Table Of Content.
UNCONDITIONAL BASES OF SUBSPACES RELATED TO NON-SELF-ADJOINT
PERTURBATIONS OF SELF-ADJOINT OPERATORS
ALEXANDERK.MOTOVILOVANDANDREIA.SHKALIKOV
ABSTRACT. AssumethatT isaself-adjointoperatoronaHilbertspaceHandthatthespectrumofT is
confinedintheunion j∈JD j, J⊆Z,ofsegmentsD j=[a j,b j]⊂Rsuchthata j+1>b jand
7
1 S inf a j+1−b j =d>0.
j
0
If B is a bounded (in general non-self-adjo(cid:0)int) pertur(cid:1)bation of T with kBk=:b<d/2 then the spec-
2
trum of the perturbed operator A=T+B lies in the union j∈JUb(D j) of the mutually disjoint closed
an b-neighborhoodsUb(D j)ofthesegmentsD j inC. LetQj beStheRieszprojectionontotheinvariantsub-
J spaceofAcorresponding tothepartofthespectrumofAlyinginUb D j , j∈J. Ourmainresultisas
4 follows:ThesubspacesLj=Qj(H),j∈J,formanunconditionalbasis(cid:0)in(cid:1)thewholespaceH.
2
]
P
S 1. INTRODUCTION AND MAIN RESULT
.
h Webeginwithrecallingsomedefinitions(see[1,Ch.6.5]). Asequenceofnonzerosubspaces{L }
k
at ofaHilbertspacesHissaidtobeabasisifanyelementx∈Hisuniquely represented bytheseries
m
x=(cid:229) x , where x ∈L , (1)
[ k k k
k
2 that converges in the norm of H. A basis of subspaces {L } is said to be unconditional if the series
v k
6 (1) converges to x after any rearrangement of its elements. An unconditional basis of subspaces is also
9 called a Riesz basis of subspaces. In the case where all the subspaces L are one-dimensional (finite-
j
2 dimensional) wecanchoose elements y ∈L =Q (H) (abasis {y }inL ). Thenthesequence ofthe
6 j j j kj j
0 subspaces {Lk}isabasisifandonlyifthecorresponding systemisabasis(abasiswithparentheses) in
. H.
1
0 Whenonedealswithbasesofsubspaces, itisconvenient toworkintermsofprojections. Wewill
7 usedefinitions andresults presented in[2,§6]. LetJ beafiniteorinfinite ordered setofindices, J ⊂Z.
1
: Asystem ofprojections {Qj}j∈J issaidtobecompleteiftheequalities
v
i (Q x,y)=0, forany x∈H andany j∈J,
X j
r imply y=0. Itis easily seen that the system {Q } iscomplete ifand only if any element x∈H can
a j j∈J
beapproximated withanarbitraryaccuracybylinearcombinations ofelementsx ∈L =Q (H),k∈J.
k k k
Asystemofprojections {Q } iscalledminimalif
j j∈J
Q Q =d Q forany j,k∈J.
j k kj j
It follows directly from the definitions that if {Q } is a minimal system of projections, then the
j j∈J
sequence of the subspaces L =Q (H) forms a basis (an unconditional basis) if and only if the series
j j
(cid:229) Q converges (converges after anyrearrangement oftheindices) inthestrong operator topology to
j∈J j
theidentityoperator.
Furtherwewillmakeuseofthefollowingresults(forthecorresponding proofssee[1,Ch. 6]and
[2,§6]).
TheoremA.Let{Q } beasystemofprojectionsinaHilbertspaceH.Thefollowingstatementsare
j j∈J
equivalent:
Keywordsandphrases. Rieszbasis,unconditionalbasisofsubspaces,non-self-adjointperturbations.
∗ThisworkwassupportedbytheDeutscheForschungsgemeinschaft(DFG)andtheRussianFoundationforBasicResearch,
grantNo.15-51-12389.
1
(1) AsequenceofsubspacesL =Q (H), j∈J,isanunconditionalbasisoftheHilbertspaceH.
j j
(2) ThereexistsanequivalentinnerproductinH suchthatasequenceofsubspacesL =Q (H),
j j
j∈J,iscompleteandmutuallyorthogonal(L ⊥L fork6= j).
k j
(3) ThereexistsaboundedandboundedlyinvertibleoperatorK inH,andacompleteandminimal
systemoforthogonalprojections{P}suchthat
j
Q =K−1PK j∈J.
j j
(4) Thesystemofprojections{Q } iscomplete,minimal,andtheseries(cid:229) Q convergesun-
j j∈J j∈J j
conditionally.
