AN ELEMENTARY OPERATOR
B. P. DUGGALReceived 25 July 2004
A range-kernal orthogonality property is established for the elementary operatorsᏱ(X)=
n
i=1AiXBiandᏱ∗(X)= n
i=1A∗i XB∗i, whereA=(A1,A2,...,An) andB=(B1,B2,...,Bn)
aren-tuples of mutually commuting scalar operators (in the sense of Dunford) in the al-gebraB(H) of operators on a Hilbert spaceH. It is proved that the operatorᏱsatisfies Weyl’s theorem in the case in whichAandBaren-tuples of mutually commuting gener-alized scalar operators.
1. Introduction
For a Banach space operatorT,T∈B(ᐄ), the kernelT−1(0) and the rangeT(ᐄ) are said to have ak-gap for some real numberk≥1, denotedT−1(0)⊥kT(ᐄ), if
y∈T−1(0)=⇒ y ≤kdisty,T(ᐄ) (1.1)
[8, Definition, page 94]. Recall from [10, page 93] that a subspaceᏹof the Banach space ᐄisorthogonalto a subspaceᏺofᐄifm ≤ m+nfor allm∈ᏹandn∈ᏺ. This def-inition of orthogonality coincides with the usual defdef-inition of orthogonality in the case in whichᐄ=His a Hilbert space. A 1-gap betweenT−1(0) andT(ᐄ) corresponds to the range-kernel orthogonalityfor the operatorT (see [1,2,8,14]). The following implica-tions are straightforward to see
T−1(0)⊥
kT(ᐄ)=⇒T−1(0)∩T(ᐄ)= {0} =⇒T−1(0)∩T(ᐄ)= {0} =⇒asc(T)≤1,
(1.2)
whereT(ᐄ) denotes the closure of T(ᐄ) and asc(T) denotes theascent ofT. Ak-gap betweenT−1(0) and T(ᐄ) does not imply thatT(ᐄ) is closed, or even whenT(ᐄ) is closed thatᐄ=T−1(0)⊕T(ᐄ) (see, e.g., [1,2,23]).
The classical Putnam-Fuglede commutativity theorem says that ifAandBare nor-mal Hilbert space operators,A andB∈B(H), and ifδAB∈B(B(H)) is thegeneralized
derivationδAB(X) :=AX−XB, thenδAB−1(0)=δ−1A∗B∗(0). Extant literature contains various
Copyright©2005 Hindawi Publishing Corporation
generalizations of the Putnam-Fuglede theorem, amongst them the two n-tuples A=
(A1,A2,...,An) andB=(B1,B2,...,Bn) of mutually commuting normal (Hilbert space)
operatorsAiandBi, 1≤i≤n. LetᏱ∈B(B(H)) be theelementary operator
Ᏹ(X) :=n
i=1
AiXBi. (1.3)
If asc(Ᏹ)≤1, thenᏱ−1(0)=Ᏹ−1
∗(0), whereᏱ∗(X) :=
n
i=1A∗iXBi∗(see [21,22,24]). The
conclusionᏱ−1(0)⊆Ᏹ−1
∗ (0) fails if asc(Ᏹ)>1 [21]; moreover, in such a case it may hap-pen that Ᏹ−1(0)∩Ᏹ(B(H))= {0}[22]. For 2-tuplesAandB of mutually commuting normal operators it is always the case that asc(Ᏹ)≤1 (see [14] or [8]).
This paper considersn-tuplesAandBof mutually commutingscalar operators(in the sense of Dunford and Schwartz [10])AiandBi, 1≤i≤n, to prove that the operatorᏱµ:=
(Ᏹ−µI)∈B(B(H)) satisfies: (i) there exists a complex numberλ=αexpiθ,α >0 and 0≤θ <2π, such that ifᏱ−1λ (0)= {0}, thenᏱ−1
λ (0)⊥kᏱλ(B(H)) andᏱ−1∗λ(0)⊥kᏱ∗λ(B(H)),
whereᏱ∗λ=(Ᏹ∗−λI). Furthermore, if the operatorsAiandBiin then-tuplesAandB
are normal, then (ii)Ᏹ−1λ (0)=Ᏹ−1
∗λ(0). This compares with the fact that the operatorᏱ
may fail to satisfy thek-gap property of (i) or the Putnam-Fuglede-theorem-type com-mutativity property of (ii). However, if we restrict the lengthnof the n-tuples Aand
Bton=1 (resp., n=2), then both (i) and (ii) hold for all complex numbersλ[7,9] (resp.,λ=0 andλ=αexpiθfor some real numberα >0; see [7] andTheorem 2.4infra). Our proof of (i) and (ii) makes explicit the relationship between the existence of ak-gap between the kernel and the range of the operatorᏱλ, and the Putnam-Fuglede commuta-tivity property forn-tuplesAandBconsisting of mutually commuting normal operators. Letting then-tuplesAandBconsist of mutually commutinggeneralized scalar operators (in the sense of Colojoar˘a and Foias¸ [5]), it is proved that (i) a sufficient condition for Ᏹλ(B(ᐄ)) to be closed is that the complex numberλis isolated in the spectrum ofᏱ; (ii)
f(Ᏹ) and f(Ᏹ∗) satisfy Weyl’s theorem for every analytic function f defined on a neigh-borhood of the spectrum ofᏱ, and the conjugate operatorᏱ∗satisfiesa-Weyl’s theorem. These results will be proved in Sections2and3, but before that, we explain our notation and terminology.
