Regularity of pairs of positive operators

REGULARITY OF PAIRS OF POSITIVE OPERATORS

SHANGQUANBU,PHILIPPE

CLIMENT

ANDSYLVIE

GUERRE-DELABRIRE

0. Introduction

In

thispaper,we considerapair(A, B)of closed operatorsona Banachspace X

with domain D(A)and D(B). The pair (A,B) iscalledregularifforevery

_f

E X, theproblem Au

+

Bu

_f

possessesoneandonlyonesolution.

Relatedto thenotion of coercively positive pair of operators, introduced in [S],

wealsoconsidertheexistenceofasolution totheproblem XAu

+

Bu

_f

forall

X > 0, with someuniformity inX. Thisstrongerpropertyiscalled X-regularity. Thesenotionsof regularity and X-regularitynaturallyarise invector-valuedCauchy problems; see [G], [DG], [S] and also [CD]. The uniformity in X, givenby the

X-regularity, isoften usefulincertain applicationstopartial differentialequations. In [G],under the hypothesis that 0E p(B)andin[DG],some sufficientconditions aregiventoensuretheregularityofapair(A,B)oncertainsubspacesofX,relatedto the operatorB. Thesesubspaces,denotedby

DR

(0, p),are real interpolationspaces

betweenD(B) andX (Theorem 1.2).

It was observed in [S] that if 0 p(A) (q _{p(B), then the pair} is X-regular on DR(O, p).

Inthispaper,weprovethe X-regularity ofthispair (A, B),considered in [G],on

DR(O,p) under the weaker assumption that 0 6 p(B)only (Theorem 2.1).

Note

that ifBisbounded, then the pairisX-regularonX.

Weconstruct anexampleofaregularpair(A, B) of operatorsin a Hilbertspace,

withBbounded, satisfying theassumptionsofthe theorem ofGrisvard[G],which is notX-regular (Example 2.2).

1. Preliminaries

Inthis section wegive precisedefinitions ofregularity andX-regularityof apair

of operators. Then, for the sake ofcompleteness,werecall aresultofDa Pratoand Grisvard [DG] (seealso[CD]),which isthe starting point ofourresults.

Let XbeaBanach spaceandAand Bbetwoclosed operators inX.

DEFINITION 1. The pair (A, B) iscalled regular, ifforall

_f

6 X,there existsa uniqueu D(A)C)_D(B) such thatAu

₊

Bu

_f

ReceivedJanuary29, 1997.

1991 Mathematics SubjectClassification.Primary 46B25, 47A05, 47A10, 47E05, 34B05. Thefirstauthorwassupported bytheNSFof China and the FokYingTungEducationFoundation.

@1998bytheBoardofTrusteesof theUniversityof Illinois Manufactured in the UnitedStatesof America

358 S.BU,P.

CLIMENT

AND S.

GUERRE-DELABRIRE

Ifthe pair (A, B)isregular,itfollows from the Banach theorem that

(1.0)

Ilull

+

IIAull

+

IlOull

MIIAu

+

Bull

forsomeM > and for allu D(A)C3 D(B). Itiseasytoverifythe following lemma.

LEMMA

l.O. Let AandBbetwoclosed operatorsinX. Then the pair(A,B)is

regular

if

andonly

_if

(1) (1.0)holds and

(2) R(A

₊

B) isdenseinX.

Moreover,

_ifO

p(A)orp(B) (wherep(.) denotes the resolventset

_of

anoperator), then(1.0) isequivalentto

[IAu[[

₊

[IBull <_

_MIIAu

₊

for

someM >_ and

_for

allu D(A)f3D(B).

Remark

I.

Theoperator A

₊

B isclosed if andonlyif

Ilull-+-IIAull

+

Ilnull

M(IIAu

+

Bull

+

Ilull)

for someM > and for allu D (A)N D (B).

In particular,if the pair(A, B) isregular, A

+

Bhastobe closed.

A

regularpair of operators (A,B)iscalledcoercivein[S].

Also, the stronger notion of coercively positive pairis introduced in [S], which motivatesour Definition 2.

