The minus order and range additivity

0.25 MB·

by  Marko Djikic

The minus order and range additivity
MarkoS.Djikic´a,1,GuillerminaFongib,2,AlejandraMaestripierib,c,3,∗
aDepartmentofMathematics,FacultyofSciencesandMathematics,UniversityofNiš,Višegradska33,18000Niš,
Serbia.
bInstitutoArgentinodeMatemática“AlbertoP.Calderón"Saavedra15,Piso3(1083),BuenosAires,Argentina.
cDepartamentodeMatemática,FacultaddeIngeniería,UniversidaddeBuenosAires.
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n
a Abstract
J
WestudytheminusorderonthealgebraofboundedlinearoperatorsonaHilbertspace. By
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givingacharacterizationintermsofrangeadditivity,weshowthattheintrinsicnatureofthe

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A minusorderisalgebraic. Applicationstogeneralizedinversesofthesumoftwooperators,to
F systemsofoperatorequationsandtooptimizationproblemsarealsopresented.
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Keywords: minusorder,rangeadditivity,generalizedinverses,leastsquaresproblems
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a 2000MSC:06A06,47A05
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[

1 1. Introduction
v
6 TheminusorderwasintroducedbyHartwig[25]andindependentlybyNambooripad[32],
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4 inbothcasesonsemigroups,withtheideaofgeneralizingsomeclassicalpartialorders.Itwas
0 extendedtooperatorsininfinitedimensionalspacesindependentlybyAntezana,Corachand
0
Stojanoff[2]andbySˇemrl[36]. Thereisnowanextensiveliteraturedevotedtothisorderand
.
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other related partial orders on matrices, operators and elements of various algebraic struc-
0
7 tures. Seeforexample,[8,30,31].
1
Themaingoalofthisworkistoobtainanewcharacterizationoftheminusorderforop-
:
v
eratorsactingonHilbertspacesintermsofthesocalledrangeadditivityproperty. Giventwo
i
X linearboundedoperatorsA and B actingonaHilbertspace(cid:72),wesaythatA and B havethe
r rangeadditivitypropertyifR(A+B)=R(A)+R(B),whereR(T)standsfortherangeofanop-
a
eratorT. Operatorswiththispropertyhavebeenstudiedin[4]and[5](seealso[10]). Recall
thatifA andB aretwoboundedlinearHilbertspaceoperatorsthenA≤− B (wherethesymbol
“≤−”standsfortheminusorderofoperators)ifandonlyifthereareobliqueprojectionsP and
Q suchthatA =PB andA∗=QB∗. Inthispaper,weprovethatthisisequivalenttotherange
∗Correspondingauthor
Emailaddresses:[email protected](MarkoS.Djikic´),[email protected]
(GuillerminaFongi),[email protected](AlejandraMaestripieri)
1SupportedbyGrantNo.174007oftheMinistryofEducation,ScienceandTechnologicalDevelopmentofthe
RepublicofSerbia
2PartiallysupportedbyCONICET(PIP4262013-2015)
3PartiallysupportedbyCONICET(PIP1682014-2016)
PreprintsubmittedtoElsevier January3,2017
ofB beingthedirectsumofrangesofA andB−A andtherangeofB∗beingthedirectsumof
rangesofA∗ and B∗−A∗. Thustheminusorderisintrinsicallyalgebraicinnature. Thisplays
anequivalentroletoaknowncharacterizationwhenA andB arematrices[25,31];thatA≤− B
ifandonlyiftherankof B−A isthedifferenceoftherankof B andtherankofA.
As a consequence, diverse concepts that have been developed for matrices and opera-
tors are in fact manifestations of the minus order. These include, for example, the notions
ofweaklybicomplementarymatricesdefinedduetoWerner[37],andquasidirectadditionof
operatorsdefinedbyLešnjakandŠemrl[28]. Althoughinthesepaperstheminusorderdoes
notappearexplicitly,thesenotionswhenappliedtooperatorsA and B areequivalenttosay-
ing that A ≤− A+B. The minus order also lurks in the papers of Baksalary and Trenkler [9],
Baksalary,SˇemrlandStyan[7],Mitra[29]andArias,CorachandMaestripieri[5].
The minus order can be weakened to what we call left and right minus orders. As with
theminusorder,theseordersareeasilyderivedfromarangeadditivitycondition. Ithappens
that they truly differ from the minus order only in the infinite dimensional setting. When
A≤− B,wegivesomeapplicationstoformulasforgeneralizedinversesofsumsA+B interms
ofgeneralizedinversesofAandB,andweshowthatcertainoptimizationproblemsinvolving
theoperatorA+B canbedecoupledintoasystemofsimilarproblemsforA and B.