The ascent ofT∈B(ᐄ), asc(T), is the least nonnegative integernsuch thatT−n(0)=
T−(n+1)(0) and the descent of T, dsc(T), is the least nonnegative integer n such that Tn(ᐄ)=Tn+1(ᐄ). We say thatT−λis of finite ascent (resp., finite descent) if asc(T− λI)<∞(resp., dsc(T−λI)<∞). Thenumerical range ofTis the closed convex set
WB(ᐄ),T=f(T) : f ∈B(ᐄ)∗,f =f(I)=1 (1.4)
of the setCof complex numbers (see [3]). Aspectral operator(in the sense of Dunford) is an operator with a countable additive resolution of the identity defined on the Borel sets ofC; a spectral operatorT is said to bescalar typeif it satisfiesT=λE(dλ), where Eis the resolution of the identity forT [11, page 1938]. IfA=(A1,A2,...,An) is an
n-tuple of mutually commuting scalar operators inB(H), then there exists an invertible self-adjoint operatorSsuch thatS−1AiS=Miis a normal operator for alli=1, 2,...,n
algebra homomorphismΦ:C∞→B(ᐄ) for whichΦ(1)=IandΦ(Z)=T, whereC∞(C) is the Fr´echet algebra of all infinitely differentiable functions onC(endowed with its usual topology of uniform convergence on compact sets for the functions and their partial derivatives) andZis the identity function onC(see [5,16]). We will denote the spectrum and theisolated points of the spectrumofTbyσ(T) and isoσ(T), respectively. The closed unit disc inCwill be denoted byD, and∂Dwill denote the boundary of the unit discD. The operator ofleft multiplication by T (right multiplication by T) will be denoted by LT (resp.,RT). It is clear that [LS,RT]=0 for allS,T∈B(ᐄ), where [LS,RT] denotes the
commutatorLSRT−RTLS. We will henceforth shorten (T−λI) to (T−λ).
An operatorT∈B(ᐄ) is said to be Fredholm,T∈Φ(ᐄ), ifT(ᐄ) is closed and both thedeficiency indicesα(T)=dim(T−1(0)) andβ(T)=dim(ᐄ/T(ᐄ)) are finite, and then theindexofT, ind(T), is defined to be ind(T)=α(T)−β(T). The operatorT isWeyl if it is Fredholm of index zero. The (Fredholm) essential spectrumσe(T) and the Weyl
spectrumσw(T) ofTare the sets
σe(T)= {λ∈C:T−λis not Fredholm},
σw(T)= {λ∈C:T−λis not Weyl}. (1.5)
Letπ0(T) denote the set ofRiesz pointsofT(i.e., the set ofλ∈Csuch thatT−λis Fred-holm of finite ascent and descent [4]), and letπ00(T) denote the set of isolated eigenvalues ofTof finite geometric multiplicity. Also, letπa0(T) be the set ofλ∈Csuch thatλis an isolated point ofσa(T) and 0<dim ker(T−λ)<∞, whereσa(T) denotes the approximate
point spectrum of the operatorT. Clearly,π0(T)⊆π00(T)⊆πa0(T). We say thatWeyl’s theorem holds forTif
σ(T)\σw(T)=π00(T), (1.6)
anda-Weyl’s theorem holds forTif
σea(T)=σa(T)\πa0(T), (1.7)
whereσea(T) denotesthe essential approximate point spectrum (i.e.,σea(T)= ∩{σa(T+
K) :K ∈K(ᐄ)} with K(ᐄ) denoting the ideal of compact operators onᐄ). If we let Φ+(ᐄ)= {T ∈B(ᐄ) :α(T)<∞andT(ᐄ) is closed} denote the semigroup of upper semi-Fredholm operatorsinB(ᐄ) and letΦ−+(ᐄ)= {T∈Φ+(ᐄ) : ind(T)≤0}, thenσea(T)
is the complement inCof all thoseλfor which (T−λ)∈Φ−
+(ᐄ). The concept ofa-Weyl’s theorem was introduced by Rakoˇcevi´c:a-Weyl’s theorem forT implies Weyl’s theorem forT, but the converse is generally false [20].
