Upper triangular operator matrices, asymptotic intertwining and Browder, Weyl theorems

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R E S E A R C H

Open Access

Upper triangular operator matrices,

asymptotic intertwining and Browder, Weyl

theorems

Bhagwati P. Duggal

1

_{, In Ho Jeon}

2

_{and In Hyoun Kim}

3*

*_{Correspondence:} [email protected] 3_{Department of Mathematics,} Incheon National University, Incheon, 406-772, Korea Full list of author information is available at the end of the article

Abstract

Given a Banach spaceX, letMC∈B(X⊕X) denote the upper triangular operator

matrixMC=

(

A C₀_B

)

, and let

δ

AB∈B(B(X)) denote the generalized derivation

δ

AB(X) =AX–XB. If limn→∞

δ

ABn (C) 1

n = 0, then

σ

_x(M_C) =

σ

_x(M₀), where

σ

_xstands for

the spectrum or a distinguished part thereof (but not the point spectrum);

furthermore, ifR=R1⊕R2∈B(X⊕X) is a Riesz operator which commutes withMC,

then

σ

x(MC+R) =

σ

x(MC), where

σ

xstands for the Fredholm essential spectrum or a

distinguished part thereof. These results are applied to prove the equivalence of Browder’s (a-Browder’s) theorem forM0,MC,M0+RandMC+R. Suﬃcient conditions

for the equivalence of Weyl’s (a-Weyl’s) theorem are also considered. MSC: Primary 47B40; 47A10; secondary 47B47; 47A11

Keywords: Banach space; asymptotically intertwined; SVEP; polaroid operator

1 Introduction

A Banach space operatorT∈B(X), the algebra of bounded linear transformations from a Banach spaceX into itself, satisﬁes Browder’s theorem if the Browder spectrumσb(T)

ofT coincides with the Weyl spectrumσw(T) of T; T satisﬁes Weyl’s theorem if the

complement ofσw(T) in σ(T) is the set(T) of ﬁnite multiplicity isolated eigenval-ues ofT. Weyl’s theorem implies Browder’s theorem, but the converse is generally false (see [–]). LetMandMC∈B(X ⊕X) denote, respectively, the upper triangular

op-eratorsM=A⊕BandMC= _{A C}

B

for some operatorsA,C,B∈B(X). It is well known thatσx(M) =σx(A)∪σx(B) =σx(MC)∪ {σx(A)∩σx(B)} forσx=σorσb, andσw(M)⊆ σw(A)∪σw(B) =σw(MC)∪ {σw(A)∩σw(B)}. The problem of ﬁnding suﬃcient conditions

ensuring the equality of the spectrum (and certain of its distinguished parts) ofMand

MC, along with the problem of finding sufficient conditions for M satisfies Browder’s theorem and/or Weyl’s theorem to implyMCsatisfies Browder’s theorem and/or Weyl’s

theorem (andvice versa), has been considered by a number of authors in the recent past (see [], and some of the references cited there). For example, if eitherA*_or_B_{has the} single-valued extension property, SVEP for short, thenσ(M) =σ(MC) =σ(A)∪σ(B).

Again, ifσw(MC) =σw(A)∪σw(B), thenσ(M) =σ(MC) =σ(A)∪σ(B) [, Proposition .]

andMsatisﬁes Browder’s theorem if and only ifMCsatisﬁes Browder’s theorem [,

The-orem .]; furthermore, in such a case,Msatisﬁes Weyl’s theorem if and only ifMC

satisﬁes Weyl’s theorem if and only if(M) =(MC) [, Theorem .]. The

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ityσw(MC) =σw(A)∪σw(B) may be achieved in a number of ways: if eitherAandA*, or AandB, orA*_and_B*_{, or}_B_and_B*_{have SVEP, then}_σ

w(MC) =σw(A)∪σw(B) [,

Proposi-tion .]. In this paper we consider condiProposi-tions of another kind, condiProposi-tions which do not assume SVEP.

Given S,T∈B(X),S andT are said to beasymptotically intertwinedby X∈B(X) if limn→∞δnST(X)



n_{= . Here}_δ_ST ∈_B₍_B₍X_{)) is the generalized derivation}_δ_ST₍_X_{) =}_SX_–_XT

andδn

ST=δST(δSTn–). Evidently,SandT asymptotically intertwined byXdoes not implyT

andSasymptotically intertwined byX. Furthermore,SandT asymptotically intertwined byXdoes not implyσ(S) =σ(T), not evenσ(S)⊆σ(T); see [, Example ..]. However, as we shall see, ifA,B,Care as in the deﬁnition ofMCabove, thenAandBasymptotically

intertwined byCimplies the equality of the spectra, and many distinguished parts thereof to spectrum ofMandMC. We prove in the following that iflimn→∞δABn (C)



n _{= , then}

MCsatisfies Browder’s theorem if and only ifMsatisfies Browder’s theorem. If, addition-ally, the isolated points ofσ(M) are poles of the resolvent ofM, thenMcsatisfies Weyl’s

theorem if and only ifMsatisﬁes Weyl’s theorem. Extensions toa-Browder’s theorem,

a-Weyl’s theorem and perturbations by Riesz operators are considered.