Thepaperisorganizedasfollows.InSection2wecollectsomeusefulknownresultsabout
rangeadditivity,whileinSection3,theminusorderisdefinedandtheconnectionwithrange
additivity is made. Motivated by the concepts of the left and the right star orders, we define
leftandtherightminusorderson L((cid:72)). Formatrices,theseareequivalenttotheminusor-
der,withdifferencesonlyemergeintheinfinitedimensionalcontext. Proposition3.13char-
acterizes the left minus order in terms of densely defined, though not necessarily bounded,
projections. Additionally,theleftminus,therightminusandtheminusordersarecharacter-
ized in terms of (densely defined) inner generalized inverses, generalizing a matricial result
(see[30]).
Finally,Section4isdevotedtoapplications.Webeginbyrelatingtheminuspartialorderto
someformulasforreflexiveinnerinversesofthesumoftwooperators. Inparticular,wegive
analternativeprooffortheFill-FishkindformulafortheMoore-Penroseinverseofasum,as
found in [21] for matrices and extended to L((cid:72)) by Arias et al. [5]. We also apply the new
characterizationoftheminusordertosystemsofequationsandleastsquaresproblems. We
include a final remark about a possible generalization of the minus order involving densely
definedprojectionswithclosedrange.
2. Preliminaries
Throughout, ((cid:72),〈·,·〉) denotes a complex Hilbert space and L((cid:72)) the algebra of linear
boundedoperatorson(cid:72),(cid:81)isthesubsetofL((cid:72))ofobliqueprojections,i.e.,(cid:81)={Q∈L((cid:72)):
Q2=Q}and(cid:80) thesubsetof(cid:81) oforthogonalprojections,i.e.,(cid:80) ={P∈L((cid:72)):P2=P=P∗}.
Given(cid:77) and(cid:78) twoclosedsubspacesof(cid:72),write(cid:77) +˙ (cid:78) forthedirectsumof(cid:77) and
(cid:78) ,(cid:77) ⊕(cid:78) theorthogonalsumand(cid:77) (cid:9)(cid:78) =(cid:77) ∩((cid:77) ∩(cid:78) )⊥. If(cid:77) +˙ (cid:78) =(cid:72),theoblique
projection with range (cid:77) and null space (cid:78) is P(cid:77)//(cid:78) and P(cid:77) = P(cid:77)//(cid:77)⊥ is the orthogonal
projectiononto(cid:77).
2
ForA ∈L((cid:72)),R(A)standsfortherangeofA,N(A)foritsnullspaceandP forP . The
A R(A)
Moore-PenroseinverseofA isthe(denselydefined)operatorA†:R(A)⊕R(A)⊥→(cid:72),defined
byA†|R(A)=(A|N(A)⊥)−1 andN(A†)=R(A)⊥. ItholdsthatA†∈L((cid:72))ifandonlyifA hasaclosed
range.
Given (cid:77) and (cid:78) two closed subspaces of (cid:72), the minimal angle between (cid:77) and (cid:78) is
α ((cid:77),(cid:78) )∈[0,π/2],thecosineofwhichis
0
c ((cid:77),(cid:78) )=sup(cid:8)|(cid:10)ξ,η(cid:11)|: ξ∈(cid:77), (cid:107)ξ(cid:107)≤1, η∈(cid:78) , (cid:107)η(cid:107)≤1(cid:9) ∈[0,1].
0
Whentheminimalanglebetween(cid:77) and(cid:78) isstrictlylessthat1,thenthesum(cid:77) +(cid:78) is
closedanddirect,moreover,wehavethefollowing.
Proposition2.1. Let (cid:77) and (cid:78) be two closed subspaces of (cid:72). The following statements are
equivalent:
(1) c ((cid:77),(cid:78) )<1;
0
(2) (cid:77) +˙ (cid:78) isclosed;
(3) (cid:72) =(cid:77)⊥+(cid:78) ⊥.
Foraproof,seeLemma2.11andTheorem2.12in[15].
ForA,B ∈L((cid:72)),italwaysholdsthatR(A+B)⊆R(A)+R(B). WesaythatA andB havethe
rangeadditivitypropertyifR(A+B)=R(A)+R(B).Inthiscase,R(A)⊆R(A+B).Conversely,if
R(A)⊆R(A+B)then,forx ∈(cid:72),Bx =(A+B)x−Ax ∈R(A+B).Wehaveprovedthefollowing.
