Standardization of technology for extraction of wild apricot

14 Pages·2010·0.11 MB·English

J.Aust. Math. Soc. 80(2006),383–396
ELEMENTS OF RINGS AND BANACH ALGEBRAS WITH
RELATED SPECTRAL IDEMPOTENTS
N.CASTRO-GONZA´LEZ andJ.Y.VE´LEZ-CERRADA
(Received3May2004;revised4February2005)
CommunicatedbyJ.Du
Abstract
Let a(cid:25) denote the spectral idempotent of a generalized Drazin invertible element a of a ring. We
characterizeelementsbsuchthat1−.b(cid:25) −a(cid:25)/2 isinvertible. Wealsoapplythisresultinringswith
involutiontoobtainacharacterizationoftheperturbationofEPelements. InBanachalgebrasweobtain
acharacterizationintermsofmatrixrepresentationsandderiveerrorboundsfortheperturbationofthe
Drazininverse. Thisworkextendsrecentresultsformatricesgivenbythesameauthorstothesettingof
ringsandBanachalgebras.Finally,wecharacterizegeneralizedDrazininvertibleoperatorsA;B2 .X/
suchthatpr.B(cid:25)/Dpr.A(cid:25)CS/,wherepristhenaturalhomomorphismof .X/ontotheCalkinaBlgebra
andS2 .X/isgiven. B
B
2000Mathematicssubjectclassification:primary16A32,16A28,15A09.
Keywordsandphrases: GeneralizedDrazininverse, spectralidempotent, EPelements, perturbation,
Fredholmoperators.
1. Introduction
The perturbations of the conventional and generalized Drazin inverse with equal
eigenprojections at zero have been studied by Castro, Koliha and Wei in [4, 5] for
matricesandclosedoperators,respectively,byRakocˇevic´ [7]forelementsinBanach
algebrasandbyKolihaandPatr´icio[15]forelementsofrings.
Recently, the authors [6] studied perturbations of the Drazin inverse of a square
complexmatrix Aforthecasewhentheperturbedmatrix Bhastheeigenprojectionat
zerosatisfying B(cid:25) D A(cid:25) CS,where Sisagivenmatrix. Theresultsobtainedtherein
canbeappliedtothecase S D 0inordertoobtainacharacterization ofmatrices for
which B(cid:25) D A(cid:25) and,so,recoverresultsof[4].
(cid:13)c 2006AustralianMathematicalSociety1446-8107/06$A2:00C0:00
383
384 N.Castro-Gonza´lezandJ.Y.Ve´lez-Cerrada [2]
Themainpurposeofthispaperistoinvestigatetheperturbationofthegeneralized
Drazin inverse for elements of rings and Banach algebras. In Section 3 we give a
characterization of elements a;b with spectral idempotents related by the condition
that1−.b(cid:25)−a(cid:25)/2isinvertible. Inparticular,ifb(cid:25) D a(cid:25)werecover[15,Theorem6.1].
InSection 4 weapplythis resultinringswithinvolution toobtainacharacterization
ofthe perturbation of EP elements, that is, elements whichhave Drazin andMoore-
Penroseinverseandbothcoincide.
In several recent articles perturbations of the Drazin inverse were studied with a
purpose to obtain explicit error bounds [3, 14, 16, 8, 17, 18]. We work in Banach
algebras in Section 5 and derive an upper bound of kbD − aDk=kaDk in terms of
kaD.b−a/kandkb(cid:25)−a(cid:25)k. Thisestimationissharperthantheerrorboundgivenin[14,
Theorem2.1],and[3,Theorem6.1]formatricesandclosedoperators,respectively.
Finally in Section 6, we give a characterization for bounded linear operators in
Banachspaces,basedonthemainresultofSection3,ofgeneralizedDrazininvertible
operators A;B 2 .X/ such that their eigenprojections at zero satisfy pr.B(cid:25)/ D
pr.A(cid:25)CS/,whereBpristhenaturalhomomorphismof .X/ontotheCalkinalgebra.
Inparticular,when S D0werecover[7,Corollary2.3B].
2. Preliminaries
Let be a ring with unit 1. The commutant and the double commutant of an
elemenRta 2 aredefinedby
R
comm.a/Dfx 2 Vax D xag;
R
comm2.a/Dfx 2 V xy D yx forall y 2comm.a/g:
R
We denote by −1 the group of invertible elements of . An element a 2
is quasinilpotent Rif 1 C xa 2 −1 for every x 2 comm.Ra/ [10]. The set of aRll
R
quasinilpotentelementsof willbedenotedby qnil.
