Volume 2009, Article ID 284526,13pages doi:10.1155/2009/284526
Research Article
Perturbation Results on Semi-Fredholm Operators and Applications
Boulbeba Abdelmoumen and Hamadi Baklouti
D´epartement de Maths, Facult´e des Sciences de Sfax, B. P. 1171, 3000 Sfax, Tunisia Correspondence should be addressed to
Boulbeba Abdelmoumen,[email protected] Received 14 July 2009; Accepted 26 September 2009
Recommended by Jozef Banas
We give some results concerning stability in the Fredholm operators and Browder operators set, via the concept of measure of noncompactness. Moreover, we prove some localization results on the essential spectra of bounded operators on Banach space. As application, we describe the essential spectra of weighted shift operators. Finally, we describe the spectra of polynomially compact operators, and we use the obtained results to study the solvability for operator equations in Banach spaces.
Copyrightq2009 B. Abdelmoumen and H. Baklouti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Throughout this paper,Xdenotes an infinite dimensional complex Banach space. We denote by LX the space of all bounded linear operators on X. The subspace of all compact operators of LX is denoted by KX. We write NT ⊆ X for the null space and RT ⊆ X for the range of T. The nullity, nT,of T is defined as the dimension of NT and the deficiency,dT,ofT is defined as the codimension ofRTinX. The set of upper lowersemi-Fredholm operators are defined, respectively byΦX {T ∈ LX;nT <
∞andRT is closed in X}and, respectively, Φ−X {T ∈ LX; dT < ∞}. We use ΦX : ΦX∩Φ−Xfor the set of Fredholm operators inLX, andΦ±X : ΦX∪ Φ−Xfor the set of semi-Fredholm operators inLX. IfT ∈Φ±X,theniT:nT−dT is called the index ofT. It is well known that the index is a continuous function on the set of semi-Fredholm operators.
Various notions of essential spectrum appear in the applications of spectral theory see, e.g.,1,2. We useσTfor the spectrum ofT ∈ LX, σeT {λ ∈C; λ−T /∈ΦX} for Wolf essential spectrum, σessT C\ {λ ∈ C; λ−T ∈ ΦX and iλ−T 0} for Schechter essential spectrum, andσaT {λ∈C; infx1λ−Tx 0}for approximate point spectrum.
Recall thataT resp.,δT, the ascentresp., the descentofT ∈ LX, is the smallest nonnegative integernsuch thatNTn NTn1 resp.RTn RTn1. If no suchnexists, then aT ∞resp.δT ∞. The sets of upper and lower semi-Browder operators are defined, respectively by BX {T ∈ LX; T ∈ ΦXandaT < ∞},B−X {T ∈ LX; T ∈ Φ−XandδT < ∞}. The set of Browder operators on X isBX BX∩ B−X.The corresponding spectrum is defined byσbT {λ∈C; λ−T /∈ BX}.
We are interested in this paper Section 2 to the study of the stability problem in Fredholm operators set and semi-Fredholm operators set. In the past few years, a lot of work has been done along these lines,3–5and others. A well-known fact is that ΦX is an open set. An important question is to characterize, for a givenS ∈ΦX, the class of bounded operatorsT onX, such thatST still belongs toΦX. Recall that ifT ∈ KX, thenST ∈ΦX see2, Theorem 16.9 . More generally, this fact holds true also forT a strictly singular operatorsee6, Proposition 2.c.10. Noncompactness measures provide advanced techniques to obtain current precise results along this line; see for example7,8.
By means of the Kuratowski measure, for a givenS ∈ΦX, we describe inTheorem 2.2a class of bounded operatorsTonX, for whichST ∈ΦX. We should notice that, in general, the size of the perturbation depends uponS. This key-result permits to prove inCorollary 2.3 some localization results about the essential spectraσe andσessof bounded operators onX.
Next, we investigate the stability in the semi-Browder operators set. In9, Grabiner proves thatBXandB−Xare closed under commuting perturbation. In4, Rakoˇcevi´c extends this result to the perturbation classes associated with the sets of semi-Fredholm operators. In Theorem 2.4, by means of the Kuratowski measure, we characteriz for a givenS ∈ BX, a class of bounded operatorsT onX, that commute withS, such that ST ∈ BX.As the corollary of this theorem we obtain the main result of Grabiner. As the application of the obtained results, we describe the essential spectra of weighted shift operators.