Lemma2.2([5,Proposition2.4]). ForA,B ∈L((cid:72)),R(A+B)=R(A)+R(B)ifandonlyifR(A)⊆
R(A+B).
Operators having the range additivity property were characterized in [5, Theorem 2.10].
CloselyrelatedisthefollowingforoperatorsA,B ∈L((cid:72))satisfyingtheconditionR(A)∩R(B)=
{0}.
Proposition2.3([5, Theorem 2.10]). Consider A,B ∈ L((cid:72)) such that R(A)∩R(B)={0} then
R(A+B)=R(A)+˙ R(B)ifandonlyif(cid:72) =N(A)+N(B).
ThenextresultwillbeusefulincharacterizingtheminusorderinSection3(see[5,Propo-
sition2.2]).
Proposition2.4. ForA,B ∈L((cid:72))considerthefollowingstatements:
(1) R(A∗)+˙ R(B∗)isclosed;
(2) thereexistsQ∈(cid:81) suchthatA∗=Q(A∗+B∗);
(3) N(A)+N(B)=(cid:72);
3
(4) R(A+B)=R(A)+R(B).
Then(1)⇔(2)⇔(3)⇒(4). Theimplication(4)⇒(3)holdsifR(A)∩R(B)={0}.
Foraproofof(1)⇔(2)see[2, Proposition4.13]. (1)⇔(3)wasstatedinProposition2.1.
Theimplication(3)⇒(4)followsfromtheproofof[3,Proposition2.8]. (4)⇒(3)followsfrom
Proposition2.3sinceR(A)∩R(B)={0}.
3. Theminusorder
Differentdefinitionshavebeengivenfortheminus(partial)order. Foroperatorsweoffer
onewhichequivalenttothoseappearingin[2]and[36].
Definition3.1. ForA,B ∈L((cid:72)),A≤− B ifthereexistP,Q∈(cid:81) suchthatA=PB andA∗=QB∗.
Proofsthat≤− isapartialorderonL((cid:72))canbefoundin[2,Corollary4.14]and[36,Corol-
lary 3]. It is easy to see that the ranges of P and Q can be fixed so that R(P) = R(A) and
R(Q)=R(A∗). Fordetails,see[2,Proposition4.13]andthedefinitionofminusorderin[36].
Inthenextpropositionwecollectsomecharacterizationsoftheminusorderintermsof
angleconditionsandsumofclosedsubspaces.
Proposition3.2. ConsiderA,B ∈L((cid:72)). Thefollowingstatementsareequivalent:
(1) A≤− B;
(2) c (R(A),R(B−A))<1andc (R(A∗),R(B∗−A∗))<1;
0 0
(3) R(B)=R(A)+˙ R(B−A)andR(B∗)=R(A∗)+˙ R(B∗−A∗);
(4) N(A)+N(B−A)=N(A∗)+N(B∗−A∗)=(cid:72);
(5) thereexistsP∈(cid:81) suchthatA=PB andR(A)⊆R(B).
Proof. The equivalences (1) ⇔ (2) ⇔ (4) follow applying the definition of the minus order
andProposition2.1totheoperatorsA, B−A,A∗and B∗−A∗,seealso[2,Proposition4.13].
For (2) ⇔ (3), suppose that c (R(A),R(B−A)) < 1 and c (R(A∗),R(B∗−A∗)) < 1. Then
0 0
R(A) +˙ R(B−A) and R(A∗) +˙ R(B∗−A∗) are closed. In this case, R(B) ⊆ R(A) +˙ R(B−A).
On the other hand, applying Proposition 2.4, there existsQ ∈ (cid:81) such that A∗ =QB∗. Then
N(B∗) ⊆ N(A∗) and N(B∗) ⊆ N(B∗−A∗), or R(A) ⊆ R(B) and R(B−A) ⊆ R(B). Then R(B) =
R(A)+˙ R(B−A). Similarly,R(B∗)=R(A∗)+˙ R(B∗−A∗). Seealso[36,Theorem2]. Conversely,
if item 3 holds, then R(A) +˙ R(B−A) and R(A∗) +˙ R(B∗−A∗) are closed or equivalently, by
Proposition2.1,item2holds.
Next consider (1)⇔(5). If A ≤− B then A =PB = BQ∗ with P,Q ∈(cid:81), so that A =PB and
R(A)⊆R(B). Conversely,supposeR(A)⊆R(B)andthereexistsP∈(cid:81) suchthatA=PB. Then
byLemma2.2itholdsthatR(B)=R(A)+R(B −A). MoreoverR(A)∩R(B −A)={0}because
R(A) ⊆ R(P) and R(B −A) ⊆ N(P), so that R(B) = R(A) +˙ R(B −A). In this case, (4) ⇒ (2)
of Proposition 2.4 can be applied so that there existsQ ∈ (cid:81) such that A∗ =QB∗. Therefore
A≤− B.