Anelementa 2 issaidRtohaveageneralizedRDrazininverseifthereexistsx 2
R R
suchthat
(2.1) x 2comm2.a/; x Dax2; a−a2x 2 qnil:
R
If a has a generalized Drazin inverse, then the generalized Drazin inverse of a is
uniqueandisdenotedbyaD. Ifinthepreviousdefinitiona−a2x isinfactnilpotent
(whichisequivalenttoakC1x Dak wherek istheindexofnilpotency),thenaD isthe
conventional Drazin inverse of a and ind.a/ D k where ind.a/ is the Drazin index
of a [9]. Moreover, in this case the condition x 2 comm2.a/ can be replaced with
x 2 comm.a/. Whenind.a/ D 1theDrazininverse ofa iscalledthegroupinverse,
[3] ElementsofringsandBanachalgebras 385
andisdenotedbyaD D a#. By gD, # wedenotethesetofallgeneralized Drazin
R R
invertibleandgroupinvertibleelementsof ,respectively.
R
TheDrazininverseinringswasoriginallydefinedbyDrazin[9]forpolarelements
andgeneralizedbyHarte[10]toquasipolarelements. ThegeneralizedDrazininverse
was studied by Koliha [12] in Banach algebras and by Koliha and Patr´icio [15] in
rings.
THEOREM2.1. Anelementa 2 hasageneralizedDrazininverseaD ifandonly
ifthereexistsanidempotent p 2 Rsuchthat
R
(2.2) p 2comm2.a/; ap 2 qnil; aC p 2 −1:
R R
There is at most one idempotent p such that conditions (2.2) hold. The unique
idempotent piscalledaspectralidempotentofa anditisdenotedbya(cid:25). Itisknown
that
(2.3) aD D.aCa(cid:25)/−1.1−a(cid:25)/D.1−a(cid:25)/.aCa(cid:25)/−1 and a(cid:25) D1−aaD:
Now, we state two lemmas which will be needed in the next section. We recall
that two idempotents elements p and p are called complementary provided that
1 2
p C p D1.
1 2
LEMMA 2.2. Let p1, p2 be nonzero complementary idempotents in a ring . If
z 2 commuteswith p and1C p z 2 −1,then R
2 2
R R
zC p 2 −1 ifandonlyif p zC p 2 −1:
2 1 2
R R
PROOF. Sincez 2 commuteswithp2,itfollowsthat.p1zCp2/.1Cp2z/D zCp2.
Hence,theequivalencReisverifiedbecause1C p z 2 −1.
2
R
Let D.p ; p /beasystemofnonzerocomplementaryidempotentselementsin
1 2
P
aring . Wedefinetheset
R
(cid:26)(cid:18) (cid:19) (cid:27)
x x
. ; /D 11 12 V x 2 p p ; forall i; j 2f1;2g :
M2 R P x21 x22 ij iR j
(cid:0) (cid:1)
Theset . ; /isaringwiththeusualmatrixoperations,withtheunit p1 0
andthemaMpp2inRg(cid:30)PV ! . ; /definedby 0 p2
2
R M R P
(cid:18) (cid:19)
p ap p ap
(cid:30).a/D 1 1 1 2
p ap p ap
2 1 2 2
isaringisomorphism.
386 N.Castro-Gonza´lezandJ.Y.Ve´lez-Cerrada [4]
LEMMA 2.3. Let p1, p2 be nonzero complementary idempotents in a ring , and
letz 2 . If p zp D0and p zp 2.p p/−1,i D1;2,thenz 2 −1. R
2 1 i i i i
R R R
PROOF. Since p2zp1 D0,usingtheisomorphism(cid:30) V ! 2. ; /weobtain
R M R P
(cid:18) (cid:19)
p zp p zp
(cid:30).z/D 1 1 1 2 :
0 p zp
2 2
Since p zp 2 .p p/−1, i D 1;2, it follows that the matrix (cid:30).z/ is invertible in
i i i i
. ; /and,coRnsequently,z isinvertiblein .