In Section 3, we are interested in the study of polynomially compact opera- tors. Consider PKX : {T ∈ LXsuch that PT ∈ KX for some nonzero complex polynomialP}.ForT ∈ PKXthere exists a unique unitary polynomialmTz of least degree such that mTT is compact. This polynomial will be called the minimal polynomial of T. In this section, we describe σeS − T, for T, S ∈ LX with compact commutator such thatT ∈ PKX.Next, we show that if there exists an analytic function f in a neighborhood ofσTsuch thatfTis compact, thenT ∈ PKX.As application, we use the obtained results to investigate the solvability for operator equations in Banach spaces,Sϕ−Tϕ ψ.ForT ∈ PKX,we give affirmative answer under several sufficient conditions onS.This result extends the analysis started in10,11and generalizes the result obtained, in caseSλI, in12, Theorem 2.2.
2. Some New Properties in Fredholm Theory by Means of the Kuratowski Measure of Noncompactness
In this section, we give some results concerning the classes of Fredholm operators and Browder operators via the concept of measures of noncompactness. General definition can be found in13. We writeMXfor the family of all nonempty and bounded subset ofX. We deal with a specific measure: the Kuratowski measure of noncompactness defined onMXas followssee14:
γA inf
ε >0 : A may be covered by finitely many set of diameter≤ε
. 2.1
ForT ∈ LX, we define the two nonnegative quantitiessee15associated withTby
αT sup
γTA
γA ;A∈MX, γA>0
, βT inf
γTA
γA ;A∈MX, γA>0
. 2.2 LetX1be an infinite dimensional subspace ofXand letJX1be the natural embedding ofX1 intoX. The discresp., circlewith center 0 and radiusris denoted byD0, r resp.,C0, r.
We writeD0, rfor the closure ofD0, rand we useCr1, r2:D0, r2\D0, r1,forr1 ≤r2. We start this section by some fundamental properties satisfied byαandβwhich will be useful in the remainder of the text. For more detail, we refer to15.
Proposition 2.1. LetT, Sbe inLX.Then one has the following.
iαtT |t|αTandβtT |t|βT, for allt∈R.
ii|αT−αS| ≤αTS≤αT αS.
iiiαT◦S≤αTαSandβT◦S≥βTβS.
ivβT−αS≤βTS≤βT αS.
vIfT is an isomorphism, thenαT−1βT 1.
viβT>0 if and only if T ∈ΦX.
viiαT≤ TandβT≥lim infx →∞Tx/x.
In the following theorem we establish a stability property in the upper semi-Fredholm operators set. This result provides, in particular, an extension of Theorem 6.1 in8.
Theorem 2.2. Let T, S be two bounded operators on X and let f be an analytic function in a neighborhoodΩofσS∪σTnot vanishing on a connected component ofσS∪σT.
iIfαT< βS, thenTS∈ΦXandiTS iS.
Suppose moreover that the commutatorT, S ∈ KXandαfT < βfS, then one has the following.
iiT−S∈ΦX.
iiifS∈ΦXimplies thatT−S∈ΦX.
ivfz zn,for somen∈N∗,implies thatiT−S iS.
Proof. ByProposition 2.1, we have for allt∈0,1, βtTS≥βS−tαT>0,thentTS∈ ΦX,for allt∈0,1,in particular,TS∈ΦX.By the continuity of the index onΦX, we getiTS iS,and this provesi.
Now, assume thatαfT < βfS,applying i, we getfT−fS∈ ΦXand ifT−fS ifS.Let ω be an open set with closureω ⊂ Ω and whose boundary
∂ω consists of finite number of simple closed curves that do not intersect, and such that σS∪σT⊂ω. Then we have
fT−fS 1 2iπ
∂ω
z−T−1−z−S−1
fzdz. 2.3
SinceS, T∈ KX,then there exist compact operatorsK1andK2such that
z−T−1−z−S−1 T−Sz−T−1z−S−1K1 z−T−1z−S−1T−S K2. 2.4
Integrating along∂ω, we get
fT−fS T−SLK3LT−S K4, 2.5
where L 1/2iπ ∂ωz−T−1z−S−1fzdz. It is easily checked that L ∈ LX and K3, K4 ∈ KX.This leads toT −S ∈ ΦX.If fS ∈ ΦX, thenfT−fS ∈ ΦX.