DEFINITION2. The pair(A, B)iscalled,k-regularinX,if for all

_f

6 Xand for

all ,k > 0, there exists auniqueu D(A)fq _D(B) such that,kAu

₊

Bu

_f

and moreover, for all,k > 0,

]l.Aull

+

IIBull

<

MII.Au

+

Bull

for someM > 1, independent

of,k

andfor all u D(A) D(B).

Remark 2. Clearlyif(l. 1)holds, then theinequality

llAull

+

zllBull

<

MIIZ.Au

+

#Bull

holds for someM> 1, for all,k,/z > 0andu D(A)(3D(B),whichshows that the definition of,k-regularityissymmetricin

A

andB.

Itisalso clear thatthisinequalityisequivalenttothe following ones:

IIAull

MIIAu

+

)null,

forsomeM > and all,k > 0 andu D(A)N D(B), and

llnull

MIIAu

+

null

forsomeM > andall,k > 0andu D(A) D(B).

LEMMA

1.0... LetAandBbetwoclosed operatorsinX (notnecessarilydensely defined).

If

0 p A), then the pair A B)is2.-regular andonly

(1) (1.1)4

holds.forall)

>0;

(2) Thereexists)o >0such thatR()oA

+

B) isdenseinX.

Proof.

Clearly, it is enough toprove that conditions (1)and(2) imply that the pair(A, B) ish-regular.

First observe that conditions (1) and (2) togetherwith Lemma 1.0, where A is

replaced by 2.0A,and the fact that 06 p(A), imply that the pair(,k0A, B)isregular.

Thus,inparticular,0 p()oA

+

B).

Next

weshow that if 06 p()A

₊

B)forsome > 0, then 06 p(A

+

B)for

all > 0 such that

6 ifM > and ifM=

(*)

L

M+

1’

M-

m+i’

Indeed, problem Au

₊

Bu

_f

isequivalentto

Au

₊

Bu 1- Bu

Settingv Au

₊

Bu,wehave

(**) v B(A

From 1.1 )z,itfollows that

IIB(kA

₊

B)

-

M.

Under assumption (,), by the Banach fixed point theorem, it is clear that there exists oneandonlyonev 6 Xsatisfying(**)and hence (A, B)isaregularpair for such

.

Noting that

IIB(Au

₊

B)

-

Malso holdsfor in this interval, we can

repeatthisargument and, since < and > 1, showbyinductionthat the

pair (A, B)isregularfor all > 0,whichtogetherwith(1.1)zimplies that the pair

360

s.

BU, P.

CLIMENT

AND S.

GUERRE-DELABRIRE

Letusrecall classical definitions on closed operators:

A

closedlinearoperator

A

D(A) C X X (notnecessarilydenselydefined)iscalled positivein(X,

_I1"

II) [Tr]if there exists C >0such that

(1.2)

Ilull

Cllu

+

)Aull, forevery,k >0 andu D(A),

and ifR(I

₊

)A) Xfor some,k > O, equivalentlyforall) > O.

Remark 3.

In

[Tr],an operator A is calledpositive if it ispositive and satisfies the additional assumptionthat0 6 p(A).

In

thispaper,it is convenienttorelaxthis

extracondition.

Observe also that

A

ispositiveif andonlyif the pair(A,I)is,k-regular.

IfAispositive, injectiveanddenselydefined, itiseasytoprovethatA

-

isalso

positive.

IfX isreflexive and Aispositive,thenAisdenselydefined[K].

Let

_E

:= {,k 6

C\{0};

arg

.1

<r} t2{0},forcr 6 [0, 7r). IfAispositive,there

exists0 6 [0,7r) such that(1.3)holds, [K p. 288]:

(1.3) (i)r(A)

_

E0

and

(ii)for each0’ c (0, 7r],thereexists

M(O’)

> such thatII(.l A)

-M(O’),

forevery,k

_C\{0}

with larg

_Xl

>_ 0’

wherer(A)denotesthespectrumofA.

The numberCOA .’-- inf{0 6 [0,7r);

(l.3)holds}

iscalled thespectral angleof the

operatorA. Clearly

coa

G [0, 3T).