2
M R P R
Thefollowingsetswillappearinthemaintheoremofthenextsection. Ifa 2 ,
R
wedefine
a Dfax V x 2 g; a D fxa V x 2 g;
R R R R
a0 Dfy 2 Vay D0g; 0a D fy 2 V ya D0g:
R R
IfM (cid:26) ,wedefineM Dfmx Vm 2 M; x 2 gandM0 Dfx 2 V Mx Df0gg.
R R R R
Thesets M and0M canbedefinedsimilarly.
R
These sets have the following properties [11, Proposition 6]. We recall that an
elementa 2 isregularifthereexists x 2 suchthataxa D a,thatis,ifithasan
R R
innerinverse x.
PROPOSITION2.4. Leta;b 2 beregularelementsand A;B (cid:26) . Then
R R
(i) a0 D. a/0.
(ii) a DR0.. a/0/D0.a0/.
(iii) RA (cid:26) B imRplies0A (cid:27)0B.
3. Characterizationofelementsofringswithrelatedspectralidempotents
In this section we give a characterization of elements a;b 2 with spectral
idempotents satisfying b(cid:25) D a(cid:25) C s, where s is a given elementRof such that
1−s2 2 −1. Weremarkthat,foranys 2 qnilwehave1−s2 2 −1. TRhisresearch
R R R
extendsarecentworkoftheauthorsformatrices[6]tothesettingofrings.
First we note the following properties involving the spectral idempotent a(cid:25) and
idempotentelementsoftheforma(cid:25) Cs.
PROPOSITION 3.1. Let a 2 gD and s 2 such that 1−s2 2 −1. If a(cid:25) Cs is
R R R
idempotent,then
(i) a(cid:25) Cs D.1−s/−1a(cid:25).1Cs/ D.1Cs/a(cid:25).1−s/−1;
[5] ElementsofringsandBanachalgebras 387
(ii) 1−a(cid:25) −s D.1−s/.1−a(cid:25)/.1Cs/−1 D.1Cs/−1.1−a(cid:25)/.1−s/;
(iii) if r D .1 C s/a(cid:25) C .1 − s/.1 − a(cid:25)/, then r 2 −1, r−1 D a(cid:25).1 − s/−1
C.1−a(cid:25)/.1Cs/−1 anda(cid:25) Cs Dra(cid:25)r−1. R
PROOF. Since.a(cid:25) Cs/2 Da(cid:25) Cs,wehavea(cid:25) Ca(cid:25)sCsa(cid:25) Cs2 Da(cid:25) Cs. Hence
a(cid:25).1Cs/ D a(cid:25) Cs −sa(cid:25) −s2 D .1−s/.a(cid:25) Cs/ and thus the first equality of (i)
holds. Thesecondequalityof(i)andbothequalitiesof(ii)canbededucedinasimilar
way.
Now,writer D.1Cs/a(cid:25)C.1−s/.1−a(cid:25)/andt Da(cid:25).1−s/−1C.1−a(cid:25)/.1Cs/−1.
Applyingproperties(i)and(ii)wecheckthat
rt D.1Cs/a(cid:25).1−s/−1C.1−s/.1−a(cid:25)/.1Cs/−1
D.a(cid:25) Cs/C.1−a(cid:25) −s/ D1:
Analogously we can check that tr D 1. Consequently, t D r−1. Onthe other hand,
ra(cid:25)r−1 D .1 C s/a(cid:25).1− s/−1 D a(cid:25) C s, where we apply property (i) in the last
equality.
REMARK. Condition .a(cid:25) Cs/2 D a(cid:25) Cs yields that s2.1−a(cid:25)/ D .1−a(cid:25)/s2 D
.1−a(cid:25)/s.1−a(cid:25)/ands2a(cid:25) D a(cid:25)s2 D−a(cid:25)sa(cid:25).
Wearenowinapositiontostatethemainresultofthispaper.
THEOREM3.2. Leta 2 gD andlets 2 besuchthat1−s2 2 −1. Ifa(cid:25) Cs is
idempotent,thenthefollowRingconditionsoRnb 2 areequivalent:R
R
(i) b 2 gD,andb(cid:25) Da(cid:25) Cs.
(ii) a(cid:25) CRs 2comm2.b/,.a(cid:25) Cs/b 2 qnil,andbCa(cid:25) Cs 2 −1.
(iii) a(cid:25) Cs 2comm2.b/,.a(cid:25) Cs/b 2Rqnil,and1Cs CaD.b−Ra/2 −1.