By2.5, we conclude thatT−S∈ΦX.
Now, iffz zn,thenαTn< βSnyieldsαtTn< βSn, ∀t ∈0,1. Therefore, byii, tT−S∈ΦX,for allt∈0,1.By the continuity of the index function onΦX,we getiT−S iS.
ForT ∈ LX,define β0T resp.,α0Tto be the limit of the sequenceβTn1/n resp.,αTn1/n. For the existence of these limits see2, Lemma 1.21.
Corollary 2.3. LetT be a bounded operator onX, the one has the following.
iσessT⊂D0, α0T.
iiIfT /∈Φ−X,thenD0, β0T⊂σeT.
iiiIfT ∈Φ−X,thenσeT⊂Cβ0T, α0T.
ivIf 0/∈σessT,thenσessT⊂Cβ0T, α0T.
vIf 0∈σessT,thenD0, β0T⊂σessT.
Proof. Letn∈N∗ and suppose that|λ|n > αTn,then, byTheorem 2.2iv, we haveλ−T ∈ ΦXandiλ−T 0.Hence, if|λ|> α0T,thenλ /∈σessT,and this provesi.
Notice that ifβT 0,then β0T 0 and the results are all trivial. Suppose that βT>0. For|λ|< β0T, there existsn∈N∗such that|λ|n< βTn.Then, byTheorem 2.2iv, we haveλ−T ∈ΦXandiλ−T iT.Hence, we get easilyii–v.
2.1. Stability in the Browder and the Semi-Browder Operators
The following theorem uses the measure of noncompactness to establish stability in the semi- Browder operators set. More precisely, we have the following.
Theorem 2.4. Suppose thatSandTare commuting bounded linear operators on the Banach spaceX.
Assume thatαT< βS,then
aS<∞implies thataTS<∞. 2.6
Proof. Fort ∈0,1,we haveαtT< βS,and then, byTheorem 2.2i,tT S∈ΦX.Set N∞T
nNTnandR∞T
nRTn.SinceSandT are commuting, then according to 16, Theorem 3, for allt∈0,1,there existsεt>0 such that, for allsin the diskDt, εt,
N∞tT S∩ R∞tT S N∞sTS∩ R∞sTS. 2.7
Hence,N∞tTS∩ R∞tTSis a locally constant function ofton the interval0,1.Since every locally constant function on a connected set is constant, then
∀t∈0,1, N∞tTS∩ R∞tTS N∞S∩ R∞S. 2.8
Now, sinceaS<∞,then from5, Proposition 1.6i
N∞S∩ R∞S N∞S∩ R∞S {0}. 2.9 Thus, N∞T S∩ R∞T S {0},and again by 5, Proposition 1.6i, it follows that aTS<∞.
Remark 2.5. Theorem 2.4 extends the results of Grabiner 9, Theorem 2. Indeed, if T is compact, we obtain 0 αT< βS βTS.Hence,Theorem 2.4yieldsaS<∞if and only ifaTS<∞.This proves thatBXis closed under commuting compact perturbation.
By duality argument, we prove the closeness ofB−X.
Corollary 2.6. LetS, Tbe commuting bounded operators onX.Suppose that there existsn∈N∗such thatαTn< βSn.
iIfS∈ BX,thenTS∈ BX.
iiIfS∈ BX,thenTS∈ BX.
Proof. iLett∈0,1.SinceαtTn< βSn,then fromTheorem 2.2,tTS∈ΦX.Arguing as in the proof ofTheorem 2.4, we get the result.
iiSinceS∈ BX,theniS 0.ByTheorem 2.2,iTS 0.On the other hand,i yieldsaTS<∞.According to17, Theorem 4.5d, we getδTS<∞.
Corollary 2.7. LetT be a bounded operator onX, thene one has the following.
iσbT⊂D0, α0T.
iiIf 0/∈σbT,thenσbT⊂Cβ0T, α0T.
Proof. iFor|λ|> α0T, there existsn∈N∗such that|λ|n> αTn.ByCorollary 2.6, we have λ−T ∈ BX. The result follows since we can choosenarbitrary large.
iiSince 0/∈σbT,thenT ∈ ΦXand henceβT > 0.For|λ| < β0T, there exists n∈N∗such that|λ|n< βTn.Corollary 2.6implies thatλ−T ∈ BXsinceT ∈ BX.