2 Notation and complementary results

For a bounded linear Banach space operator S∈B(X), letσ(S),σp(S),σa(S), σs(S) and

isoσ(S) denote, respectively, the spectrum, the point spectrum, the approximate point

spectrum, the surjectivity spectrum and the isolated points of the spectrum ofS. Letα(S) andβ(S) denote the nullity and the deﬁciency ofS, deﬁned by

α(S) =dimS–() and β(S) =codimS(X).

If the rangeS(X) ofSis closed andα(S) <∞(resp.β(S) <∞), thenSis called anupper semi-Fredholm(resp. alower semi-Fredholm) operator. IfS∈B(X) is either upper or lower semi-Fredholm,Sis called asemi-Fredholmoperator, andind(S), theindexofS, is then de-ﬁned byind(S) =α(S) –β(S). If bothα(S) andβ(S) are ﬁnite, thenSis aFredholmoperator. Theascent, denotedasc(S), and thedescent, denoteddsc(S), ofSare given by

asc(S) =infn:S–n() =S–(n+)(), dsc(S) =infn:Sn(X) =Sn+(X)

(where the inﬁmum is taken over the set of non-negative integers); if no such integern

exists, thenasc(S) =∞, respectivelydsc(S) =∞. Let

+(S) ={λ∈C:S–λis upper semi-Fredholm},

–(S) ={λ∈C:S–λis lower semi-Fredholm},

(S) ={λ∈C:S–λis Fredholm},

σSF+(S) =

λ∈σa(S) :λ∈/+(S)

,

σSF–(S) =

λ∈σa(S) :λ∈/–(S)

,

σe(S) =

λ∈σ(S) :λ∈/(S),

σw(S) =

λ∈σ(S) :λ∈σe(S) or ind(S–λ)= 

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σaw(S) =

λ∈σa(S) :λ∈σSF+(S) or ind(S–λ) > 

,

σsw(S) =

λ∈σs(S) :λ∈σSF–(S) or ind(S–λ) < 

,

σb(S) =

λ∈σ(S) :λ∈σe(S) orasc(S–λ)=dsc(S–λ)

,

σab(S) =

λ∈σa(S) :λ∈σSF+(S) or asc(S–λ) =∞

,

σsb(S) =

λ∈σs(S) :λ∈σSF–(S) or dsc(S–λ) =∞

,

(S) =

λ∈isoσ(S) :  <dim(S–λ)–() =α(S–λ) <∞,

p(S) =

λ∈isoσ(S) :λ∈(S),asc(S–λ) =dsc(S–λ) <∞,

H(S) =

x∈X: lim

n→∞S

n_x/n

= 

.

Hereσw(S) is the Weyl spectrum,σaw(S) denotes the Weyl (essential) approximate point

spectrum,σsw(S) the Weyl (essential) surjectivity spectrum,σb(S) the Browder spectrum,

σab(S) the Browder (essential) approximate point spectrum,σsb(S) the Browder (essential)

surjectivity spectrum, andH(S) the quasi-nilpotent part ofS[]. Recall, [], thatH(S) andK(S), whereK(S) denotes theanalytic core

K(S) =x∈X: there exists a sequence{xn} ⊂X andδ>  for which

x=x,S(xn+) =xnandxn ≤δnxfor alln= , , . . .

,

are hyper-invariant (generally non-closed) subspaces of S such thatS–p_()_⊆_H

(S) for every integerp≥ andSK(S) =K(S). Recall also that if ∈isoσ(S), thenX =H(S)⊕

K(S).

We say thatShas thesingle valued extension property, or SVEP, atλ∈Cif for every open neighborhoodUofλ, the only analytic solutionf to the equation (S–μ)f(μ) =  for all μ∈U is the constant functionf ≡; we say thatShas SVEP ifShas a SVEP at every λ∈C. It is well known that ﬁnite ascent implies SVEP; also, an operator has SVEP at every isolated point of its spectrum (as well as at every isolated point of its approximate point spectrum).

S∈B(X) satisﬁes Browder’s theorem, shortened toSsatisﬁes Bt, ifσw(S) =σb(S) (if and

only ifσ(S)\σw(S) =p(S), see [, p.]);Ssatisfies Weyl’s theorem, shortened toS sat-isfies Wt, ifσ(S)\σw(S) =(S) (if and only ifSsatisfies Bt andp(S) =(S)) [, p.]. The implication Wt⇒Bt is well known.