(iv) b 2 gD,1CsCaD.b−a/2 −1R,andbD D.1CsCaD.b−a//−1RaD.1−s/.
(v) b 2RgD,and.1Cs/bD−aD.R1−s/ DaD.a−b/bD.
(vi) b 2RgD,a(cid:25) Cs 2comm.b/,and1−.b(cid:25) −a(cid:25) −s/2 2 −1.
(vii) bD.1RCs/ (cid:26) .1−s/aD ,and.bD.1Cs//0 (cid:26)..1−sR/aD/0.
R R
PROOF. (i)ifandonlyif(ii): ThisequivalencefollowsbyTheorem2.1.
(ii)ifandonlyif(iii): Letr D.1Cs/a(cid:25) C.1−s/.1−a(cid:25)/. ByProposition3.1(iii),
we have that r 2 −1 and a(cid:25) Cs D ra(cid:25)r−1. Then the conditions (ii) and (iii) are
R
equivalentto
(ii)0 a(cid:25) 2comm2.r−1br/,a(cid:25)r−1br 2 qnil,andr−1br Ca(cid:25) 2 −1.
(iii)0 a(cid:25) 2comm2.r−1br/,a(cid:25)r−1br 2Rqnil,andra(cid:25) CaDbr 2R−1.
R R
388 N.Castro-Gonza´lezandJ.Y.Ve´lez-Cerrada [6]
Wewillshowthatundertheassumptionsa(cid:25) 2comm2.r−1br/anda(cid:25)r−1br 2 qnil,
R
(3.1) r−1br Ca(cid:25) 2 −1 ifandonlyif ra(cid:25) CaDbr 2 −1:
R R
Assumefirstthatr−1br Ca(cid:25) 2 −1. Letz Dra(cid:25) CaDbr. Thenwecanwrite
R
z Da(cid:25)ra(cid:25) CaDr.r−1br Ca(cid:25)/.1−a(cid:25)/C.1−a(cid:25)/.r CaDbr/a(cid:25):
We prove that z is invertible. For this, first we observe that p D 1 − a(cid:25) and
1
p D a(cid:25) are commuting complementary idempotents; we may assume that they are
2
nonzero. Recall that p , p commute with a,r−1br,and1−s2. Wecanverify bya
1 2
directcalculationthat
p zp D0; p zp D p .aC p /−1p r.r−1br C p /p ; and p zp D p rp :
2 1 1 1 1 2 1 2 1 2 2 2 2
Inthering p p withunit p ,wecheckthat
1 1 1
R
(cid:0) (cid:1)
p .aC p /−1p r.r−1br C p /p −1 D p .r−1br C p /−1.1−s2/−1.aC p /−1p ;
1 2 1 2 1 1 2 2 1
and in the ring p p with unit p , we check that .p rp /−1 D p .1 − s2/−1p .
2 2 2 2 2 2 2
R
AccordingtoLemma2.3,z isinvertible.
Conversely,assumethatrp CaDbr 2 −1. First,wewillprovethat
2
R
u D.aDC p /.1−s2/.p r−1br C p /
2 1 2
isinvertible, whichwillimply that p r−1br C p 2 −1. Since1C p r−1br 2 −1
1 2 2
because p r−1br 2 qnil,Lemma2.2ensuresthatr−R1br C p 2 −1. R
2 2
Weverifythat pRup D0andbyadirectcalculation,whichtaRkesintoaccountthe
2 1
commutative relations mentioned above, weobtain that p up D p .rp CaDbr/p
1 1 1 2 1
and p up D p .1−s2/p . Further,
2 2 2 2
.p .rp CaDbr/p /−1 D p .rp CaDbr/−1p ;
1 2 1 1 2 1
.p .1−s2/p /−1 D p .1−s2/−1p ;
2 2 2 2
in the rings p p , p p , respectively. By Lemma 2.3, u is invertible in , and
1 1 2 2
consequentlyrR−1br C pR2 −1. R
2
R
(iii)implies(iv): Sincecondition(i)istrueifandonlyif(ii)istrue,andcondition(ii)
istrueifandonlyif(iii)istrue,wehaveb(cid:25) Da(cid:25) Cs and1CsCaD.b−a/ 2 −1.
R
Byequation(2.3),
.1Cs CaD.b−a//bD D .a(cid:25) Cs CaDb/.1−.a(cid:25) Cs//.bCa(cid:25) Cs/−1
D aDb.1−a(cid:25) −s/.bCa(cid:25) Cs/−1
D aD.1−s/:
[7] ElementsofringsandBanachalgebras 389
Therefore(iv)holds.