4
The following is a key result that will be useful on many occasions throughout the pa-
per. Itgivesanewcharacterizationoftheminuspartialorderintermsoftherangeadditivity
property,showingthattheminusorderhasanalgebraicnature.
Theorem3.3. ConsiderA,B ∈L((cid:72)). Thenthefollowingassertionsareequivalent:
(1) A≤− B;
(2) R(B)=R(A)+˙ R(B−A)andR(B∗)=R(A∗)+˙ R(B∗−A∗).
Proof. Suppose that A ≤− B. By Proposition 3.2, it follows that R(A) +˙ R(B−A) and R(A∗) +˙
R((B−A)∗)areclosed. Inparticular,R(A)∩R(B−A)=R(A∗)∩R(B∗−A∗)={0}. Also,itfollows
fromProposition2.4thatR(B∗)=R(A∗)+R(B∗−A∗)andR(B)=R(A)+R(B −A). Therefore,
R(B)=R(A)+˙ R(B−A)andR(B∗)=R(A∗)+˙ R(B∗−A∗).
Conversely,supposethatR(B)=R(A)+˙ R(B−A)andR(B∗)=R(A∗)+˙ R(B∗−A∗).Applying
(4)⇒(1)inProposition2.4,itfollowsthatR(A)+˙ R(B−A)andR(A∗)+˙ R(B∗−A∗)areclosed.
Hence,byProposition3.2andProposition2.1,A≤− B.
Let A ∈ L((cid:72)) for 1 ≤ i ≤ k. Lešnjak and Šemrl [28] give the following definition: the
i
k
operatorA=(cid:80)A isthequasidirectsumiftherangeofA isthedirectsumoftherangesofthe
i
i=1
A sandtheclosureoftherangeofA isthedirectsumoftheclosuresoftherangesoftheA s.
i i
Thenextresultmayberestatedassayingthat B isthequasidirectsumofA and B−A ifand
onlyifA≤− B.
Corollary3.4. IfA,B ∈L((cid:72)),thefollowingconditionsareequivalent:
(1) A≤− B;
(2) R(B)=R(A)+˙ R(B−A)andR(B)=R(A)+˙ R(B−A).
Proof. (1)⇒(2)followsfromProposition3.2andTheorem3.3.
Fortheconverse,sinceR(A)+˙ R(B−A)isclosed,fromProposition2.4wehavethatR(B∗)=
R(A∗)+R(B∗−A∗). Toseethatthissumisdirect,applyingProposition2.4againandusingthe
fact that R(B)=R(A)+˙ R(B −A) we get that R(A∗)∩R(B∗−A∗)={0}. Thus R(B∗)=R(A∗)+˙
R(B∗−A∗),andsobyTheorem3.3,A≤− B.
The next result shows the behavior of the minus order when the operators have closed
ranges.
Corollary3.5. ConsiderA,B ∈L((cid:72))suchthatA ≤− B. ThenR(B)isclosedifandonlyifR(A)
andR(B−A)areclosed.
Proof. If A ≤− B, then by Corollary 3.4, R(B) = R(A) +˙ R(B−A) and R(B) = R(A) +˙ R(B−A).
IfR(B)isclosedthenR(A)+˙ R(B−A)=R(A)+˙ R(B−A). HenceR(A)=R(A)andR(B−A)=
R(B −A). Infact, givenx ∈R(A), thenx ∈R(A)+˙ R(B −A), sothatthereexistx ∈R(A)and
1
x ∈R(B−A)suchthatx =x +x . Butx −x =x ∈R(A)∩R(B)={0},andsox =x ∈R(A);
2 1 2 1 2 1
thatis,R(A)=R(A). Similarly,R(B−A)=R(B−A). TheconversefollowsbyCorollary3.4.
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3.1. Theleftandrightminusorders
Inthissectionwedefinetheleft andrightminusordersandshowthattheyareageneral-
izationoftheleftandrightstarorders. Aswewillsee,theseordersarereallyonlyinteresting
oninfinitedimensionalspaces. Formatrices,theycoincidewiththeminusorder.
Webeginanalyzingthepropertiesoftheleftandrightstarorders. Originally,Drazin[20]
introduced the star order on semigroups with involutions, Baksalary and Mitra [6] defined
the left and right star orders for complex matrices, and later, Antezana, Cano, Mosconi and
Stojanoff[1]extendedthestarordertothealgebraofboundedoperatorsonaHilbertspace.