An isolated pointλ∈isoσ(S) is a pole (of the resolvent) of S∈B(X) ifasc(S–λ) = dsc(S–λ) <∞. In such a case we say thatSis polar atλ; we say thatSis polaroid (resp., polaroid on a subsetFof the set of isolated points ofσ(S)) ifSis polar at everyλ∈isoσ(S) (resp., at everyλ∈F). Letp(S) denote the set of poles ofS.

Throughout the following,M∈B(X⊕X) shall denote the diagonal operatorM=A⊕

BandMC∈B(X⊕X) shall denote the upper triangular operator matrix _{A C}

B

, for some operatorsA,B,C∈B(X). Recall, [, Exercise , p.], thatasc(A)≤asc(MC)≤asc(A) +

asc(B) anddsc(B)≤dsc(MC)≤dsc(A) +dsc(B).

Lemma . Ifσ(M) =σ(MC),then p(M) =p(MC).

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works just as well for the case in whichλ∈ρ(A) (=C\σ(A)) orλ∈ρ(B). Letλ∈p(MC).

Then

asc(A–λ)≤asc(MC–λ) <∞ and dsc(B–λ)≤dsc(MC–λ) <∞.

Ifλ∈isoσ(B) anddsc(B–λ) <∞, thenasc(B–λ) =dsc(B–λ) <∞andBis polar atλ[, Theorem .]. Now letλ∈isoσ(A). SinceMCis polar atλ,H(MC–λ) = (MC–λ)–p()

for some integerp≥. Observe that

H(A–λ) =H(MC–λ)∩X= (MC–λ)–p()∩X= (A–λ)–p().

Hence, ifλ∈isoσ(A), then

X=H(A–λ)⊕K(A–λ) = (A–λ)–p()⊕K(A–λ)

⇒ (A–λ)pX= ⊕(A–λ)pK(A–λ) =K(A–λ)

⇒ X= (A–λ)–p()⊕(A–λ)pX,

i.e.,Ais polar atλ. Now, since

asc(M–λ)≤asc(A–λ) +asc(B–λ) and dsc(M–λ)≤dsc(A–λ) +dsc(B–λ),

we have

asc(M–λ) =dsc(M–λ) <∞,

i.e.,Mis polar atλ. Conversely, ifλ∈p(M), thenasc(M–λ) =max{asc(A–λ),asc(B–λ)} anddsc(M–λ) =max{dsc(A–λ),dsc(B–λ)}impliesasc(MC–λ)≤asc(A–λ) +asc(B–λ)

anddsc(MC–λ)≤dsc(A–λ) +dsc(B–λ) are both ﬁnite, hence equal. ThusMCis polar

atλ.

Remark . A number of conditions guaranteeing (the spectral equality)σ(MC) =σ(M) are to be found in the literature. Thus, for example, ifA*_or_B_{has SVEP, or if}_σ

w(MC) =

σw(A)∪σw(B), orσaw(MC) =σaw(A)∪σaw(B) [, (I) p. and Proposition .], thenσ(MC) =

σ(M). Compact operators have SVEP; hence, if either ofAorBis compact, thenσ(MC) =

σ(M).

Lemma . shows that ifBis a compact operator thenp(M) =p(MC). A proof of the

following lemma may be obtained from that of Lemma .: we give here an independent proof, exploiting the additional information contained in the hypothesis.

Lemma . Ifσ(M) =σ(MC),then p(M) =p(Mc).

Proof Once again we consider pointsλ∈isoσ(A)∩isoσ(B). Letλ∈p(MC). Thenα(MC–

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ifλ∈p(M), thenA–λandB–λ∈, and hence (sinceλis isolated inσ(A) andσ(B))

λ∈p(A)∩p(B). This, as above, impliesλ∈p(MC).

The following technical lemma will be required in the sequel.

Lemma . If A is polaroid on(MC)andσ(MC) =σ(M),then(MC)⊆(M).

Proof Evidently, (MC–λ)–()=∅implies (M–λ)–()=∅, andα(MC–λ) <∞implies

α(A–λ) <∞. Letλ∈(MC); thenλ∈isoσ(M). We prove thatα(B–λ) <∞. Suppose to the contrary thatα(B–λ) =∞. Since

(MC–λ)(x⊕y) =

(A–λ)x+Cy⊕(B–λ)y,

either dim(C(B–λ)–_{()) <}_∞ _or _dim(_C₍_B_–_λ₎–_{()) =}_∞_{. If} _dim(_C₍_B_–_λ₎–_{()) <}_∞_, then (sinceα(B–λ) =∞) (B–λ)–_{() contains an orthonormal sequence}_{_y

j}such that

(MC–λ)(⊕yj) =  for allj= , , . . . . But thenα(MC–λ) =∞, a contradiction. Hence

dim(C(B–λ)–_{()) =}_∞_{. Since}_λ_∈_ρ₍_A₎_∪_iso_σ₍_A_{) and}_A_{is (by hypothesis) polar at}_λ_(with, as observed above,α(A–λ) <∞)α(A–λ) =β(A–λ) <∞. Thusdim{C(B–λ)–_()_∩₍_A_– λ)X}=∞, and so there exists a sequence{xj}such that (A–λ)xj=Cyjfor allj= , , . . . .

