestothe choices
f=
N’,
and e.
FixI
and e(0,).
SincePisnondecreasing onI,itfollows that
/,)
dr <
T,,,(P)
ef(t) P(’)/"
P(t)Similarly,
T,
(W) < The choicef
ismade so thatcertainupperbounds on w(t)S,(e)
andS2(e)
are equal:Sk(e)
<, sup/zj.(t)
tl (k 1, 2).By
(2.1), there exists a,constant C independentof such that
Sk()
< for k 1, 2 and (0,).
Hence byTheoremA,there is aconstant
K
such thatfor all y D(T).
By
the same calculations usedto obtain (1.8)in the proofof Theorem 1.1,IIyll-< K, ’-+’-"" IlYll
/K, :-"" IITMI, K, K ’’p,
for all y D(T). Since p > 1, the coefficientofIITMI
canbe madearbitrarily smallby choosinge
e (0,)
sufficiently small.Therefore,BisT-boundedwithrelativebound0.
(ii)
Suppose
(2.2)holds for somed
> 0.T-compactness
ofB
followsbytheargumentused in proving sufficiency in Theorem 1.1 (ii). 1-I
EXAMPLE
2.1. Let W(t)=-
and P(t)e’.
Then T e’/’D" and BbD
j--0. In
thiscase, condition
(2.1)precludes
exponential growth ofbj. Suppose
Ibm(t)
<Cjt a’,
a < < 00, 0 < j < n-I,A
> O,for some constants
C
andA.
Fix j and letA A
and CC:.
Thenbythe definitionof/z.
inTheorem 2.1,
e(S/,,
/(,,,))[.: z
dz(Ap
+
I)e0/" ’/"))’+ de’/’))
For
sufficiently large,we obtain(witha differentconstant) e(p i)tl(np) Ce(a-J)t/n/1. (t)
<n-I
Hence
(2.1)holds ifA
_<j,and(2.2)
holds ifA < j.For
example,the Euler operatorD’
is1--0
T-bounded, and the operator
[
j-D
(e >0)
isT-compact. I
j=0
Westatehere another part of Theorem 2.1 fromBrownand Hinton[3]mentionedearlier.
THEOREMB.
Let _< p < ,,o,I
[a,,), and 0 <_ j <_ .-1. Let N, W, andP
be positive measurablefunctions
such that N eLt.(1); for
p > 1, W-"1,
P-qI.(1)
where+ 1;for
p 1, W-,
P-|arelocallyessentially boundedonI.Define
P q
T,. (t’)
p=l
<p<,,
withsimilar
definitions for T.
(W).Suppose
thereexistse > 0 andapositivecontinuousfunction
f
f(t)onIsuch thatf’(t)
> O,R,(e)
:=sup{f(,)
("-)N(t)T.t(P)}
<tl
and
R2(e)
:=sup{f(t)
-’N(t) T,. ,(W)}
<t!
for
alle (0,e0).
Then thereexistsK >
0 such thatfor
alle (0,Co)and
y D,I, rCly’ ’l <- r I, W[Y] + et’-"P R,(e)It P
where
D {y" y"-" ACto(’), t WlY
<**, andI, <**}
This result canbe usedtoprovethefollowing theorem,whichgivea pointwiseconditions sufficientfor relative boundedness and relative compactness.
THEOREM
2.2.Suppose
the conditionsin Theorem2.1aresatisfied
with thedefinition of lz.
replaced by
In
addition, supposeP
AC(I)
with d P(t)>-
0for
I. Then the conclusions inw L-j
Theorem 2.1hold
for
<_ p < providedthatfor
the case p 1,W -
and P-| are locallyessentiallybouledonI.
PROOF.
(i)Suppose sup
uj(t)<
for 0 _< j _< n-1.We
willshowthatTheoremB
appliestothe choicesf--
N=’,
andanye
> 0. FixtI
ande
> 0. SinceP and W are nondecreasing on I,
T,.(P)
< andT,.(W)
<Hence
P(t) W(t)
R,(E)
<_Tpt f(t)
("-’)"[0,(t) l’ e--SS
andP(t;)
_<sup f(t)-’" [0,(t)[" By
the choiceoff, R(e)
_< suppj(t)
< (k I, 2). Therefore, TheoremB
applies. Therestof theproof,tel
including (ii),follows as in theproofof Theorem2.1.