SeealsoDolinarandMarovt[18],DengandWang[14]andDjikic´ [16].
Given A,B ∈ L((cid:72)), the star order, left star order and right star order are respectively de-
finedby
∗
• A≤B ifandonlyifA∗A=A∗B andAA∗=BA∗,
• A∗≤B ifandonlyifA∗A=A∗B andR(A)⊆R(B),and
• A≤∗B ifandonlyifAA∗=BA∗andR(A∗)⊆R(B∗).
∗
IfA,B ∈L((cid:72)),thenA≤B ifandonlyifthereexistP,Q∈(cid:80) suchthatA=PB andA∗=QB∗
(see[1,Proposition2.3]or[18,Theorem5]). WecanalwaystakeP=PA andQ=PA∗.
Thenextresultisastraightforwardconsequenceof[14,Theorem2.1].Weincludeasimple
proof.
Proposition3.6. LetA,B ∈L((cid:72)). IfA∗≤B thenA≤− B.
Proof. If A ∗≤ B , then A∗A = A∗B, or equivalently A∗(A −B) = 0. Hence P (A −B) = 0, or
A
A = P B. Conversely, if A = P B, then A∗A = A∗B. So A ∗≤ B is equivalent to A = P B and
A A A
R(A)⊆R(B). ByProposition3.2(5),thisgivesA≤− B.
Thefollowingresultscharacterizetheleftandrightstarordersintermsofanorthogonal
rangeadditivityproperty.
Proposition3.7. ForA,B ∈L((cid:72)),A∗≤B ifandonlyifR(B)=R(A)⊕R(B−A).
Proof. FromtheproofofProposition3.6,A∗≤B ifandonlyifA=P B andR(A)⊆R(B). Thus
A
R(B)=R(A)⊕R(B−A)sinceR(B−A)⊆N(P )=R(A)⊥.
A
Conversely, if R(B) = R(A) ⊕ R(B −A), then R(A) ⊆ R(B) and R(B −A) ⊆ R(A)⊥, so that
A=P B. HenceA∗≤B.
A
Corollary3.8. ForA,B ∈L((cid:72)),A≤∗B ifandonlyifR(B∗)=R(A∗)⊕R(B∗−A∗).
The next characterization of the star order follows from the previous results (or alterna-
tively,fromTheorem3.3).
Corollary3.9. GivenA,B ∈L((cid:72)),thefollowingstatementsareequivalent:
6
∗
(1) A≤B;
(2) A∗≤B andA≤∗B;
(3) R(B)=R(A)⊕R(B−A)andR(B∗)=R(A∗)⊕R(B∗−A∗).
∗
Proof. Obviously, A ∗≤ B and A ≤∗ B. On the other hand, if A ≤ B, then by the proof of
Proposition3.6,A =PAB andA∗ =PA∗B∗. HenceR(A∗)⊆R(B∗)andR(A)⊆R(B). Thus(1)⇔
(2). Theequivalenceoftheseto(3)followsfromProposition3.7.
Asageneralizationoftheleftandrightstarorders,wenowdefinetheleft andrightminus
orders.
Definition3.10. ForA,B ∈L((cid:72)),
• A−≤B ifandonlyifR(B)=R(A)+˙ R(B−A),and
• A≤−B ifandonlyifR(B∗)=R(A∗)+˙ R(B∗−A∗).
Proposition3.11. Therelations−≤and≤−definepartialorders.
Proof. Weonlygivetheprooffor−≤,sincetheprooffor≤−isidentical.
First of all, −≤ is clearly reflexive. So consider A,B ∈ L((cid:72)) such that A −≤ B and B −≤A.
Then R(B) = R(A) +˙ R(B −A) and R(A) = R(B) +˙ R(B −A). From the last equality R(B −
A) ⊆ R(A). But R(B −A)∩R(A) = {0}, so that R(B −A) = {0}. Therefore A = B thus −≤ is
antisymmetric.
To prove −≤ is transitive, consider A,B,C ∈ L((cid:72)) such that A −≤ B and B −≤ C. Then
R(B)=R(A)+˙ R(B−A)andR(C)=R(B)+˙ R(C−B). SinceR(A)⊆R(B)⊆R(C),byLemma2.2,
R(C)=R(A)+R(C −A). ItremainstoshowthatR(A)∩R(C −A)={0}. SinceR(A)∩R(C −A)⊆
R(A)∩(R(C−B)+R(B−A))wecanwritex ∈R(A)asx =x +x ,x ∈R(C−B)andx ∈R(B−A).