2.2. Application: Weighted Shift Operators
Let ω ωnn∈N be a bounded complex sequence. Consider the unilateral backward weighted shift operator Wω, pdefined on X lrN,C, r ≥ 1,byWω, px0, x1, . . .
ωpxp, ωp1xp1, . . .. In 18, Proposition 1.6.15, the authors give a localization results for the spectrum and the approximate point spectrum of unilateral backward weighted shift operator. In this section, we investigate the Wolf essential spectrum ofWω, p.
Proposition 2.8. The following statements hold true.
iαWω, p≤ω:lim supn→∞|ωn|andβWω, p≥ω−:lim infn→∞|ωn|.
iiσeWω, p⊂Cω−, ω.
Proof. Forε >0,the setE{n∈N; |ωn|> εω or|ωn|<−εω−}is finite. Consider
X1{xnn∈X;xn0, ∀n∈E}, X2{xnn∈X; xn0, ∀n /∈E}. 2.10
We haveXX1⊕X2. SinceX2is finite dimensional subspace, then
α W
ω, p α
W ω, p
JX1
≤W ω, p
JX1≤εω. 2.11 Otherwise, byProposition 2.1,
β W
ω, p β
W ω, p
JX1
≥ lim inf
x →∞
W ω, p
JX1x
x ≥ −εω−. 2.12
Since we can chooseεarbitrary small, then we geti.
We should notice that if 0 is a cluster point for the sequence|ωn|n, thenω− 0 and iifollows fromCorollary 2.3i. If not, thenF0{n≥p such thatωn0}is a finite set and Wω, pis a Fredholm operator with indexp.More precisely,nWω, p pcardF0and dWω, p cardF0, here cardF0denotes the cardinal ofF0.Now, byCorollary 2.3iii, we getσeWω, p⊂Cω−, ω, which proves the proposition.
Remark 2.9. Notice that if |ωn|n converges to l, then according to Proposition 2.8, we get αWω, p βWω, p l and σeWω, p ⊂ C0, l. SinceiWω, p/0,then by the continuity of the index function onΦX, we obtainσeWω, p C0, l.This is a well- known factsee, e.g.,19, Proposition 27.7, page 139.
In what follows, we investigate more precisely the essential spectrum ofWω, p. For this end defineA0|ω|to be the limit set of|ω|n, that is, the set of all cluster points of the sequence|ωn|n, andAk1|ω|to be the limit set ofAk|ω|fork≥0.
Proposition 2.10. Suppose thatA0|ω| {0≤l1<· · ·< lN}is finite, then
σe
W ω, p
⊂
1≤i≤N
C0, li. 2.13
Proof. For 0< ε <1/2infi /j|li−lj|,considerAi {n∈N;||ωn| −li|> ε}, i 1, . . . , N, A0 N
i1AiandXi {xnn ∈X; xn 0, ∀n ∈Ai}, i 0, . . . , N.We can writeX ⊕Ni0Xi.For i0, . . . , N,define the operatorSiby
SiW ω, p
onXi,
Si0 on⊕j /iXj. 2.14
Since, For alli /j, Si◦Sj 0 andWω, p n
i0Si,then
i∈{0,...,N}
λ−Si λN−1 λ−W
ω, p
. 2.15
This yields
σe
W ω, p
\ {0}
0≤i≤N
σeSi
\ {0}. 2.16
Observe that X0 is finite dimensional and then S0 is finite rank. Hence, σeS0 {0}. It remains to prove that, For alli1, . . . , N, σeSi⊂ C0, li.Consider the operatorSi:X → X defined by
Si0 onXi
SiliS on⊕j /iXj, 2.17
whereSW1, pis the corresponding un-weighted shift operator. We haveSiSiWv, p, withv vnnbeing the sequence defined by
vnωn for n /∈Ai,
vnli for n∈Ai. 2.18
Observe that|vn|nconverges and limn→∞|vn|li,thenσeWv, p C0, li.SinceSi◦Si Si◦Si 0,then, as above,σeWv, p\ {0} σeSi∪σeSi\ {0}.Hence,σeSi⊂ C0, li, and this completes the proof.
Now, we prove the following result.