But then (MC–λ)(xj⊕–yj) =  for allj= , , . . . , and henceα(MC–λ) =∞. This

contra-diction implies that we must haveα(B–λ) <∞. Sinceα(M–λ)≤α(A–λ) +α(B–λ), we

conclude thatλ∈(M).

LetδST∈B(B(X)) denote the generalized derivationδST(X) =SX–XT, and deﬁneδnST

byδ_STn–(δST). The operatorsS,T∈B(X) are said to be asymptotically intertwined by the

identity operator I∈B(X) iflimn→∞δ_STn (I)n _{= ;} _S_,_T _{are said to be}_{quasi-nilpotent} equivalentiflimn→∞δSTn (I)



n₌_lim_n_→∞_δn TS(I)



n _{=  [, p.]. Quasi-nilpotent}

equiv-alence preserves a number of spectral properties [, Proposition ..]. In particular:

Lemma . Quasi-nilpotent equivalent operators have the same spectrum,the same ap-proximate point spectrum and the same surjectivity spectrum.

3 Results

Let K(X) denote the ideal of compact operators inB(X). The following construction, known in the literature as the Sadovskii/Buoni, Harte and Wickstead construction [, p.], leads to a representation of the Calkin algebraB(X)/K(X) as an algebra of op-erators on a suitable Banach space. Let S∈B(X). Let∞(X) denote the Banach space of all bounded sequencesx= (xn)∞n= of elements ofX endowed with the normx∞:= sup_n_∈Nxn, and writeS∞,S∞x:= (Sxn)∞n=for allx= (xn)∞n=, for the operator induced byS on∞(X). The setm(X) of all precompact sequences of elements ofXis a closed subspace of∞(X) which is invariant forS∞. LetXq:=∞(X)/m(X), and denote bySqthe operator S∞onXq. The mappingS→Sqis then a unital homomorphism fromB(X)→B(Xq) with

kernelK(X) which induces a norm decreasing monomorphism fromB(X)/K(X) toB(Xq)

with the following properties (see [, Section ] for details):

(i) Sis upper semi-Fredholm,S∈+, if and only ifSqis injective, if and only ifSqis

bounded below;

(ii) Sis lower semi-Fredholm,S∈–, if and only ifSqis surjective;

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Lemma . For every S∈B(X),σe(S) =σ(Sq),σSF+(S) =σa(Sq)andσSF–(S) =σs(Sq).

Proof The following implications hold:

λ∈/σSF+(S) ⇐⇒ S–λ∈+ ⇐⇒ (S–λ)qis bounded below

⇐⇒ λ∈/σa(Sq),

λ∈/σSF–(S) ⇐⇒ S–λ∈– ⇐⇒ (S–λ)qis onto

⇐⇒ λ∈/σs(Sq) and

λ∈/σe(S) ⇐⇒ S–λ∈ ⇐⇒ (S–λ)qis invertible ⇐⇒ λ∈/σ(Sq).

The following theorem is essentially known [] we provide here an alternative proof, using quasi-nilpotent equivalence and the construction above. Letdenote either ofσe,

σSF+,σSF–,σw,σaw,σsw,σb,σabandσsb.

Theorem . Let S,R∈B(X).If R is a Riesz operator which commutes with S,thenσx(S+ R) =σx(S),whereσx∈.

Proof It is clear from the deﬁnition of a Riesz operatorR∈B(X) thatR–μis Browder (i.e., μ∈/σb(R)), anda-Browder ands-Browder, for all non-zeroμ∈σ(R) (see, for example, [,

Theorem .]). Henceσ(Rq) ={},i.e.,Rq∈B(Xq) is quasi-nilpotent. Lett∈[, ]; then Scommutes withtRand (S+tR)q=Sq+tRq. It follows that

lim

n→∞δ n

(S+tR)qSq(Iq)  n₌ _lim

n→∞δ n

Sq(S+tR)q(Iq)  n_{= ,}

i.e.,SqandSq+tRqare quasi-nilpotent equivalent operators for allt∈[, ]. Thusσx((S+ R)q) =σx(Sq), whereσx=σorσaorσs. Hence

σx(S+R) =σx(S); σx=σeorσaeorσse.

The semi-Fredholm index being a continuous function, we also have from the above that

σx(S+R) =σx(S); σx=σworσaworσsw.

To complete the proof, we prove next thatσb(S+R) =σb(S); the proof forσabandσsbis

similar, and left to the reader. It would suﬃce to prove that ∈σb(S)⇐⇒∈σb(S+R).