(5) Thesystemofprojections{Q } iscomplete,minimal,and
j j∈J
(cid:229) (Q x,x) <¥ forany x∈H. (2)
j
j∈J
(cid:12) (cid:12)
Now we are ready to formula(cid:12)te the m(cid:12)ain result of the paper. In what follows it is assumed that
eitherthesetofindicescoincides withNorwithZ.
Theorem 1. LetT beaself-adjointoperatoronaHilbertspaceH. AssumethatthespectrumofT is
confinedintheunionD := D ofthesegmentsD =[a ,b ]⊂Rsuchthata >b forall j∈J.
j∈J j j j j j+1 j
Assumeinadditionthat
S
inf(a −b )=d>0. (3)
j+1 j
j∈J
LetBbeabounded(generallynon-self-adjoint)operatoronHwithkBk=:b<d/2.Thenthespectrum
theoperatorA=T +B liesintheunion U (D ) ofthemutuallydisjointclosedb-neighborhoods
j∈J b j
U (D )ofthesegmentsD inC.IfQ ,j∈J,aretheRieszprojectionsontotheinvariantsubspacesL of
b j j j S j
AcorrespondingtotheisolatedcomponentsofitsspectrumlyinginU (D ),thentheinvariantsubspaces
b j
L ,L =Q (H), j∈J,formanunconditionalbasisinH.
j j j
There are many papers devoted tothe Riesz basis property of the root vectors of non-self-adjoint
operators, whichareperturbations ofself-adjoint ones. Thecorresponding results andreferences canbe
found in the book of Markus [3] and in the paper of Shkalikov [2]. First results related to Theorem 1
wereobtained byMarkus [4]and Kato(see [5, Ch.5, Theorem 4.15a]): LetT beaself-adjointoperator
withdiscretespectrumonaHilbertspaceH suchthatitseigenvalues{l } aresimpleandsubjectthe
j
conditionl −l →¥ ,as j→¥ . Thenforanybounded(generallynon-self-adjoint)operatorBthe
j+1 j
operatorT+BhasdiscretespectrumanditsrootvectorsformaRieszbasisinthespaceH.
A more general result follows from the Markus-Matsaev theorem [3, Ch.1, Theorem 6.12]: Let
T beaself-adjointoperatorinaHilbertspaceH,havingafiniteorder(i.e. itseigenvalues,counting
multiplicities,aresubjecttothecondition|l |≥CjpwithsomeconstantsC,p>0),andthereexistgaps
j
inthespectrumofT withthelengths≥d. IfkBk≤d/2 thenthetherootsubspacesoftheperturbed
operatorT+BformaRieszbasiswithparenthesesH(oraRieszbasisconsistingoffinite-dimensional
rootsubspaces).
The condition for T to be of finite order was dropped in [2]. However, the compactness of the
resolvent (T −l )−1 was essentially used in the proof. To the best of our knowledge, Theorem 1 is,
apparently, the first result in this topic which deals with an unperturbed operator T possibly having a
non-discrete spectrum.
We have an additional motivation to prove Theorem 1. We expect that this result might help to
resolve some open problems concerning bounded perturbations of self-adjoint operators in spaces with
indefinitemetric(see[6,7]).
2. PROOF OF THEOREM 1
We divide the proof into two parts that are called below Step 1 and Step 2, respectively. At Step
1 we prove that the sequence of subspaces {L } forms a basis of H. At Step 2 we prove that it is,
j j∈J
actually, anunconditional basis.
2
Step 1. For the sake of definiteness, we assume that the index set J coincides with the set of all entire
numbers Z={...,−2,−1,0,1,2,...}. Thiscorresponds tothe mostgeneral case wheninfinitely many
segmentsD liebothonR− andR+.