An

operatorA issaidtobe oftype(09, M) [Tan],ifAispositive,coisthespectral angleofA and

M := inf{C > 0; (1.2)holds min{C >0; (1.2)holds}. NotethatM isalso the smallestconstantin(1.3) ii)for0’

Two

positive operators A and B in X are said to be (resolvent) commuting if the bounded operators (I 4-

.A)-

and (I 4-/z

B)-

commute for some

.,

_tt > 0, equivalently for all.,/z >0.

IfA and B arecommuting positiveoperatorsthen A4- B (withdomain D(A) D(B))isclosable[DG].

The following theorem,which is aconsequence ofatheoremofDaPrato-Grisvard

[DG]and ofGrisvard [G]willbeessential inthesequel.

THEOREM 1.1. LetAandBbetwocommutingpositiveoperators inXsuch that (i) D(A)

₊

D(B) isdenseinX,

Then the closure

_of

A

₊

Bis

_of

type (o, M)with09 < max(wA, wn).

If

moreover

(iii) 0E p(A)orp(B) (resolventset

_of

AorB), then

(a) thereexistsM > such that

(1.4)

Ilull

MIIAu

+

Bull,

_for

allu D(A)fq_D(B),

andO p(A

+

B),

(b) R(A

₊

B)

_

D(A)

₊

D(B),

(c) A

+

B isclosed

_if

andonly i(R(A

+

B) X Oandonly if(l.l)holds, (d) theinverse

_of

A

₊

B

isgivenby

(.) (A

+

B)-x

f(A

+

z)

-I(B

z)-xdz,

27ri

_J

where y is anysimplecurve in p(B) f)

p(

A)from

e-iO _{to e}

_iO,

_with

o)_B

Oo

mO)A

Remark4. (1) Under hypotheses (i)-(iii) of Theorem I.1, assumption 2) of

Lemma

1.0 isalwayssatisfied. Therefore, in ordertoprovethe regularity of a pair

(A, B),itis sufficienttoverify inequality (1.1), whichmeansthatA(A

₊

B)

-

is a boundedoperator.

(2) Similarly, under hypotheses (i)-(iii) of Theorem 1.1, assumption (2) of

Lemma1.0.)isalwayssatisfied. Therefore,inordertoprovethe,k-regularityof a pair (A, B),it is sufficient toverify inequality (1.1),whichmeans that)A()A

₊

B)-1

is auniformlybounded operator for all,k > 0.

Inthispaper,weshallalwaysbeinthe situation of(i)-(ii)of Theorem 1.1,which means that we will consider the following threehypotheses for a pair ofpositive

operatorsAand BinXoftype respectively (OOa, MA)and (OB,Mn):

Ho"

D(A)

₊

D(B) isdense in X.

H"

AandB areresolventcommuting.

H2"

OOA WO)B <

.

Inordertoobtainresultsonthe regularity and the -regularity ofapair ofoperators,

we need to introduce the interpolation spaces

DA

(0,p), associated with a closed operatorA,for 0 6 (0, 1)and p 6 1,+cx]. ThesespacesaresubspacesofXwhich

aredenseinXfor thenorm

_II.

wheneverAisdenselydefined.

For0 6 (0, 1)and p E [1,+),

DA

(0,p)isthesubspaceofX consistingof all x such that

362 S.BU, P.

CLIMENT

AND S.

GUERRE-DELABRIRE

P

where

L.

isthespaceofp-integrableBorel functionson(0, +cxz)equippedwith its invariant measuredt

/

t.

For0 6]0, 1[,

DA

(0, 0) isthesubspaceofX consistingofallx 6 X such that

sup{lltA(A

q-

t)-xll

(0, -t-a)} <

+.

When0belongstop(A),

_DA

(0,p) equippedwiththenorm

IIxlIzgAO.Z,)

IItOA(A

+

t)--xll’.

becomes a Banach space.

When0 6 p(A) andA isbounded,

II.llDa(0.p)

isequivalenttothenormofX.