Suppose that  /∈σb(S). ThenS∈(andasc(S) =dsc(S) <∞), henceS+tR∈for all t∈[, ]. For an operatorT, letN∞(T) andT∞(X) denote, respectively, the closure of the hyper kernel and the hyper range ofT. ThenN∞(S+tR)∩(S+tR)∞(X) is constant on [, ], and so, sinceN∞(S)∩S∞(X) =N∞(S)∩S∞(X) ={}, it follows thatN∞(S+R)∩ (S+R)∞(X) ={}. Consequently,S+Rhas SVEP at  [, Corollary .]. But then since

S+R∈,S+Ris Browder. ConsideringS= (S+R) –Rproves  /∈σb(S+R)⇒ /∈σb(S).

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Lemma . If S,R∈B(X),and R is a Riesz operator which commutes with S,thenf(S+ R)∩σ(S)⊆isoσ(S).

Let=∪σ∪σa∪σs.

Theorem . Iflimn→∞δABn (C) 

n _{= ,}_then_σ_x₍_M_C_{) =}_σ_x₍_M__),_where_σ_x∈_.

Proof A straightforward calculation shows that

δn_M

CM(I) = –δ n

MMC(I) =

 δn_AB–(C)

 

.

Hence

lim

n→∞δ n MCM(I)

 n₌ _lim

n→∞δ n MMC(I)

 n≤ _lim

n→∞δ n–

AB(C)  n_{= ,}

i.e.,MC andMare quasi-nilpotent equivalent. Similarly, writing MC(q) for (MC)q and M(q)for (M)q,

lim

n→∞δ

n

MC(q)M(q)(Iq)  n ₌ _lim

n→∞δ

n

M(q)MC(q)(Iq)  n

≤ lim

n→∞δ n–

AqBq(Cq)  n

= lim

n→∞δ n–

AB(C)  n _{= ,}

i.e.,MC(q) andM(q) are quasi-nilpotent equivalent (in B((X ⊕X)q)). Hence σx(MC) =

σx(M), whereσx=σ orσaorσsorσeorσSF+orσSF–. Since

M=

A   I I   B

= I 

 B A   I and

MC= I   B I C  I A   I ,

where _I C

I

is invertible, and sinceλ∈/ σe(MC)⇐⇒λ∈/ σe(M)⇒A–λ,B–λ∈ (similarly,λ∈/ σSF+(MC)⇒A–λ,B–λ∈+ andλ∈/ σSF–(MC)⇒A–λ,B–λ∈–),

ind(MC –λ) =ind(A–λ) +ind(B–λ) =ind(M–λ). Hence σx(MC) =σx(M), where σx=σworσaworσsw. Observe that

σb(MC) =

λ∈σ(MC) :λ∈σw(MC) orλ∈/isoσ(MC)

=λ∈σ(M) :λ∈σw(M) orλ∈/isoσ(M)

,

σab(MC) =

λ∈σa(MC) :λ∈σaw(MC) orλ∈/isoσa(MC)

=λ∈σa(M) :λ∈σaw(M) orλ∈/isoσa(M)

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and

σsb(MC) =

λ∈σs(MC) :λ∈σsw(MC) orλ∈/isoσs(MC)

=λ∈σs(M) :λ∈σsw(M) orλ∈/isoσs(M)

[, Corollary ., Theorem . and Theorem .]. Henceσx(MC) =σx(M), whereσx=

σborσaborσsb.

Remark . IfM∈B(X⊕X) is the operatorM=A C_{D B}such that the entriesA,B,Cand

Dmutually commute, thenσx(M) ={λ∈C: ∈σx((A–λ)(B–λ) –CD)}[, Theorem .],

whereσx=σorσe. Dispensing with the mutual commutativity hypothesis and assuming

instead that CD=DC= , Ccommutes with AandB, and lim_n_→∞δn_AB(D)n = , an

argument similar to that used to prove Theorem . shows thatσx(M) =σx(MC), where

σx=σorσaorσsorσeorσSF±.

Theorem . Suppose thatlim_n_→∞δn_AB(C)n= .Then:

(a) MCsatisﬁes Bt if and only ifMsatisﬁes Bt.

(b) LetRi∈B(X),i= , ,be Riesz operators such thatR=R⊕Rcommutes withMC.

ThenMsatisfies Bt⇐⇒MC+Rsatisfies Bt⇐⇒M+Rsatisfies Bt⇐⇒MC

satisﬁes Bt.

Proof The hypothesisRcommutes withMCimpliesRcommutes withM,RC=CRand δn₍_M

C+R)(M+R)(I) =δ n MCM(I).

(a) Recall that an operatorSsatisﬁes Bt if and only ifσw(S) =σb(S). Hence the following

implications hold:

Msatisﬁes Bt ⇐⇒ σw(M) =σb(M)

⇐⇒ σw(Mc) =σb(MC) (Theorem .)

⇐⇒ MCsatisﬁes Bt.