j
UnderthehypothesiskBk=bwith0≤b<d/2thatweassume,theclosedneighborhoodsU (D )
b j
ofthesegmentsD , j∈J,aredisjointand,surely,U (D )= U (D ). Thefirstassertionofthetheorem
j b j∈J b j
ontheinclusionofthespectrums (A)ofAintheunion U (D )isaconsequence ofthewell-known
jS∈J b j
estimate
S
1 1
(T−l )−1 ≤ ≤ , D := D , (4)
dist(l ,s (T)) dist(l ,D ) j
(cid:13) (cid:13) [j∈J
where s (T), s (T)⊂(cid:13)(cid:13)D , is the sp(cid:13)(cid:13)ectrum of the self-adjoint operator T. For any l lying outsideUb(D )
wehave
d :=dist(l ,D )>b. (5)
Thencombining(4)and(5)withthebound
1
(1+B(T−l )−1 −1 ≤ (6)
1−b/d
(cid:13)(cid:0) (cid:1) (cid:13)
fortheresolvent(A−l )−1=(T+(cid:13)B−l )−1onefinds (cid:13)
k(A−l )−1k= (T−l )−1 1+B(T−l )−1 −1
(cid:13)(cid:13)≤ 1 <¥(cid:0), (cid:1) (cid:13)(cid:13) (7)
(cid:13) d −b (cid:13)
where the quantity d =d (l ) is defined in (5). Hence, any l ∈C\U (D ) belongs to the resolvent set
b
of the perturbed operator A=T +B and then s (A)⊂U (D )= U (D ). Since the neighborhoods
b j∈J b j
U (D )fordifferent j∈J aredisjoint, namely,
b j S
dist(U (D ),U (D ))≥d−2b, j6=k,
b j b k
the spectral sets of A confined inU (D ) are also disjoint. The Riesz projections Q for these sets are
b j j
welldefined. Inparticular, givenanarbitrary b′∈(b,d/2),onemaywriteQ intheform
j
1
Q =− (A−l )−1dl , G =¶ U (D ), (8)
j 2p i G j b′ j
Z j
wheretheintegration alongcontourG isperformed intheanti-clockwise direction.
j
Belowwewillalsousetherepresentation
(A−l )−1=(T−l )−1−G(l ), l ∈/U (D ), (9)
b
where
G(l )=(A−l )−1B(T−l )−1=(T−l )−1M(l )B(T−l )−1 (10)
and
M(l )= 1+B(T−l )−1 −1.
NowdenotebyR therectangle inCwhoseverticalsidespassthroughthepoints
n (cid:0) (cid:1)
c =(b +a )/2 andc =(b +a )/2 (11)
−n −n−1 −n n n n+1
whilethehorizontal sidescoincide withthesegments[c ±ig ,c ±ig ]where
−n n n n
g =max{|c |,|c |}. (12)
n −n n
Clearly,thesidesofR donotintersect thesetU (D ). Byvirtueof(9)oneobtains
n b
(cid:229) n Q x=− 1 (A−l )−1xdl =(cid:229) n Px+I x, (13)
j=−n j 2p iZ¶ Rn −n j n
3
whereI aretherespectivecontour integralsoftheoperator-valued functionG(l )along¶ R ,
n n
1
I := G(l )dl , (14)
n 2p iZ¶ Rn
and P are the spectral projections onto the spectral subspaces of the self-adjoint operator T associated
j
withthepartsofitsspectrum insidethecorresponding segmentsD .
j
Givenanarbitrary x∈Hwehave
n
(cid:229) Px→x as n→¥ . (15)
j
j=−n
n
Thus, in order to prove that (cid:229) Q x → x one only needs to show that the sequence of I x in (13)
j n
j=−n
converges tozeroasn→¥ .
First, let us show that the operators I are uniformly bounded. We have I = I1+I2, where I1
n n n n n
and I2 are the integrals along the horizontal and vertical sides of the rectangles R , respectively. For l
n n
varyingonthehorizontal sidesoftherectangle ¶ R wehavetheestimate
n
1
k(T −l )−1k≤ , l =x ±ig , x ∈[c ,c ],
g n −n n
n
whereg isdefinedby(12). Hence,byvirtueof(7)and(10),itfollows
n
b
kG(l )k≤ , l =x ±ig , x ∈[c ,c ]. (16)
(g −b)g n −n n
n n
Takinginto account thatthelengths ofthehorizontal sidesdonotexceed 2g andg →¥ asn→¥ ,we
n n
getkI1k→0asn→¥ .
n
LetusestimatethenormsoftheoperatorsI2. Forthesakeofdefinitenessconsidertherightvertical
n
sideoftherectangle R anddivideitinthreeparts
n
w ∪w +∪w −, wherew =(c −id,c +id), w ±=[c ±id,c ±ig ].
n n n n n n n n n
Thelengths oftheintervalsw equal2d anddueto(7)and(10)
n
2bd
G(l )dl ≤ . (17)
w (d/2−b)(d/2)
(cid:13)Z n (cid:13)
Forl ∈w ± wehave (cid:13)(cid:13) (cid:13)(cid:13)
n
(cid:13) (cid:13)
d (l )−b≥|Iml |−b. (18)
Thus,againapplying (7)and(10),oneobtains
gn b gn b b
G(l )dl ≤ dt ≤ dt ≤ <1. (19)
(cid:13)Zw n± (cid:13) Zd t (t −b) Zd (t −b)2 d−b
(cid:13) (cid:13)
Therefore, (cid:13) (cid:13)
(cid:13) (cid:13)
kI k≤C, (20)
n
wheretheconstantCdepends onlyond andb.