The following fundamental result, due to Grisvard (Theorem 2.7 of[G]) is the startingpointofthispaper.

THEOREM 1.2. LetXbeacomplexBanach space, and letAandBbetwopositive operatorsin X,

_of

type (O)A,MA) and (w,MI) respectively, satisfying hypotheses

Ho, H,

H2.

If

0 p(B), the pair(A, B) isregularin

Dt

(0, p). 2. Results

The firstresult ofthis paperis the following theorem which is an extension of Theorem 1.2tothe case of ,k-regularity.

THEOREM2.I. Let X be a complex Banach space, and let A and B be two positive operators in X,

_of

type (O.)A,MA) and ((_OB, MB) respectively, satisfying

hypotheses

Ho, H,

H2.

If

0 p(B), thepair (A, B) is )-regularin DB(O,p)

for

everyO <0 < and < p <_ (x.

Remark5. If moreoverB isbounded,it isclear that the pair(A, B)is,k-regular

inX.

The nextexample shows thatin particular, even ifX is aHilbert space, the

hy-pothesis 0 6 p(B)cannotbe omitted in Theorem 2.I.

Example2.2. There exists aHilbertspace Gand thereexist twopositiveoperators

A and B inG satisfying hypotheses H0,

_H

and H2, with Bbounded, such that the pair(A, B) isregular,butnot)-regularinG.

Remark 6.

In

[L,Theorem2.4] (seealso[CD]),anotherexampleisgiven, where

Aisthe derivativeactingon LP ([0, T]; Y)for somenonreflexivespace Y,such that thepair (A, B)isnot,k-regularin

_DA

(0, p).

Proof of

Theorem2.1. Fix ,k > 0.

By

Theorem 1.2, we know that the pair (A, ,kB)isregularinDB(O,p). Inparticular, for all x E DB(O,p),

yz (A

₊

)B)-x

D(A)f3D(B)

andwehave

Byz

DR(O, p)togetherwiththeinequality

II)Byzllo,O,p)

CIIxllD,O,p).

We

shall show thatCisindependentof

..

Forthis, we aregoingtouseequality (,)

of Theorem 1.1, appliedto A and.B. Withoutloss ofgenerality, since0 6 p(B), we cansupposethat_Vconsistsof the halfline (oe

-iO,

ge-i],

the arc of the circle

C

{z

Izl

,

larg(z)l <

00}

and the half line

leei, cxei"),

for some fixed

00, (_O_{B <} (90 _< 7r O)_{z and for} sufficiently small e inorder toinsure that _?, is in p(-A)f3p(,kB). SinceAisof type (COA,MA),by (1.3)thereexists

M

such that for allzsuch thatlarg

_zl

< 00,

II(m

₊

z)

-As

intheproofof Theorem 3.1 of[DG],forevery > 0we canwrite

()B

+

t)-yz

()B

+

t)-(A

+

)B)-x

f

2rri (A

+

z)

(.B

+

t)

(.B

z)-x

dz

f

dz 2rri (A

+

z)

(.B

z)

X

+

z

f

dz 2rri (A

+

z)

-

()B

+

t)-x

₊

z

f

dz 2rri (A

+

z)

(.B

z)

X

+

z (,kB

+

t)

-

f

(A

+

z)-x

dz

+

z

f

dz 2rri (A

+

z)

-

()B

z)-x

+

z by (,)and

_H

(A+z)

-

M’

by analyticityof thefunction

(A+z)-’t+z.

and the fact that _t+z. <

Iz(z+t)l for larg

zl

_<

_0o.

Hence

LB(;B

₊

t)-y

y t(kB

₊

t)-Iy

364 S.BU, P.

CLIMENT

ANDS.

GUERRE-DELABRIIRE

Then 2zri (A

+

z)

()B

z)-x

dz

l

(A

+

z)

-()B

z)-xdz

2rci

_Jr

+z

I

z (A+z)

-()B-z)-xdz

2rri

_Jy

t+z

)B(,kB

+

t)-IYz z

4-

(A

+

Z)-!

(,kB

z)-xdz.

First,weclaimthat

lim

I

z (A

₊

z)

-()B

z)-xdz

O.