(b) The hypothesislimn→∞δABn (C) 

n _{=  implies that}_M_C₊_R_and_M_₊_R_are

quasi-nilpotent equivalent (⇒by Theorem . thatσx(MC+R) =σx(M+R), whereσx∈).

The operatorRbeing Riesz, Theorem . impliesσx(T+R) =σx(T), whereT=MCorM andσx=σworσb. The (two way) implications

Msatisﬁes Bt ⇐⇒ σw(M) =σb(M) ⇐⇒ σw(M+R) =σb(M+R)

(⇐⇒M+Rsatisﬁes Bt)

⇐⇒ σw(MC+R) =σb(MC+R) ⇐⇒ MC+Rsatisﬁes Bt ⇐⇒ σw(MC) =σb(MC) ⇐⇒ MCsatisﬁes Bt

now complete the proof.

Remark . (i)S∈B(X) satisﬁesa-Browder’s theorem,a-Bt, if and only ifσaw(S) =σab(S)

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S–λ∈+,asc(S–λ) <∞}[, Theorem .]). Theorem . holds with Bt replaced bya-Bt. (Thus, if eitherMorMCsatisﬁesa-Bt, thenM,MC,M+RandMC+Rall satisfya-Bt.)

Furthermore, sinceSsatisfies generalized Browder’s theorem, gBt, if and only if it satisfies Bt andSsatisfies generalizeda-Browder’s theorem,a-gBt, if and only if it satisfiesa-Bt [], Bt may be replaced by gBt ora-gBt in Theorem .. Here, we refer the interested reader to consult [, ] for information about gBt anda-gBt.

(ii) The equivalenceSsatisfies Bt⇐⇒S*_{satisfies Bt is well known. This does not hold} fora-Bt:S satisfies a-Bt does not implyS* _satisfies_a_{-Bt (or}_{vice versa}_{). We say that}_S satisfiess-Bt ifS*_satisfies_a_{-Bt (equivalently, if}_σ

sb(S) =σsw(S)). It is easily seen, we leave

the veriﬁcation to the reader, if eitherMorMCsatisﬁess-Bt, then (in Theorem .)M,

MC,M+RandMC+Rall satisfys-Bt.

We consider next a suﬃcient condition for the equivalence of Weyl’s theorem for opera-torsMandMCsuch thatlimn→∞δABn (C)



n _{= . We say in the following that an operator}

Sis finitely polaroid on a subsetF⊆isoσ(S) if everyλ∈Fis a finite rank pole ofS. Evi-dently,Mis finitely polaroid if and only ifAandBare finitely polaroid.

Theorem . Suppose thatlim_n_→∞δ_ABn (C)n = .

(a) IfAis polaroid,thenMCsatisﬁes Wt if and only ifMsatisﬁes Wt.

(b) LetRi∈B(X),i= , ,be Riesz operators such thatR=R⊕Rcommutes withMC.

A sufficient condition for the equivalenceMC+Rsatisfies Wt⇐⇒M+Rsatisfies Wt is thatMis finitely polaroid.

Proof (a) IfMC satisﬁes Wt, thenσ(MC)\σw(MC) =p(MC) =(MC). Sinceσ(M) = σ(MC) andσw(MC) =σw(M) (Theorem .) and since Wt implies Bt, Theorem .(a) impliesσ(M)\σw(M) =p(M)⊆(M). Consequently,(MC)⊆(M). Letλ∈ (M). Thenλ∈isoσ(MC),α(A–λ) <∞andα(B–λ) <∞. Hence, sinceα(A–λ)≤

α(MC–λ)≤α(A–λ) +α(B–λ),α(MC–λ) <∞. Evidently,λ∈isoσ(A)∪ρ(A). Ifλ∈

isoσ(A), thenApolaroid implies  <α(A–λ), and hence  <α(MC–λ). If insteadλ∈ρ(A),

then –(A–λ)–_Cx_⊕_x_∈₍_M

C–λ)–() for everyx∈(B–λ)–(); once again,  <α(MC–λ).

Consequently,λ∈(MC–λ) =p(MC–λ) =p(M–λ) and hence(M) =p(M)⇒

Msatisﬁes Wt. Conversely, ifMsatisﬁes Wt, thenσ(MC)\σw(MC) =p(MC) =p(M) = (M) =σ(M)\σw(M) and(M)⊆(MC). SinceAis polaroid (hence polar on

(MC)) andσ(M) =σ(MC), Lemma . implies(M) =(MC). ThusMC satisﬁes

Wt.

(b) Start by observing thatσ(M) =σ(MC), and henceMCis ﬁnitely polaroid if and only

ifMis ﬁnitely polaroid (Lemma .). SupposeM+Rsatisﬁes Wt. Then the implication Wt⇒Bt combined with Theorem .(b) implies that bothM+RandMC+Rsatisfy Bt.