Let us show that kI xk→0 as n→¥ for any fixed x∈H. To this end, choose some e >0 and,
n
first,findN ∈Nsuchthat
e
kx−x k< , (21)
N
2C
where
N
(cid:229)
x = Px.
N j
j=−N
Obviously, from(20)and(21)itfollows
e
kI (x−x )k< forany n∈N. (22)
n N
2
4
Let us estimate kI x k as n→¥ . Denote e(t):=(E(t)x,x) where E(t) is the spectral function of the
n N
self-adjoint operatorT,andobservethat,bythespectral theorem,
cN
2 de(t)
(T−l )−1x = , l =x +it 6∈s (T). (23)
N (t−x )2+t 2
(cid:13) (cid:13) c−ZN
(cid:13) (cid:13)
ForN fixedtheequalit(cid:13)y (23)implies(cid:13)
(T−l )−1x =O |l |−1 as |l |→¥ . (24)
N
(cid:13) (cid:13)
Nowwecanmodifyestimate((cid:13)17)andgetfro(cid:13)m(7)a(cid:0)nd(24(cid:1))
(cid:13) (cid:13)
G(l )x dl =O |c |−1 →0 asn→¥ .
N n
w
(cid:13)Z n (cid:13)
(cid:13) (cid:13) (cid:0) (cid:1)
Analogously, bytakingintoaccount(18)wecanmodifytheestimate(19)andobtain
(cid:13) (cid:13)
(cid:13) (cid:13)
G(l )x dl ≤ gn dt =O lncn =o(1) asn→¥ .
(cid:13)Zw n± N (cid:13) Zd (t −b) (cn−cN)2+t 2 (cid:18) cn (cid:19)
(cid:13) (cid:13)
Thelasttw(cid:13)oestimatestoge(cid:13)ther give p
(cid:13) (cid:13)
I x →0 as n→¥ .
n N
Hence, there is N ∈N (N ≥N) such that kI x k<e /2 whenever n>N . Taken together with (22)
1 1 n N 1
thisyieldskI xk<e forn>N . Therefore,wehaveproventhatthesequenceofI stronglyconvergesto
n 1 n
zeroasn→¥ andthenfrom (13)and(15)itfollowsthatforanyx∈H thetwo-sided series(cid:229) n Q x
j=−n j
converges tox. Thus,thesystemofsubspaces L =Q (H), j∈Ziscomplete. Thatthesesubspaces are
j j
linearly independent follows from the mutual orthogonality of the Riezprojections (8) in the sense that
Q Q =d Q forany j,k∈Z(see, e.g. [1,Ch. I§1.3]). Thus, thesystem {L }¥ represents abasis
j k jk j j j=−¥
ofsubspaces inH.
Step 2. By Theorem A, in order to prove that the above basis of subspaces {L }¥ is unconditional,
j j=−¥
it suffices to show that the series (cid:229) ¥j=−¥ (Qjx,x) converges for any x∈H. First, let us transform the
contoursG intheintegrals(8)intothecontoursG ,whichsurroundtherectangleswiththeverticalsides
j (cid:12) (cid:12)j
l =c +it andl =c +it wheret isv(cid:12)aryingin(cid:12)[−d,d],andthehorizontalsidesx ±id,x ∈[c ,c ].
j−1 j j−1 j
Aspreviously, thenumbers c aredefinedby(11e)andcoincide withthecenters ofthegapsbetweenthe
j
neighboring intervals D andD .
j j+1
Takingintoaccount therepresentation (10)andestimate(6),weimmediately conclude that
¥ ¥ ¥
2p (cid:229) |(Q x,x)|≤ (cid:229) (T−l )−1x,x dl + (cid:229) (G(l )x,x)dl .
j
j=−¥ j=−¥ (cid:12)ZG j (cid:12) j=−¥ (cid:12)ZG j (cid:12)
(cid:12) (cid:16) (cid:17) (cid:12) (cid:12) (cid:12)
Thefirstseriesconverges sinceitsimp(cid:12)(cid:12)lyecoincides withtheseri(cid:12)(cid:12)es(cid:229) j∈Z((cid:12)(cid:12)Pejx,x)=kxk2;we(cid:12)(cid:12)recallthatPj
are the spectral projections of the self-adjoint operator T associated with the segments D . Byvirtue of
j
(6)and(10),thesecond seriescanbeestimatedasfollows:
¥ ¥
(cid:229) (G(l )x,x)dl ≤C (cid:229) (T−l )−1x 2|dl |
1
j=−¥ (cid:12)ZG j (cid:12) j=−¥ ZG j
(cid:12) (cid:12) (cid:13) (cid:13)
(cid:12)(cid:12) e (cid:12)(cid:12)≤C e (cid:13)(T−l )−1x(cid:13) 2|dl |+2(cid:229) (T−l )−1x 2|dl | , (25)
1
ZG +∪G − j∈ZZw j !