1 2 1 2
Thenx −x =x ∈R(B)∩R(C −B)={0},andsox =x . Hencex ∈R(A)∩R(B−A)={0};that
2 1 2
isx =0andR(A)∩R(C −A)={0}. ThisimpliesthatR(C)=R(A)+˙ R(C −A),orequivalently,
A−≤C,andso−≤istransitive.
ThenextcorollaryisaconsequenceofTheorem3.3.
Corollary3.12. ForA,B ∈L((cid:72)),A−≤B andA≤−B ifandonlyifA≤− B.
It follows from Proposition 2.4 that A −≤ B if and only if A∗ =QB∗ forQ ∈ (cid:81) and R(A)∩
R(B−A)={0}.Thereisalsoacharacterizationoftheleftminusordersimilartothatoftheleft
starorderasfoundintheproofofProposition3.6. Weleavetheobviousversionfortheright
minusorderunstated.
Proposition3.13. For A,B ∈ L((cid:72)), A −≤ B if and only if there exists a (possibly unbounded)
denselydefinedprojectionP suchthatA=PB andR(A)⊆R(B).
7
Proof. IfA−≤B thenR(B)=R(A)+˙ R(B−A)sothatR(A)⊆R(B).DefineP=PR(A)//R(B−A)⊕N(B∗).
Then P is a densely defined projection and it is easy to check that A = PB. Conversely, if
A = PB for a densely defined projection and R(A) ⊆ R(B) then R(B) = R(A)+R(B −A) by
Lemma2.2,andthesumisdirectsinceR(A)⊆R(P)andR(B−A)⊆N(P).
Remark 3.14. Theminusordercanbeseenasastarorderafterapplyingsuitableweightsto
theHilbertspacesinvolved. Recallthat,ifA,B ∈L((cid:72),(cid:75))aresuchthatA≤− B thenthereexist
projectionsP ∈L((cid:75))andQ ∈L((cid:72))suchthatA =PB = BQ. TheoperatorsW =Q∗Q+(I −
1
Q∗)(I−Q)∈L((cid:72))andW =P∗P+(I−P∗)(I−P)∈L((cid:75))arepositiveandinvertible. Hencethe
2
innerproductsin(cid:72) and(cid:75) respectively,
(cid:10)x,y(cid:11) =(cid:10)W x,y(cid:11), forx,y ∈(cid:72) and 〈z,w〉 =〈W z,w〉, forz,w ∈(cid:75)
W1 1 W2 2
give rise to equivalent norms. With these new inner products, the projections P andQ are
∗
orthogonalin(cid:75) =((cid:75),〈·,·〉 )and(cid:72) =((cid:72),〈·,·〉 ),respectively,andsoA≤B.
W2 W2 W1 W1
On the other hand, A −≤ B if and only if there exists a densely defined projection P such
thatA=PB andR(A)⊆R(B).Inthiscase,itispossibletofindapositiveandinvertibleweight
W on(cid:75) suchthatP issymmetricwithrespectto〈·,·〉 (orequivalentlyA∗≤B inL((cid:72),(cid:75) ))
ifa2ndonlyifP admitsaboundedextensionP˜∈(cid:81) (orWe2quivalentlyA≤− B). W2
Hereisaproofofthelaststatement: supposethatthereexistsaweightW on(cid:75) positive
2
andinvertiblesuchthatP issymmetricwithrespectto〈·,·〉 . SinceP isa(denselydefined)
W2
idempotent then (cid:68)(P) = R(P) +˙ N(P), where (cid:68)(P) is the domain of P. Moreover, given x ∈
R(P)andy ∈N(P)wehave(cid:10)x,y(cid:11) =(cid:10)Px,y(cid:11) =(cid:10)x,Py(cid:11) =0becauseP issymmetricwith
W2 W2 W2
respectto〈·,·〉 andy ∈N(P). Hence(cid:68)(P)=R(P)⊕ N(P),andconsequently(cid:72) =R(P)⊕
W2 W2 W2
N(P), where the closures are taken with respect to 〈·,·〉 . Then P = P is a bounded
W2 R(P)//N(P)
extensionofP.
Conversely,supposethatthereexistsP˜∈(cid:81)suchthatP⊆P˜,andletW =P˜∗P˜+(I−P˜)∗(I−
2
P˜), which is positive and invertible and satisfies W P˜ = P˜∗W . Finally, P is symmetric with
2 2
respectto〈·,·〉 . Infact,ifx,y ∈(cid:68)(P)then(cid:10)Px,y(cid:11) =(cid:172)P˜x,y(cid:182) =(cid:172)W P˜x,y(cid:182)=(cid:172)x,W P˜y(cid:182)=
(cid:172)W x,P˜y(cid:182)=(cid:172)Wx2,P˜y(cid:182) =(cid:10)x,Py(cid:11) . W2 W2 2 2
2 W2 W2
Corollary3.15. LetA,B ∈L((cid:72))besuchthatA−≤B. ThenR(B∗)=R(A∗)+˙ R(B∗−A∗).