Theorem 2.11. Suppose that there existsk≥0 such thatAk|ω|is a finite set, then
σe
W ω, p
⊂
l∈A0|ω|
C0, l. 2.19
Proof. by induction. For k 0, the result follows by Proposition 2.10. Let k ≥ 0 be an integer and suppose that ifAk|ω|is a finite set, then2.19holds true. Suppose now that
Ak1|ω| {l1, . . . , lN}is a finite set. For 0 < ε < 1/2infi /j|li−lj|andi 1, . . . , N, we considerBi{n∈N;||ωn|−li|> ε}andB0N
i1Bi.Define the sequenceui uinn, 0≤i≤N by
uin ωn, ∀n /∈Bi, uin0, ∀n∈Bi.
2.20
SinceWω, p N
i0Wui, pandWui, p◦Wuj, p 0,for alli /j,then
σe W
ω, p
\ {0} ⊂ N
i0
σe W
ui, p
\ {0}. 2.21
Observe thatAk|u0|is a finite set andA0|u0|⊂ A0|ω|∪ {0}.Hence σe
W
u0, p
⊂
l∈A0|ω|
C0, l∪ {0}. 2.22
Now, consider the sequencevi vinn, i1, . . . , Ndefined by
vinli, ∀n∈Bi, vin0, ∀n /∈Bi.
2.23
Clearly, lim infn→∞|uinvni| ≥li−εand lim supn→∞|uinvni| ≤liε.Hence, byProposition 2.8, σeWuinvin, p⊂ Cli−ε, liε.SinceWui, p◦Wvi, p Wvi, p◦Wui, p 0 and Wui, p Wvi, p Wuivi, p,then
σe
W
uivi, p
\ {0}
σe
W
ui, p
∪σe
W
vi, p
\ {0}. 2.24
Hence, we get, fori1, . . . , N,
σe
W
ui, p
⊂ Cli−ε, liε. 2.25
Since we can chooseε >0 arbitrary small, then by2.21,2.22, and2.25, we get2.19.
Finally, consider the superposition of two weighted shift operatorsWω, p Wu, k.
Suppose that lim supn→∞|un| < lim infn→∞|ωn|, then, by Proposition 2.8, αWu, k <
βWω, p. By Theorem 2.2, Wω, p Wu, k ∈ ΦX and iWω, p Wu, k iWω, p.
To close this section, we define a special class of bounded operators on a Banach space X,that presents some interesting properties. Set
L0X:
T ∈ LX; αT βT
. 2.26
First, we observe thatKX ⊂ L0X,andλI ∈ L0Xfor allλ ∈C.Also, we notice that if ω ωnnis a complex sequence that converges, then the weighted shift operatorWω, pis a nontrivial element ofL0lrN,C.Now, we prove the following result.
Proposition 2.12.
iFor allT1, T2∈ L0Xone hasT1T2∈ L0X,andαT1T2 αT1αT2. iiIfT ∈ L0Xis invertible, thenT−1∈ L0X.
Proof. We observe, byProposition 2.1, that
αT1αT2 βT1βT2≤βT1T2≤αT1T2≤αT1αT2. 2.27
This proves the statementi. Again, byProposition 2.1, forT invertible, we haveαT−1 1/βT 1/αT βT−1.This provesii.
As an immediate result we get, for allTbeing inL0X, α0T αT β0T βT. In the following proposition we describe the essential spectra for a givenT∈ L0X.
Proposition 2.13. LetTbe inL0Xand suppose that 0∈σessT,then one has the following.
iσbT σessT D0, αT.
iiIfT /∈Φ−X,thenσeT D0, αT. iiiIfT ∈Φ−X,thenσeT C0, αT.
Proof. According to Corollary 2.3, we have D0, αT ⊂ σessT. By Corollary 2.7, we get σbT ⊂ D0, αT. SinceσessT ⊂ σbT, then we get i. The assertioniifollows from Corollary 2.3i–ii. Foriii, on one hand, byCorollary 2.3iii, we haveσeT⊂ C0, αT, on the other hand, the boundary∂σessT⊂σeT.
Notice that if ω ωnn is a complex sequence that converges to l, then by Proposition 2.13i,
σb W
ω, p σess
W ω, p
D0, l. 2.28
3. Fredholm Theory for Polynomially Compact Operators
In this section, we present a spectral analysis for polynomially compact operators. We begin by proving an important result about perturbation by polynomially compact operators in the general context of normed spaces. First, we make the following definition.