(cid:13) (cid:13) (cid:13) (cid:13)
where G ± are the lines l =x ±id, x ∈R(cid:13)(cid:13), w j are the v(cid:13)(cid:13)ertical segments l(cid:13)(cid:13)=cj+it , −(cid:13)(cid:13)d ≤t ≤d, and
C = const. Convergence of the integrals and the series in (25) can be proven by using the spectral
1
theorem. Asbefore,denotee(t):=(E(t)x,x)whereE(t)standsforthespectral function ofT. Then
2 de(t)
(T−l )−1x |dl |= |dl |
G G R|t−l |2
Z ± Z ± Z
(cid:13) (cid:13)
5
(cid:13) (cid:13)
(cid:13) (cid:13)
dx
= de(t)
R R|t−x |2+d2
Z Z
p
= de(t)
d R
p Z
= kxk2. (26)
d
Further, noticethatforc givenby(11),thelowerbound(3)yields
j
d/2+d(j−k) forall t ∈D , k≤ j,
|t−c |≥ k
j d/2+d(k− j−1) forall t ∈D , k≥ j+1.
k
(cid:26)
Hence,forl =c +it ,t ∈[−d,d],wehave
j
d
(cid:229) (T−l )−1x 2|dl |= (cid:229) dt de(t)
j∈ZZg± j∈ZZ ZR|t−cj−it |2
(cid:13) (cid:13) −d
(cid:13) (cid:13)
(cid:13) (cid:13) d
= (cid:229) dt (cid:229) de(t)
j∈ZZ k∈ZZD k |t−cj−it |2
−d
(cid:229) (cid:229) de(t)
≤2d
j∈Zk∈ZZD k |t−cj|2
2
≤ (cid:229) kPxk2 2 2 + (cid:229) 1
k d d2|k− j|2
k∈Z (cid:18) (cid:19) j∈Z,j6=k !
≤ 2C2 (cid:229) kPxk2 = 2C2kxk2, (27)
d2 k d2
k∈Z
¥
whereC =4+ (cid:229) 1 =4+p 2. Theestimate(25)togetherwith(26)and(27)entailsthebound(2). This
2 j2 6
j=1
completes theproofofTheorem1.
Thefollowingstatementisasimplecorollary ofTheorem1(combinedwithTheoremA).
Corollary 2. AssumethehypothesisofTheorem1.Thenthereexistsaninnerproducth , iinHwith
thefollowingproperties.
(1) Theproducth, iisnorm-equivalenttotheoriginalinnerproduct(, )inH.
(2) ThesubspacesL =Q H aremutuallyorthogonalwithrespecttotheinnerproducth , i and,
j j
withrespecttoh, i,theHilbertspaceHadmitstheorthogonaldecomposition
H= L . (28)
j
j∈J
M
(3) ThesubspacesL , j∈J,arereducingfortheperturbedoperatorA=T+Band,withrespectto
j
thedecomposition(28),thisoperatoradmitsablockdiagonalmatrixrepresentation
A=diag(...,A A ,A ,A ,A ,...),
−2 −1 0 1 2
whereA =A ,j∈J,denotesthepartofAinthereducingsubspaceL .
j L j
j
(cid:12)
(cid:12) REFERENCES
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(2016)(inpress).
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1988.
6
[4] A.S.Markus Expansion in root vectors of a slightly disturbed self-adjoint operator, Dokl. Acad. Nauk USSR 142:3
(1962),538–541.
[5] T.Kato,PerturbationTheoryforLinearOperators,Springer–Verlag,Berlin,Heidelberg,NewYork,1980.
[6] S.Albeverio,A.K.Motovilov,andA.A.Shkalikov,BoundsonvariationofspectralsubspacesunderJ-self-adjointper-
turbations,Integr.Equ.Oper.Theory64(2009),455–486.
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ALEXANDERK.MOTOVILOV
JOINTINSTITUTEFORNUCLEARRESEARCHANDDUBNASTATEUNIVERSITY,DUBNA,RUSSIA
E-mailaddress: [email protected]
ANDREIA.SHKALIKOV
FACULTYOFMATHEMATICSANDMECHANICS,LOMONOSOVMOSCOWSTATEUNIVERSITY,MOSCOW,RUSSIA
E-mailaddress: [email protected]
7