Proof. FromProposition3.13(3),ifA−≤B,thenA=PB andN(B)⊆N(A)orR(A∗)⊆R(B∗)and
inthesameway,R(B∗−A∗)⊆R(B∗). ThenbyProposition2.4(1),R(A∗)+˙ R(B∗−A∗)⊆R(B∗).
Ontheotherhand,R(B∗)⊆R(A∗)+R(B∗−A∗)⊆R(A∗)+˙ R(B∗−A∗). Hence
R(B∗)⊆R(A∗)+˙ R(B∗−A∗)⊆R(B∗).
ButR(A∗)+˙ R(B∗−A∗)isclosedbyProposition2.4. Therefore,R(B∗)=R(A∗)+˙ R(B∗−A∗).
Corollary3.16. LetA,B ∈L((cid:72))suchthatA−≤B. IfR(B)isclosedthenR(A)andR(B−A)are
closedandA≤− B.
8
Proof. Since A −≤ B, by Corollary 3.15, R(B∗) = R(A∗) +˙ R(B∗−A∗). If R(B) is closed, then
R(B∗)isclosedand
R(A∗)+˙ R(B∗−A∗)=R(B∗)=R(B∗)⊆R(A∗)+˙ R(B∗−A∗).
ThereforeR(A∗)+˙ R(B∗−A∗)=R(A∗)+˙ R(B∗−A∗).ThisimpliesthatR(A∗)=R(A∗)andR(B∗−A∗)=
R(B∗−A∗)andA≤− B.
Theabovecorollaryshowsthat,unliketheleft(right)starorder,theleft(right)minusorder
coincideswiththeminusorderwhenappliedtomatrices.Howeverforoperatorstheseorders
arenotthesame.
Example3.17(Seealso[7]). LetA∈L((cid:72))beanoperatorsuchthatR(A)(cid:54)=R(A)andthatthere
existsx ∈R(A)\R(A)whichisnotorthogonaltoN(A). Forexample,consider(cid:72) =l2((cid:78))the
spaceofallsquare-summablesequences,operatorA definedasA :(xn)n∈(cid:78)(cid:55)→((1/n)xn+1)n∈(cid:78),
and take x to be x = (1/n)n∈(cid:78). Define operator B as B = A+Px, where Px is the orthogonal
projection onto the one-dimensional subspace spanned by {x}. Since N(A) (cid:54)⊆ N(P ), and
x
N(P )isofco-dimensionone,wehave(cid:72) =N(A)+N(P ),whichaccordingtoProposition2.4
x x
showsthatA andP arerange-additive; thatis,R(A)+R(B −A)=R(B). WealsohaveR(A)∩
x
R(B −A)={0}showingthatA −≤ B. Ontheotherhand,R(A)∩R(B−A)(cid:54)={0}soA ≤− B does
nothold.
ApplyingTheorem3.3itispossibletodefinetheminusorderintermsoftheinnergener-
alizedinversesoftheoperatorsinvolved. ByaninnerinverseofanoperatorA ∈L((cid:72),(cid:75))we
meanadenselydefinedoperatorA−:(cid:68)(A−)⊆(cid:75) →(cid:72) satisfyingR(A)⊆(cid:68)(A−)andAA−A=A.
Proposition3.18. ForA,B ∈L((cid:72)),thefollowingconditionsareequivalent:
(1) A−≤B;
(2) there exists an inner inverse A− of A such that A−A =A−B and AA−x = BA−x for every
x ∈(cid:68)(A−).
Proof. SupposethatA −≤ B. If(cid:78) isacomplementofR(B),then(cid:72) =R(A)+˙ R(B−A)+˙ (cid:78) .
From Proposition 2.4 we know that N(A)+N(B −A)=(cid:72); so if (cid:77) =N(B −A)(cid:9)N(A), then
(cid:72) = N(A) +˙ (cid:77). Let A be the restriction of A to (cid:77), and define A− as A−1 on R(A), and
1 1
as the null operator on R(B −A) +˙ (cid:78) . Then A− is densely defined and the domain of A− is
(cid:68)(A−)=R(B)+˙ (cid:78) .Inthiscase,(A−B)A−x =0foreveryx ∈(cid:68)(A−),becauseR(A−)⊆N(A−B).