Definition 3.1. LetY be a normed space, letT ∈ PKY, mT be the minimal polynomial of T,and letS∈ LY.We say thatT andScommunicate if There exists a continuous mapϕ : 0,1 → C; ϕ0 0 andϕ1 1,such that, for allλzero ofmT, ϕtλ ∈ρeS, for allt∈ 0,1.
Theorem 3.2. LetT, Sbe two bounded operators on a normed space Y with compact commutator.
Suppose thatT ∈ PKYandmTλ/0,for allλ∈σeS.ThenT−S∈ΦY. If moreover,T andScommunicate, theniT−S iS.
Proof. Since mTλ/0 for all λ ∈ σeS, then we can write mTS N
i1S− λi, with λi/∈σeS.This yieldsmTS ∈ ΦY.On the other hand mTTis compact, thenmTS− mTT∈ΦY.WritingmTS−mTT S−TLK1 LS−T K2,withL∈ LYand K1, K2∈ KY,we conclude thatS−T ∈ΦY.
Now, consider Qtz N
i1z − λiϕt, then QtϕtT ϕtNmTT. Thus, QtϕtTis compact and, for allλ∈σeS, Qtλ/0.This yields
ϕtT−S∈ΦY, ∀t∈0,1. 3.1
By the continuity of the index function onΦY,we getiϕtT−Sconstant for allt∈0,1.
In particular,iT−S iS.
Remark 3.3. Theorem 3.2is an improvement of12, Theorem 2.1. Indeed, ifσeSis a discrete set ofC,thenTandScommunicate. In the particular case whereSλI,we haveσeS {λ}.
Therefore,λI−Tis a Fredholm operator of index zero.
We notice that if, for some p ∈ N∗, mTz zp,then, for all S ∈ ΦY, S and T communicate. Hence, we obtain the following.
Corollary 3.4. LetT, Sbe two bounded operators on a normed spaceY with compact commutator.
Suppose thatTp∈ KY,for somep∈N∗.IfS∈ΦY,thenT−S∈ΦYandiT−S iS.
Corollary 3.5. LetT, Sbe two commuting bounded operators on the Banach spaceX. Suppose that T ∈ PKX,S∈ BX,and assume thatTandScommunicate, thenTS∈ BX.
Proof. As in the proof ofTheorem 3.2,3.1we obtain S−ϕtT ∈ ΦX.Arguing as in the proof ofTheorem 2.4, we getaS−T<∞. Now, byTheorem 3.2, we haveiT−S iS 0.
Therefore, according to17, Theorem 4.5dwe getδS−T<∞.
The following proposition is a well-know result, see12,20. Here, we present a simple proof for this fact.
Proposition 3.6. LetT ∈ PKXand letmT be the minimal polynomial ofT.Then
σeT σbT {λ∈Csuch that mTλ 0}. 3.2
Proof. Since mTT is compact, then σbmTT {0}. By 3, Theorem 1, σbmTT mTσbT.Hence,σeT⊂σbT⊂ {λ∈C; mTλ 0}.Letλ∈Cbe such thatmTλ 0,we can writemTT T−λQT QTT−λ.SincemTTis compact and, by the minimality ofmT,QTis not compact, thenT−λ/∈ΦX. Hence,{λ∈C;mTλ 0} ⊂σeT.
Proposition 3.7. LetT, Sbe two bounded operators onXwith compact commutator.
iIf T ∈ PKX, thenσeS−T⊂σeS−σeT.
iiIf there existsp∈N∗such thatTp∈ KX,thenσeS−T σeS−σeT.
Proof. iIfλ∈σeS−T,thenS−T−λ /∈ΦX.On the other handTλI ∈ PKX,and T λI, S T, Sis compact. According toTheorem 3.2, there existsλS ∈ σeSsuch that
mTλλS 0,wheremTλz mTz−λis the minimal polynomial ofTλ.HenceλλS−λT, wheremTλT 0.Finally, the result follows fromProposition 3.6
iiByi,σeS−T⊂σeS−σeT.SinceTp∈ KX,thenσeT {0},and we obtain σeS−T⊂σeS σeS−T−−T⊂σeS−T.
Notice that in general, the converse inclusion inidoes not hold.