As noted in the proof of Theorem .(b),σw(T+R) =σw(T),T=MorMC. Furthermore,

sinceM+RandMC+Rare quasi-nilpotent equivalent,σx(M+R) =σx(MC+R),σx=

σ orσw(Theorem .). Hence

(M+R) =σ(M+R)\σw(M+R) =σ(MC+R)\σw(MC+R)

=p(MC+R)⊆(MC+R).

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thenλ∈isoσ(MC) (Lemma .)⇒λ∈isoσ(M)⇒λ∈p(M) (since Mis finitely polaroid) ⇒λ∈p(MC) (Lemma .)⇒MC–λ∈, and this as above implies λ∈ p(Mc+R). Hence(MC+R) =p(MC+R), andMC +Rsatisfies Wt. The converse, MC+Rsatisfies Wt⇒M+Rsatisfies Wt follows from a similar argument (recall that

MCis ﬁnitely polaroid follows from the hypothesis thatMis ﬁnitely polaroid).

Remark . The equivalence of Theorem .(b) extends to

Msatisfies Bt ⇐⇒ M+Rsatisfies Wt ⇐⇒ MC+Rsatisfies Wt

⇐⇒ MCsatisﬁes Bt.

This is seen as follows. The implicationM+Rsatisﬁes Wt⇒Msatisﬁes Bt andMC+R

satisfies Wt⇒MCsatisfies Bt are clear from Theorem .(b). IfMsatisfies Bt, then the hypothesisMis finitely polaroid impliesMsatisfies Wt. By Theorem .(b),M+R sat-isfies Bt,i.e.,σ(M+R)\σw(M+R) =p(M+R)⊆(M+R). Letλ∈(M+R). Ifλ∈/ σ(M), then (M–λ∈⇒)M+R–λ∈⇒λ∈p(M+R) (sinceλ∈isoσ(M+R)); if λ∈σ(M), thenλ∈isoσ(M) (by Lemma .) and so (sinceMis finitely polaroid) λ∈p(M)⇒M–λ∈⇒M+R–λ∈⇒λ∈p(M+R). Thus, in either case, (M+R)⊆p(M+R), and henceM+R satisfies Wt. The proof forMC satisfies

Bt⇒MC+Rsatisfies Wt is similar: recall from Lemma . thatMfinitely polaroid im-pliesMCfinitely polaroid.

a-Wt.T∈B(X) satisﬁesa-Weyl’s theorem,a-Wt for short, ifTsatisﬁesa-Bt andpa

(T) = a_(T) (equivalently, if σa(T)\σaw(T) =pa(T) =a(T)), wherea(T) ={λ∈isoσa(T) :

 <α(T–λ) <∞}[]. We say in the following thatT isa-polaroid ifT is polar at every λ∈isoσa(T). Trivially,a-polaroid implies polaroid (indeed,pa(T) =p(T) in such a case), but the converse is not true in general. Theorem . has an a-Wt analogue, which we prove below. We note, however, that the perturbation of an operator by a commuting Riesz operator preserves neither its spectrum nor its approximate point spectrum: this will,per se, force us into making an assumption on the approximate point spectrum ofMand

M+Rin the analogue of Theorem .(b).

Theorem . Suppose thatlimn→∞δABn (C)  n _{= .}

(a) IfMisa-polaroid,thenMCsatisﬁesa-Wt if and only ifMsatisﬁesa-Wt.

(b) LetRi∈B(X),i= , ,be Riesz operators such thatR=R⊕Rcommutes withMC.If

σa(M) =σa(M+R),then a sufficient condition for the equivalenceMC+Rsatisfies a-Wt⇐⇒M+Rsatisfiesa-Wt is thatMis finitelya-polaroid.

Proof (a) We provea_(M) =a(MC): the proof of (a) would then follow from the fact

that ifMsatisfiesa-Wt (⇒Msatisfiesa-Bt⇐⇒MCsatisfiesa-Bt), then

a_(M) =σa(M)\σaw(M) =σa(MC)\σaw(MC) =pa(MC)⊆a(MC)

and ifMCsatisﬁesa-Wt, then

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Ifλ∈a_(M), then

λ∈isoσa(M),  <α(M–λ) <∞

⇐⇒ λ∈p(M) (sinceMisa-polaroid)

⇐⇒ λ∈p(A)∪p(B)

∪p(A)∪ρ(B)

∪ρ(A)∪p(B)

⇒ α(MC–λ)≤α(A–λ) +α(B–λ) <∞,

asc(MC–λ)≤asc(A–λ) +asc(B–λ) <∞,

dsc(MC–λ)≤dsc(A–λ) +dsc(B–λ) <∞

⇒ asc(MC–λ) =dsc(MC–λ) <∞,  <α(MC–λ) <∞ ⇒ λ∈p(MC)⊆(MC)⊆a(MC);

if insteadλ∈a_(MC), then

λ∈isoσa(MC),  <α(MC–λ) <∞

⇐⇒ λ∈isoσa(M),  <α(MC–λ) <∞ ⇒ λ∈p(M),  <α(MC–λ) <∞ ⇐⇒ λ∈p(Mc) (Lemma .) ⇐⇒ λ∈p(M) (Lemma .)