I
EXAMPLE
2.2. Let W(t) and P(t) e ct > 0. ThenT e=’/’D
and Bbj/Y.
Let < p <,. Suppose lb,</)l
_<c, ca,’
a < < 0 < j < n-1 forj=O
Ib,</)l
<c;
e(#j’-’’")’ ThusbyTheorem 2.2, someconstantsCj
andfls"
Then/s(t)
e,,/,fl
<otj
BisT-bounded andfl
<ctj =>
BisT-compact.np np
So the pointwiseconditionson b in Theorem 2.2 allow
b
to grow exponentially.In
contrast, theintegral
average
conditionsofTheorem 2.1 appliedtoExample
2.1 allow polynomial, butnot exponential,growth ofb.
1,3.
INTEGRAL AVERAGE CONDITIONS FOR THE CASEp
1Thefollowingtheoremgivessufficient conditions forT-boundednessfor the case p for integral averages.
THEOREM
3.1. LetP
andWbenondecreasing,positivecontinuousfunctions
such that1and1__
P
Warelocallyessentially boundedon [a, **). Let T, B:
l(a,
**) -->lJw(a,
**) be themaximal.
operatorscorrespondingto
and
"c
PD"
W
n-I
V= W
bDJ
respectively, where each
b
is ameasurablefunction
on [a, ,). For0 <j < n- and > O,define
r,,,,,,,:,:,l<,+,,,: ,,+.,r,.<,,1 ,’,..":’:>: j,
,-’<,>-,If
thereexists>
0 such thatsup
/z,.(t)
< (0 <j < n-l),thenBisT-bounded.
If
in additionb_ =-
O, then therelativeboundof
Bwithrespectto T is O.PROOF.
Weshow that TheoremA
appliestothe choicesf
p 1,N
andany
e t.
Fix e [a, .,,) ande
(0,t).
Usingthehypothesis
thatP
is nondecming, Similly, (W)< These inequalities we have.,(P)
-.It.,+ef(,)l P(t) W(t)
yield
upper
bounds forS(t)
dS(t).
The choicef
is made so that theseupr
bounds eequal: fork or 2,S(e)
supg.n(t)
<M
where the last inequalityE at<- E
followsbyhypothesis(forsomeconstant
M > 0). By
TheoremA,there existsK
> 0 such that for all e (0,iS)
and y e D(T),Let denote the norm
of/2w(a,
,,,,). Thenj=O j=O
n-1 n-!
< g
y.
j=O{ - S=<> IM
/- s,<> IITM! }
< g g1=0{ --’ IlYil
/--’ IITMI }
where wehave used the estimates on
St
andS=. Hence B
isT-bounded.If
bn_
0, then the previous sum can be truncated at j n- 2:KMf e*"/IITy{I
for ally
D(T), where C(e)is independentofC(e) y.
\j--0
Restrict e (0,
d)
such that e < I. ThenIIB3I
<C(e)I13I + K M
(nI)ellT3l
for ally D(T), fromwhich itfollows that the relative bound of
B
with respecttoTis 0.ACKNOWLEDGEMENTS.
Supportedinpart bythe UniversityofTennessee
Knoxville and Oak RidgeNationalLaboratoryScience AllianceProgram.
The author would like to thankProfessor Don B.Hinton for his advice.REFERENCES
T.G. Anderson.
A
Theoryof
RelativeBoundednessand RelativeCompactness for
OrdinaryDifferential
Operators. Ph.D..thesis, UniversityofTennessee
Knoxville(1989).
E.
Balslev andT.W.
Gamelin. The EssentialSpectrum
of aClassofOrdinaryDifferentialOperators. Pacific J.
Math.,14(1964),
755-776.R.C.
Brown
andD.B.
Hinton. Sufficient Conditions forWeighted InequalitiesofSumForm.
J.
Math. Anal.Appl., 112 (1985), 563-578.S.