--,o+

_Jc

_z

+

Since B is invertible, II(,kB z)

-

is uniformly boundedwithrespecttozin a neighborhood of the origin. Sothere existse0such thatII(kB z)

-

< 211(B)

-Then fore < _e0wehave for

_Izl

_< e0.

We

can supposethate0 < _7"

z

(A

₊

z)

-()B

z)-x

dz

z+t

<

II(A

4-z) II(ZB-z)

_{Ilxll Idzl}

Iz/tl

dO

_8M

(,n)

-

IIx

I10o

<

_2M

II(.B)

-

_IIIIxll

a-o,, 4-ecosO

whichtendsto zerowhene

--

0

+.

Theclaim isproved;hencewehave

fr,

z

(A+z)

-()B-z)-xdz

)B()B

₊

t)

-

yz

z

+

where Voconsistsof the half-lines{z’arg(z)

-0o}

and{z’arg(z) -0o}.

By

hypotheses

H

and H2, B()B

+

t)

-)Byz

2rci z

₊

(A

+

z)-)B()B

z)-x

dz

andso II*B()B 4- t)

-

)Byzll < II(m/z) II,n(,n z) 2rr

,,

Iz

+

tl

xll Idzl

+ _r

z(r)dr

< K

v/t

2_4-r2_4-

_2trcosOo

r

where Kis a constantdependingonlyonAand B,and

x(r) max{ll.B(.B

rei")-xll,

II,kB(B

re-i")-xll}

dp

The hypothesis x e_ _{DB(O, p)}meansthat

r(r)

E

Lt,’(R

+)

(see [DG]);thuswe have

t

II,kB()B

-I-

t)-

Byzll

+ _rt (r)dr < K

v/t

2

₊

_r2

₊

_2trcosOo

r

(rt-)

-

dr

rx(r)

K

v/1

+

(rt-)2

₊

2rt-cosOo

r

K.f

g(t) where

tl-O

f(t)

v/l +

2

₊

_2tcosOo

g(t)

tz(t)

L,P(R+)

6

L(R

+)

By Young’s

theorem,we canwrite

366

s.

BU,P.

CLIMENT

AND S.

GUERRE-DELABRIRE

hence

or

L]lLBy)llo(O.p)

<_

IILB(A

_{+ LB)-IxIIDu(O,p)}

<_

This is theinequalitythat we wanted. It impliesthat

I]LB(A

+

LB)-I

]ID(O,p)

<_

whichshows theL-regularityof thepair (A, B) onDs(O, p)byRemark4.2.

Letusmentionanothercaseof L-regularity whichis aconsequenceofTheorem1.2

appliedinthe context of[DV],namelywhenBc isbounded for alls 6 [-1,

+

]"

COROLLARY2.3. Let H be a Hilbert space and let A and B be twopositive operators inHsatisfying

Ho,

H1

and

H2.

If

0 p(B) andsup{

Bi’

_Is

<_ <

+cx,

then the pair(A, B)isL-regularinH.

Proof

of

Corollary2.3.

As

mentioned in [DV], under the hypothesis that

sup{llBi’llllsl

< 1} < -t-cx, D8(0,2)

D(B).

Thus Theorem 2.1 implies

that(A, B) isaL-regularpairin

D(B).

Then

Dore

and Venni show that, under the hypothesis ofCorollary 2.3, (A, B) is aregular pair in H.

An

adaptation of their

proofcanbe done toprovethatinfact, the pairisL-regular. Indeed, forx 6 H, by

Theorem 2.1, observing that

B-x

D(O, 2),wehave

IILB(A

₊

LB)-xll

IIBLB(A

₊

LB)

-

B-xll

< C

B

B-x

C

_Ilx

whereC > 0isindependent ofL > 0. 12]

Construction

_of

Example2.2.

Let

G be acomplex Hilbertspace and letA and

B be two positive operators with B bounded, satisfying hypotheses

H1

and

H2.