(iv) implies (v): From bD D .1 C s C aD.b − a//−1aD.1 − s/ it follows that
.1Cs/bDCaD.b−a/bD DaD.1−s/andthus(v)holds.
(v)implies(i): Firstmultiplytheequalitygivenin(v)bya(cid:25) fromtheleftandthen
multiplythesameequalitybyb(cid:25) fromtherighttoget
a(cid:25).1Cs/bD D0 and aD.1−s/b(cid:25) D0:
Thenwecanwritea(cid:25).1Cs/bDb D aaD.1−s/b(cid:25). Hence,
a(cid:25).1Cs/ D.a(cid:25).1Cs/C.1−a(cid:25)/.1−s//b(cid:25)
D.a(cid:25).1−s/−1C.1−a(cid:25)/.1Cs/−1/.1−s2/b(cid:25):
AccordingtoProposition3.1(iii)and(i),
b(cid:25) D.1−s2/−1..1Cs/a(cid:25) C.1−s/.1−a(cid:25)//a(cid:25).1Cs/
D.1−s/−1a(cid:25).1Cs/Da(cid:25) Cs:
(i) if and only if (vi): The right implication is obvious. We will check the left
implication. From1−.b(cid:25) −a(cid:25) −s/2 2 −1 itfollowsthat1−b(cid:25) Ca(cid:25) Cs 2 −1
and 1Cb(cid:25) −a(cid:25) −s 2 −1. Since a(cid:25) RCs 2 comm.b/ and bb(cid:25) D b(cid:25)b, we hRave
.a(cid:25) C s/b(cid:25) D b(cid:25).a(cid:25) CRs/. Then .1 − a(cid:25) − s C b(cid:25)/.a(cid:25) C s/.1 − b(cid:25)/ D 0 and
consequently.a(cid:25) Cs/.1−b(cid:25)/D0:Hence,a(cid:25) Cs D.a(cid:25) Cs/b(cid:25). Ontheotherhand,
wehave.1−b(cid:25)Ca(cid:25)Cs/b(cid:25).1−a(cid:25)−s/ D0andsob(cid:25).1−a(cid:25)−s/D0. Therefore,
b(cid:25) Db(cid:25).a(cid:25) Cs/ Da(cid:25) Cs.
(i)implies(vii): AsbDb D1−b(cid:25) D1−a(cid:25)−s,wehavebbD D .1−s/aaD.1Cs/−1
byProposition3.1(ii). Then
bD.1Cs/ D bbD.1Cs/ D.1−s/aaD D.1−s/aD :
R R R R
Similarly, bD.1Cs/D .1−s/aD.
SincebDR.1Cs/and.1R−s/aD areregularelements,byProposition2.4,
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
bD.1Cs/ 0 D bD.1Cs/ 0 D .1−s/aD 0 D .1−s/aD 0:
R R
(cid:0) (cid:1) (cid:0) (cid:1)
(vii)implies(i): As bD.1Cs/ 0 (cid:26) .1−s/aD 0,byProposition2.4,
bD.1Cs/ D0..bD.1Cs//0/(cid:27)0...1−s/aD/0/D .1−s/aD:
R R
TheaboveinclusionimpliestheconsistencyoftheequationxbbD.1Cs/ D.1−s/aDa.
ThenxbbD D.1−s/aDa.1Cs/−1 DaaD−sbyProposition3.1(ii). Thislastequation
isequivalentto.aaD−s/.1−bbD/D0. Thus
(3.2) aaD−s D.aaD−s/bbD:
390 N.Castro-Gonza´lezandJ.Y.Ve´lez-Cerrada [8]
FrombD.1Cs/ (cid:26).1−s/aD theconsistencyoftheequation
R R
bbD.1Cs/ D.1−s/aaDy
follows, which in turn is equivalent to .1 − aaD/.1 − s/−1bbD D 0. Hence, by
Proposition 3.1 (i), .1Cs/−1.1−aaD −s/bDb D 0. Then, bbD D .aaD −s/bDb.
ThusbbD DaaD−s inviewof(3.2),and(i)holds.
Specializing the equivalence of conditions (i)–(vi) to matrices we get the equiva-
lenceofconditions(a)–(b)and(d)–(f)givenin[6,Theorem2.3].