Ontheotherhand,sinceR(A−B)⊆(cid:68)(A−),wefindthatA−(A−B)=0sinceR(A−B)⊆N(A−).
Fortheconverse,supposethatthereexistsaninnerinverseA− ofA suchthatA−A =A−B
and AA−x = BA−x for every x ∈ (cid:68)(A−). In particular, if z ∈ (cid:72) then Az ∈ R(A) ⊆ D(A−), so
thatAz =AA−Az =BA−Az. HenceR(A)⊆R(B),showingthatR(B)=R(A)+R(B−A). From
A−A=A−B wehaveR(A−B)⊆N(A−),whileN(A−)∩R(A)={0},andsoR(B)=R(A)+˙ R(B−A).
Therefore,A−≤B.
Corollary3.19. ForA,B ∈L((cid:72)),thefollowingconditionsareequivalent:
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(1) A≤− B;
(2) thereexistinnerinversesA−ofA and(A∗)−ofA∗suchthat
(i) A−A=A−B andAA−x =BA−x foreveryx ∈(cid:68)(A−),
(ii) (A∗)−A∗=(A∗)−B∗andA∗(A∗)−x =B∗(A∗)−x foreveryx ∈(cid:68)((A∗)−).
4. Applications
4.1. GeneralizedinversesofA+B
InthissectionwestatetheformulasforarbitraryreflexiveinversesofA+B intermsofthe
inversesofAandB,whenA−≤A+B.Forthesakeofsimplicity,webeginbygivingtheformula
for the Moore-Penrose inverse. Theorem 4.7 states the result in the most general form, and
fromthistheoremmanyexistingresultsinthesubjectcanberecovered.
IfA ≤− A+B thenA =P(A+B)forsomeP ∈(cid:81). UsingtheprojectionP wecanconstruct
a projection E ∈ (cid:81) onto R(A+B) that will be useful in stating the formula for the Moore-
PenroseinverseofA+B.
Lemma4.1. LetA,B ∈L((cid:72))besuchthatA≤− A+B,andP∈(cid:81) besuchthatA=P(A+B). Set
E =P P+P (I −P).
A B
Then E ∈ (cid:81) and R(E) = R(A+B). Moreover, E is selfadjoint if and only if P = P(cid:77)//(cid:78) where
(cid:77) =R(A)⊕(cid:77) ,(cid:78) =R(B)⊕(cid:78) with(cid:77) and(cid:78) closedsubspacessuchthat(cid:77) ,(cid:78) ⊆N(A∗)∩
1 1 1 1 1 1
N(B∗).
Proof. If A = P(A+B) then R(A) ⊆ R(P) and R(B) ⊆ N(P). Therefore P P and P (I −P) are
A B
projections,withR(P P)=R(A)andR(P (I −P))=R(B). Moreover,
A B
P PP (I −P)=P (I −P)P P=0.
A B B A
Therefore E =P P+P (I −P)isaprojection. Also,R(A)=R(P P)=R(EP)⊆R(E). Applying
A B A
Lemma2.2,R(E)=R(A)+R(B)=R(A+B)becauseA≤− A+B.
Finally,ifP =P(cid:77)//(cid:78) thenthereexistclosedsubspaces(cid:77)1,(cid:78)1 suchthat(cid:77) =R(A)⊕(cid:77)1
and (cid:78) = R(B) ⊕ (cid:78) . Hence P P = P , P (I −P) = P and E =
1 A R(A)//R(B)+˙(cid:78)1+˙(cid:77)1 B R(B)//R(A)+˙(cid:77)1+˙(cid:78)1
P . SinceA ≤− A+B,itfollowsthat E∗=E ifandonlyif(cid:77) +˙ (cid:78) =(R(A+B))⊥=
R(A+B)//(cid:77)1+˙(cid:78)1 1 1
(R(A)+˙ R(B))⊥=N(A∗)∩N(B∗),orequivalently,(cid:77) and(cid:78) areincludedinN(A∗)∩N(B∗).
1 1
Definition4.2. LetA,B ∈L((cid:72))suchthatA ≤− A+B. ConsiderP,Q ∈(cid:81) suchthatA =P(A+
B)=(A+B)Q,thenP willbecalledoptimalforA andB ifE =P P+P (I−P)isselfadjoint. In
A B
asymmetricway,sinceA∗=Q∗(A∗+B∗),Q willbecalledoptimalforA and B ifQ∗ isoptimal
forA∗and B∗,i.e.,F =PA∗Q∗+PB∗(I −Q∗)isselfadjoint.
10