Observe thatsince Bisbounded,

_H0

isalsosatisfied. If moreover 0 6 p(A),thenby

Theorem 1.1, the pair(A, B)isregularandG D(O, p) forevery0 6 (0, 1) and p 1,cxz]. Henceifthe pair(A, B)is notL-regular,wearedone.

Inordertoconstructsuchapair, we consider, as in [BC],thespace

G e2(H) x (xk)er xk Hand

Ilxll

2

where (H,

II.ll)

isacomplexHilbertspace.

A

family (Ak)kNofbounded operators on H definesthe following closeddenselydefinedoperatorAonG"

(2.1) D(A) := {x- (x,),r, x, H,

-,r

IlA,x:]]

2

< cx:}

(Ax)k

A:xk

k E

N

forx (x,)kN 6 D(A).

Moreover

A isbounded if andonly ifsupksN

IlAk

< and ifthis isthe case, we have A supN

A,

II.

If 0 6 p(A) for all k 6

N

and supkN

IIA

-

<

,

then 0 6 p(A). Asin[BC],

we shall say that thefamilyof positive operators (A)N of type (0, M) satisfies

property (P) ifforeveryk 6

N,

(i) r(Ak) C [0, xz) and

(ii)forevery0 6 [0,7r[,thereisM (0),independent of k, such that (I/

zA)-ll

_< M(0), forevery z 6

_E0.

Wewillneed the following slightextensionofLemma4.1 of[BC],which we state withoutproof.

LEMMA2.4. Let (A,),r, (B,),rbetwo

_families

ofboundedpositiveoperators on H, satisfying property (P) and such that

_{Ak Bk}

_{Bk Ak}

_for

all k

N.

Then the operators AandB

_defined

by (2.1)aredensely

_defined

and

_of

type (0, MA)and

(0, Me) respectively. Moreover,the pair(A, B)

_satisfies

hypotheses H0,

H,

H2.

Now supposethat (A,)kr and (B,)ks are twofamiliesofoperatorsin H asin

Lemma2.4 satisfying(2.2)and

(2.3):

(2.2) 0 6 p(Ak) foreveryk 6

N

and

sup"

_-IlAk

k6N

(2.3) ’v’l >_ qxl 6 H,

_IIx/ll

1, such that

lllAlxl

+

lXlll

IIAIxlll.

Set

Bk

/zk/k,

with/zk > 0, k 6

N

such that

_B

_< for allk 6

N.

Thenthe

families (Ak)kN and (Bk)kN also satisfy the assumptions ofLemma2.4. Thepair (A, B) definedby (2.1) satisfies H0,

H,

H2.

Moreover0 6 p(A) by (2.2)andB is boundedwith B _< 1.

Weclaimthat the regular pair (A, B) is not ,k-regular. Clearlyforevery ,k > 0, the pair (A,,kB) is regularand if (A, B) is ,k-regular, then there exists M >_ 1, independent of,k _{such that for all y} 6 G,

(2.4) IIA(A

₊

,kn)-Yll

<

_MllYll

Choose y

y/)

(yt))kN

with

yl)

0 for k

:/:

368 S. BU, P.

CLIMENT

AND S.

GUERRE-DELABRIRE

Hence

withL

_#-,

from(2.4)weobtain

(2.5) MII(A!

+/z)xzll

_IIAzxzll

lll(Ai

+/z)xzll

forevery E

N,

acontradiction since (At

+/t)xt

=

0.

Itremains to constructtheoperators

At

and/t.

For

thispurpose, weshall need thefollowinglemma,whichcanbe essentially found in[BC].

LEMMA

2.5.

Let

HbeacomplexseparableHilbert spacewith aSchauderbasis

e*

(e,,),,rand let

,,),,r

be the correspondingcoordinate

_functionals.

Let (Cn)nrbe anondecreasing sequence ofpositive real numbers and let

Ct

bethelinearoperators

defined

by (2.6)

where

_Nt

N

_for

all k

N.

Ctx

Z

_cte

(x)et

/=0

Then theoperators

Ct

arebounded positive operators

_of

type (0,

Mk

satisfying property(P). Moreover,0 p(Ct)

_for

all k

N

andsuptr

I[C-

< o.