Whenweconsiderthecases D0intheprecedingtheoremwegetacharacterization
ofelementsof withequalspectralidempotents. So,werecover[15,Theorem6.1].
R
REMARK. Ifwedonotassumecondition1−s2 2 −1,thenwecanfindelements
a;b 2 suchthatb(cid:25) D a(cid:25) Cs,but1Cs CaD.b−Ra/isnotaninvertible element.
WeconRsidertherealmatrices4(cid:2)4,
0 1 0 1 0 1
−1=2 1=2 0 0 1 1 0 0 2 0 0 0
s DBB@ 1=2 −1=2 0 0CCA; a DBB@1 1 0 0CCA; b DBB@0 2 0 0CCA:
0 0 0 0 0 0 0 0 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0
Then
0 1 0 1
1=2 −1=2 0 0 0 0 0 0
a(cid:25) DBB@−1=2 1=2 0 0CCA; b(cid:25) DBB@0 0 0 0CCA
0 0 1 0 0 0 1 0
0 0 0 1 0 0 0 1
and
0 1
1=2 1=2 0 0
1Cs CaD.b−a/DBB@1=2 1=2 0 0CCA:
0 0 1 0
0 0 0 1
So,b(cid:25) Da(cid:25) Cs and1Cs CaD.b−a/isnotinvertible.
4. PerturbationofEPelementsinringswithinvolution
Let bearingwithinvolution. Recallthatanelementx 2 istheMoore-Penrose
inverseRofa 2 providedthat R
R
(4.1) xax D x; axa Da; .ax/(cid:3) Dax; .xa/(cid:3) D xa:
[9] ElementsofringsandBanachalgebras 391
Thereisatmostonex suchthatconditions(4.1)hold. Theuniquex isdenotedbya†.
ThesetofallMoore-Penroseinvertibleelementsof willbedenotedby †.
An element a is said to be EP if a 2 gD \ †Rand aD D a†. It is wRell-known
that a 2 is EP if and only if aa† D a†Ra. NecRessary and sufficient conditions for
anelemenRtofa (cid:3)-algebrastocommutewithitsMoore-Penroseinversearegivenin
C
[13].
In [15, Theorem 7.2 (i)–(ii)] it was established that a 2 is EP if and only if
a 2 # and a(cid:25) D .a(cid:3)(cid:25)/. We will use this characterization anRd our main theorem to
giveRequivalentconditionsensuringthatifa isEPthenanelementb DaCeisagain
EPwithspectralidempotentb(cid:25) Da(cid:25) Cs wheres isgiven.
THEOREM 4.1. Let be a ring with involution and let s 2 be such that
1−s2 2 −1. If a is REP and a(cid:25) Cs is idempotent, then the folloRwing conditions
onb DaRCe 2 areequivalent:
R
(i) bisEPandb(cid:25) Da(cid:25) Cs.
(ii) s(cid:3) Ds,e.a#a−s/−as De D.a#a−s/e−sa,and1CsCaCe−a#a 2 −1.
(iii) s(cid:3) Ds,e.a#a−s/−as De D .a#a−s/e−sa,and1Cs Ca#e 2 −R1.
(iv) b 2 gD\ †,andb† DbD D.1Cs Ca†e/−1a†.1−s/. R
(v) b 2RgD\R†,b† DbD,and.1Cs/b†−a†.1−s/ D−a†eb†.
R R
PROOF. Since a is EP, we have a 2 # and .a(cid:3)/(cid:25) D .a(cid:25)/(cid:3) D a(cid:25). Now, if
b(cid:25) D a(cid:25) Cs then .b(cid:3)/(cid:25) D .b(cid:25)/(cid:3) D .a(cid:25)/(cid:3)RCs(cid:3) D a(cid:25) Cs(cid:3). Hence, condition (i) is
equivalentto
(i)0 b 2 #,s isselfadjoint,andb(cid:25) Da(cid:25) Cs.
R
Using a(cid:25) D 1−aa# and aa(cid:25) D 0, conditions (ii) and (iii) can be written in the
followingequivalentform
(ii)0 s(cid:3) Ds,.aCe/.a(cid:25) Cs/D 0D.a(cid:25) Cs/.aCe/,anda(cid:25) Cs CaCe 2 −1.