An operatorT∈B(ᐄ) hasthe single-valued extension property(SVEP) atλ0∈Cif for every open discᏰλ0centered atλ0, the only analytic function f :Ᏸλ0→ᐄwhich satisfies
(T−λ)f(λ)=0 ∀λ∈Ᏸλ0 (1.8)
defined, respectively, by
H0(T)= x∈ᐄ: lim
n−→∞T
n(x)1/n=0,
K(T)=x∈ᐄ:∃a sequencexn
⊂ᐄ,δ >0 for whichx=x0,T
xn+1
=xn,xn≤δnx ∀n=1, 2,...
.
(1.9)
We note thatH0(T−λ) andK(T−λ) are (generally) nonclosed hyperinvariant subspaces ofT−λsuch that (T−λ)−q(0)⊆H0(T−λ) for allq=0, 1, 2,...and (T−λ)K(T−λ)=
K(T−λ) [18]. IfThas SVEP atλ, thenH0(T−λ) andK(T−λ) are closed. The operator T∈B(ᐄ) is said to besemiregularifT(ᐄ) is closed andT−1(0)⊂T∞(ᐄ)= ∩n∈NTn(ᐄ);
T admits ageneralized Kato decomposition, (GKD), if there exists a pair of T-invariant closed subspaces (ᏹ,ᏺ) such thatᐄ=ᏹ⊕ᏺ, the restrictionT|ᏹis quasinilpotent and T|ᏺ is semiregular. An operatorT∈B(ᐄ) has a (GKD) at everyλ∈isoσ(T), namely ᐄ=H0(T−λ)⊕K(T−λ). We say thatT−λis ofKato typeif (T−λ)|ᏹis nilpotent in theGKDforT−λ. Fredholm operators are Kato type [13, Theorem 4], and operators T∈B(ᐄ) satisfying the following property:
H(p)H0(T−λ)=(T−λ)−p(0),
for some integerp≥1, are Kato type at isolated points ofσ(T) (but not every Kato type operatorTsatisfies propertyH(p)).
2.k-gap and the Putnam-Fuglede theorem
Let, as before,A=(A1,A2,...,An) andB=(B1,B2,...,Bn) ben-tuples of mutually
com-muting scalar operators, and letᏱλandᏱ∗λ denote the elementary operatorsᏱλ(X)=
n
i=1AiXBi−λX andᏱ∗λ(X)= n
i=1A∗i XB∗i −λX. We say in the following that the
n-tupleAisnormally constitutedifAiis normal for all 1≤i≤n.
The following theorem is the main result of this section.
Theorem 2.1. (i) There exists a real numberα >0 such that if(0=)X∈Ᏹ−1
α (0), then
X ≤kᏱα(Y) +XandX ≤kᏱ∗α(Y) +Xfor some real numberk≥1and allY∈
B(H).
Furthermore, ifAandBare normally constituted, then (ii)Ᏹ−1α (0)=Ᏹ−1∗α(0).
Proof. There exist invertible self-adjoint operatorsT1,T2∈B(H) and normal operators Mi,Ni∈B(H), 1≤i≤n, such that Mi=T1−1AiT1, Ni=T2−1BiT2, and [Mi,Mj]=0=
[Ni,Nj] for all 1≤i,j≤n. Define scalarsα1andα2 byα1= ni=1Mi∗Mi1/2 andα2= n
i=1Ni∗Ni1/2, define the scalar α by α=√α1α2, and define the operators Ci and
Di, 1≤i≤n, byCi=(1/√α1)MiandDi=(1/√α2)Ni. Then (C1,C2,...,Cn) and (D1,D2, ...,Dn) aren-tuples of mutually commuting normal operators. LetE(X)=ni=1CiXDi.