In

view ofthis lemma, if(a,,),,r and (b,,),,r are twonondecreasing sequences

ofpositive numbersand

At,

/t

aredefinedby (2.6) where (Nt)tr is an arbitrary

sequenceof natural numbers, then theoperators

At,/k

satisfy allrequiredproperties

except

(2.3). In

ordertosatisfythis condition, wechoose for(en),r aconditional basis of

e2

asin[BC]andwechoose for(a,,),,r, (b,,),,Nthesequencesdenotedby

f(n)

andg(n) in[BC], havingthe property that

sup (x)et

xGo, Ilxll--! k=0ak

+

bt

ek

where

Go

span{e,, n 6 N}. Itfollows that forevery E

N,

there exists

_Nt

6

N

andott,t 6 Cfor0 <k < such that

t=oak

+

bt

et

>l

where

y(t)

-,’=o

ott,tet,0 <

Ilya)ll

1. Setting

U _(X)em

Atx

y

=0

am

em

k

x

Y,n=O

N,

bm

e*(X)em

weobtain

REGULARITY OF PAIRS OF POSITIVE OPERATORS 369

orequivalently

wheret/) (At

+/t)

-

yt)

_

0. Setting

X!)

1)

I111

weobtain(2.3). Thisconcludes the construction ofExample2.2. U]

Remark7. Inthisconstruction, wecanobtainaboundedoperator

A’

by defining A

k-vkAkwithv,

>0, kEN

inordertoensurethat

_A,

1. Then,similararguments show that the pair

(A!,

B)

doesnotsatisfy (1.1)zalthoughit satisfies(1.1).

Itfollows from Theorem 2.1 that 0 p(A’)U p(B). Henceone cannot assert as inExample2.2 that the pair(A

!,

B)isregular.

REFERENCES

[BC] J. B.Baillonand Ph.CI6ment, Examplesofunbounded imaginary powersofoperators,J.Funct.

Anal.100, (1991), 419-434.

[CD] Ph. Cl6ment andS. Guerre-Delabribre, On the regularityofabstract Cauchy problems_oforderone andboundaryvalueproblems_ofordertwo,Publ. Math. deUniv.P.etM.Curie119 (1997), 1-44.

[DG] G. Da Prato et P. Grisvard, Sommes d’oprateurs linaires et quations dlTrentielles

oprationneiles,J.Math.PuresetAppl.54 (1975), 305-387.

[DV] G.DoreandA.Venni,On the closedness_ofthesumoftwoclosedoperators,Math.Zeitschr.1911 (1987), 189-201.

[G] P.Grisvard,Equationsdffrentiellesabstraites,Ann.Sci.lcoleNorm. Sup.(1969), 311-395.

[K] H. Komatsu,Fractionalpowersofoperators, PacificJ.Math. 19 (1966), 285-346.

[L] Ch.Lemerdy,CounterexamplesonLt’-maximalregularity,preprint.

IS] P.E.Sobolevski, Fractionalpowersofcoercively positivesums_ofoperators,Dokl.Akad. Nauk

SSSR225 (1975), 6, 1638-1641.

[Tan] H.Tanabe, Equationsofevolution,Pitman,SanFrancisco,1979.

[Tr] H.Triebel,Interpolation theory,functionspaces,dfferentialoperators,North Holland, 1978.

Shangquan Bu,

Department

of Mathematics, UniversityofTsinghua,Beijing 100084, China

Equiped’Analyse, Case186, Universit6 Paris6, 4 Place Jussieu, 75252 Paris, Cedex 05,France

sbu @ math.tsinghua.edu.cn

Philippe Cl6ment,

Department

ofTechnical Mathematicsand Informatics, Delft

Uni-versityofTechnology,

E

O.Box5031, 2600GADelft, The Netherlands clement @ twi.tudelft.nl

370 S.BU,P.

CLIMENT

AND S.

GUERRE-DELABRIRE

Sylvie Guerre-Delabrire, Equiped’Analyse, Case 186, Universit6 Paris6, 4 Place Jussieu,75252Paris,Cedex05,France