(iii)0 s(cid:3) Ds,.aCe/.a(cid:25) Cs/D 0D.a(cid:25) Cs/.aCe/,and1Cs Ca#e 2 −R1.
R
Applying Theorem 3.2 we conclude the equivalence of conditions (i)–(v) of the
theorem.
Given s 2 , we obtain the following characterization of elements a 2 such
thatthespectraRlidempotentsofa anda(cid:3) satisfy.a(cid:3)/(cid:25) Da(cid:25) Cs. R
THEOREM 4.2. Let be a ring with involution and let s 2 be such that
1−s2 2 −1. ThentheRfollowingconditionsona 2 areequivalenRt:
R R
(i) a(cid:3) 2 #,and.a(cid:3)/(cid:25) Da(cid:25) Cs.
(ii) a(cid:3) 2RgD,.a(cid:3)/#a(cid:3) DaaD−s,and.1Cs/a(cid:3) DaDaa(cid:3).
(iii) a(cid:3).aDRa −s/ D a(cid:3) D .aDa −s/a(cid:3), 1Cs Ca(cid:3) −aDa 2 −1, and s2 Cs D
saaDCaaDs. R
392 N.Castro-Gonza´lezandJ.Y.Ve´lez-Cerrada [10]
(iv) s2Cs DsaaDCaaDs,1CsCaD.a(cid:3)−a/ 2 −1,anda(cid:3).aDa−s/ Da(cid:3) D
.aDa−s/a(cid:3). R
PROOF. From a(cid:25) D 1 − aDa we easily see that the conditions (ii)–(iv) can be
formulatedinthefollowingequivalentform
(ii)0 .a(cid:3)/(cid:25) D a(cid:25) Cs,anda(cid:3).a(cid:25) Cs/ D0.
(iii)0 .a(cid:25) Cs/2 Da(cid:25) Cs,a(cid:3).a(cid:25) Cs/ D.a(cid:25) Cs/a(cid:3) D0,anda(cid:3)Ca(cid:25) Cs 2 −1.
(iv)0 .a(cid:25)Cs/2 Da(cid:25)Cs,a(cid:3).a(cid:25)Cs/D .a(cid:25)Cs/a(cid:3) D0,and1CsCaD.a(cid:3)−a/2R −1.
R
Now,weapplyTheorem3.2(i)–(iii)toa(cid:3) inplaceofb.
Ifwechooses D0intheprecedingtheorem,thencondition(i)isequivalenttothe
factthata isEP.So,inthiscase,weobtainacharacterizationofEPelementsinrings
withinvolution. Inparticular,formatriceswerecover[4,Theorem5.2].
5. PerturbationofDrazininvertibleelementsinBanachalgebras
Let be a complex Banach algebra with unit 1. The generalized Drazin inverse
B
forelements ofBanachalgebras is definedasin(2.1)wherethecondition ofdouble
commutativity x 2 comm2.a/canbereplacedby x 2 comm.a/. Let gD denotethe
B
setofallelementsin whichhaveageneralizedDrazininverse.
B
The equivalence of conditions (i)–(vii) in Theorem 3.2 is also valid in Banach
algebras. Moreover,in(ii)and(iii),thecondition a(cid:25) Cs 2comm2.b/canberelaxed
toa(cid:25) Cs 2comm.b/.
In this section, first we will give a new characterization of elements a;b 2 gD
such that 1−.b(cid:25) −a(cid:25)/2 is invertible in terms of its matrix representation. SecBond,
wewillapplyTheorem3.2togetanormestimationoftheperturbationoftheDrazin
inverse.
Leta 2 gD. Weconsiderasystemofcomplementaryidempotents D.p ; p /,
1 2
where p1 DB1− a(cid:25) and p2 D a(cid:25). The set of matrices 2. ; /,Pdefine(cid:0)d as i(cid:1)n
M B P
Section2,isaBanachalgebrawiththeusualmatrixoperations,withtheunit p1 0 .
0 p2
Themapping(cid:30) V ! . ; /definedby
2
B M B P
(cid:18) (cid:19)
p ap p ap
(cid:30).a/D 1 1 1 2
p ap p ap
2 1 2 2
is an isometric Banach algebra isomorphism [2]. This fact will allow us to work
with matrix representations of elements in by identifying a 2 with its image
(cid:30).a/ 2 . ; /. For brevity, we denoteB D p p which iBs an algebra with
2 i i i
unit p,iMD1B;2.P B B
i