⇒ λ∈(M)⊆a(MC).

(b) Ifσa(M+R) =σa(M), then it follows from Lemma . and Theorem . that

σx(M) =σx(M+R) =σx(MC+R) =σx(MC); σx=σaorσaw.

Recall from Remark . that if either ofM+RorMC+Rsatisﬁesa-Bt, thenM,M+R,

MCandMC+Rall satisfya-Bt. Hence, in view of the spectral equalities above,

pa_(M) =pa(MC) =pa(MC+R) =pa(M+R),

whenever either ofM,M+R,MCandMC+Rsatisﬁesa-Bt. Observe that the

hypoth-esis M is ﬁnitely a-polaroid implies pa(M) =p(M) =p(MC) =pa(M+R); hence (since pa

(M) =pa(MC) =pa(MC+R) =pa(M+R))pa(S) =pa(T) for every choice of

S,T=MorMCorM+RorMC+R. We prove now that if either ofM+RandMC+R

satisﬁesa-Wt, thena

(M+R) =a(MC+R): this would then imply that if one satisﬁes a-Wt, then so does the other.

Suppose M+R satisﬁes a-Wt. Then p(M+R) =pa(M+R) =a(M+R) (⇒ a_(M+R) =(M+R)) anda(M+R)⊆a(MC+R). Let λ∈a(Mc+R); then

λ∈isoσa(MC+R) =isoσa(M) impliesλ∈p(M) =pa(MC+R). Thusa(MC+R)⊆ pa

(MC+R) =pa(M+R) =a(M+R). Consequently,a(M+R) =a(MC+R) in this

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and a_(MC +R)⊆a(M+R). Letλ∈a(M+R); thenλ∈isoσa(M) implies λ∈

pa

(M) =pa(MC+R). As above, this impliesa(M+R) =a(MC+R).

The following corollary is immediate from Theorem .(b).

Corollary . Suppose that lim_n_→∞δ_ABn (C)n = . If R_i ∈B(X), i= , , are quasi-nilpotent operators such that R=R⊕Rcommutes with MC,then a sufficient condition for the equivalence MC+R satisfies a-Wt⇐⇒M+R satisfies a-Wt is that Mis finitely

a-polaroid.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally.

Author details

1_{8 Redwood Grove, Northﬁeld Avenue, Ealing, London, W5 4SZ, United Kingdom.}2_{Department of Mathematics} Education, Seoul National University of Education, Seoul, 137-742, Korea. 3_{Department of Mathematics, Incheon National} University, Incheon, 406-772, Korea.

Authors’ information

Work carried out together whilst the ﬁrst author was visiting Korea.

Acknowledgements

This work was supported by the Incheon National University Research Grant in 2012.

Received: 8 February 2013 Accepted: 7 May 2013 Published: 29 May 2013

References

1. Aiena, P: Fredholm and Local Spectral Theory with Applications to Multipliers. Kluwer Academic, Dordrecht (2004) 2. Duggal, BP: SVEP, Browder and Weyl theorems. In: Jiménez, PMA, Bustamante, GJ, Djordjevi´c, SV (eds.) Tópicas de

Theoría de la Approximación. Textos Cientíﬁcos BUAP Puebla, vol. III, pp. 107-146 (2009). http://www.fcfm.buap.mx/CA/analysis-mat/pdf/LIBRO_TOP_T_APPROX.pdf

3. Duggal, BP: Browder and Weyl spectra of upper triangular operator matrices. Filomat24(2), 111-130 (2010). doi:10.2298/FIL1002111D

4. Laursen, KB, Neumann, MN: Introduction to Local Spectral Theory. Clarendon, Oxford (2000) 5. Taylor, AE, Lay, DC: Introduction to Functional Analysis. Wiley, New York (1980)

6. Müller, V: Spectral Theory of Linear Operators. Operator Theory Advances and Applications, vol. 139. Birkhäuser, Basel (2003)

7. Rakoˇcevi´c, V: Semi-Browder operators and perturbations. Stud. Math.122, 131-137 (1997) 8. Oudghiri, M: Weyl’s theorem and perturbations. Integral Equ. Oper. Theory53, 535-545 (2005) 9. Harte, RE: Invertibility and singularity of operator matrices. Proc. R. Ir. Acad. A88(2), 103-118 (1988)

10. Amouch, M, Zguitti, H: On the equivalence of Browder’s and generalized Browder’s theorem. Glasg. Math. J.48, 179-185 (2006)

doi:10.1186/1029-242X-2013-268