Example 3.8. Consider the unweighted shift operatorS W1, p.According toRemark 2.9, we haveσeS C0,1,the unit circle. Letλ λnnbe a bounded complex sequence and let Kλ:lrN,C → lrN,Cbe defined byKλxnn λnxnn.Suppose thatλipλi, for alli≥ 0,then KλS SKλ.ConsiderPz p−1
i0z−λi,then PKλ 0 ∈ KX.Suppose that
|λi|/1, for alli ∈ {0, . . . , p−1},then, applyingTheorem 3.2, we get thatSλ S−Kλ is a Fredholm operator. ByProposition 3.7, we getσeSλ⊂p−1
i0 C−λi,1.
The index of Sλ depends on the position of λi with respect to C0,1. If |λi| <
1, for alli∈ {0, . . . , p−1},thenKλandScommunicate and byTheorem 3.2,iSλ iS p.If we suppose thatλi λ, for alli∈ {0, . . . , p−1}with|λ|>1,thenKλλI,andSλis invertible.
In this caseiSλ 0/iS.Observe that in this case,KλandSdo not communicate.
Theorem 3.9. LetT be a bounded operator on X.Suppose that there exists an analytic function f in a neighborhood of σT which does not vanish on a connected component ofσTsuch that fT∈ KX,thenT ∈ PKX.
Proof . From 3, Theorem 1 , we have σbfT fσbT. Since fT ∈ KX, then σbfT {0}.Hence,σbT ⊂ σT∩ {λ ∈ C;fλ 0}and therefore,σbTis a finite set {λ1, . . . , λn}. Writefz Pzgz,wherePz:n
i1z−λiαiandgis an analytic function withgλi/0, for alli∈ {1, . . . , n}.Sincegdoes not vanish onσbT,then 0/∈σbgT.Thus, gT∈ΦXandPT∈ KX.
3.1. Application: Solvability of Operator Equations
In the following theorem, we treat the question of the solvability of operator equations. We will prove, under several sufficient conditions, that if the homogeneous equationSϕ−Tϕ0 only has the trivial solutionϕ0,then for allψ∈Xthe nonhomogeneous equationSϕ−Tϕ ψhas a unique solutionϕ∈X, and this solution depends continuously onψ.
Theorem 3.10. Let Y be a normed space and letT, Sbe two communicating commuting bounded operators onY. Suppose thatT ∈ PKYand letmT be the minimal polynomial ofT.Assume that 0/∈σessS∪σamTS.
IfF:S−Tis injective, then the inverse operatorF−1 :Y → Yexists and is bounded.
Proof. F is injective, then NF {0}, thus nF 0. Applying Theorem 3.2, we get iF iS 0.It follows thatdT 0 and therefore, the operatorF is surjective. Hence, the inverse operator F−1 S−T−1 : Y → Y exists. Since Y is not necessary a Banach space, we have to prove that F−1 is bounded. Suppose that it is not so, then there exists fnn ⊂ X with fn 1 and the sequenceϕn F−1fnsatisfies:ϕn → ∞asn → ∞.
Setgn:fn/ϕn andψn:ϕn/ϕn , n∈N.Thengn → 0 asn → 0,andψn1.Since FψngnandFSSF,then there existsL∈ LYsuch that
mTTψnmTSψnL gn
. 3.3
SincemTTis compact, we can choose a subsequenceψnkk such thatmTTψnk → ψ as k → ∞. Using 3.3, we observe that mTSψnk → ψ as k → ∞. On the one hand, FmTSψnk mTSFψnk mTSgn → 0 as k → ∞. On the other hand, FmTSψnk → Fψ.Hence,Fψ 0 which implies thatψ 0.This is in contradiction with 0/∈σamTS.
Theorem 3.11. LetT ∈ PKXandS ∈ BXbe communicating, commuting operators on the Banach space X. Suppose that 0/∈σamTS,and set F S−T. Then the projection P : X → NFaFdefined by the decompositionX NFaFRFaFis compact, and the operatorF−P is bijective.
Proof. First we notice that byCorollary 3.5,F ∈ BX, thenF ∈ΦXandaF < ∞. Thus, FaF ∈ ΦX,which implies thatNFaFis finite dimensional. Hence, the projectionP is continuous and compact. Now, we claim that F −P is bijective. Let ϕ ∈ NF −P.Since P ϕ ∈ NFaF,then FaF1ϕ 0,which implies thatFaFϕ 0.ThusPϕ ϕ.Since Fϕ Pϕ,thenFϕ ϕ.We get by iterationFaFϕ ϕ 0.On the other hand, from Theorem 3.10applied to the operatorTP,we conclude thatF−Pis surjective.
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