Set
U=C1 C2 ··· Cn
, V=D1 D2 ··· Dn
t
where [···]tdenotes the transpose of the row matrix [···]. ThenUandVare
contrac-tions. RepresentingE(X) by
E(X) :=UXIn
V, (2.2)
whereIndenotes the identity ofMn(C), it then follows thatEis a contraction. Hence,
WBB(H),E⊆D. (2.3)
Forµ∈C, letEµdenote the operatorEµ=E−µ. Then
WBB(H),E1
⊆λ∈C:|λ+ 1| ≤1. (2.4)
In particular, 0∈∂W(B(B(H)),E1). Notice that if 0 is an eigenvalue ofᏱα, then 0 is an
eigenvalue ofE1. It follows from Sinclair [23, Proposition 1] that
X ≤E1(Y) +X (2.5)
for allX∈E1−1(0) andY∈B(H). In particular, asc(E1)≤1, which by a result of Shulman [21] implies thatE−11 (0)⊆E−1∗1(0) (whereE∗1∈B(B(H)) is the operatorE∗1=E∗−1 : X→n
i=1C∗i XD∗i −X). RepresentingE∗by E∗(X)=U1
XIn
V1, (2.6)
where
U1=C1∗ C∗2 ··· Cn∗
, V1=D1∗ D∗2 ··· Dn∗
t, (2.7)
it follows thatE∗is a contraction and 0 is an eigenvalue ofE∗1 in∂W(B(B(H)),E∗1). Hence,
X ≤E∗1(Y) +X (2.8)
for allX∈E−1∗1(0) andY∈B(H), which implies thatE∗1−1(0)⊆E−11 (0). Hence,E1−1(0)= E−1∗1(0). The proof of (ii) is now a consequence of the observation that
X∈E−11 (0)⇐⇒
n
i=1
MiXNi−αX=0⇐⇒X∈E−1∗1(0)⇐⇒
n
i=1
Mi∗XNi∗−αX=0. (2.9)
To prove (i), we letT1T1−1T2T2−1 =k. Since αX ≤αE1(Y) +X
=T1−1
n
i=1 Ai
T1YT2−1
Bi−αT1YT2−1+αT1XT2−1
T2
=⇒αT1XT2−1≤αT1T2−1X ≤kᏱα
T1YT2−1
+αT1XT2−1
for all X∈E−1
1 (0) and Y ∈B(H) (equivalently, all T1XT2−1∈Ᏹ−1α (0) and T1YT2−1∈ B(H)), it follows thatᏱ−1α (0)⊥kᏱα(B(H)). A similar argument, applied this time toX ≤
E∗1(Y) +X, implies thatᏱ∗−1α(0)⊥kᏱ∗α(B(H)). This completes the proof of the
theo-rem.
The following corollary is an immediate consequence of the fact that asc(Ᏹα)≤1.
Corollary2.2. The range ofᏱαis closed if and only ifᏱ−α1(0) +Ᏹα(B(H))is closed.
For the proof see [16, Proposition 4.10.4].
The pointαofTheorem 2.1is not unique. Since 0∈∂W(B(B(H)),Eµ) for everyµ∈
C such that|µ| =1, the argument of the proof ofTheorem 2.1 implies the following theorem.
Theorem2.3. (i)There exists a complex numberλ=αexpiθ,α >0and0≤θ <2π, such that if(0=)X∈Ᏹ−1
λ (0), thenX ≤kᏱλ(Y) +XandX ≤kᏱ∗λ(Y) +Xfor some
real numberk≥1and allY∈B(H).
Furthermore, ifAandBare normally constituted, then (ii)Ᏹ−1λ (0)=Ᏹ−1
∗λ(0)for allλas in part (i).
Let the Hilbert spaceH be separable, and letᏯp denote the von Neumann-Schatten
p-class, 1≤p <∞, with norm · p. ThenTheorem 2.3has the followingᏯpversion.
Theorem2.4. (i)There exists a complex numberλ=αexpiθ,α >0and0≤θ <2π, such that if(0=)X∈Ᏹ−1
λ (0)∩Ꮿp, thenXp≤kᏱλ(Y) +XpandXp≤kᏱ∗λ(Y) +Xp
for some real numberk≥1and allY∈Ꮿp.
Furthermore, ifAandBare normally constituted, then (ii)Ᏹ−1λ (0)∩Ꮿp=Ᏹ−1∗λ(0)∩Ꮿpfor allλas in part (i).
Proof. Define the real numbersαi,i=1, 2, as in the proof ofTheorem 2.1, define the
normal operatorsCi and Di byCi=(1/
α1n1/2p)Mi andDi=(1/
α2n1/2p)Ni. Letα=
α1α2n1/ p. ThenE∈B(Ꮿp) is a contraction. Now argue as in the proof ofTheorem 2.1.
As we will see in the following section,H0(Ᏹλ)=Ᏹ−λp(0) for allλ∈Cand some integer
p≥1 (i.e.,Ᏹλsatisfies propertyH(p)), which implies that asc(Ᏹλ)≥1 for allλ∈C. (Here,
as also elsewhere, the statement asc(T)≥1 is to be taken to subsume the hypothesis that Tis not injective.) However, if then-tuplesAandBare of lengthn=1, then asc(Ᏹλ)≤1
for allλ∈Cand for a number of classes of not necessarily scalar or normal operators A1 and B1 (see [7,9]). Ifn=2 andB1=A2=I, then asc(Ᏹλ)≤1 (once again for A1 andB2belonging to a number of classes of operators more general than the class of scalar operators [7]). Again, ifn=2, then asc(Ᏹλ)≤1 forλ=0 andλ=αexpiθ, as follows from Theorem 2.1and the following argument. Define the normal operatorsMi andNi,i=
1, 2, as in the proof ofTheorem 2.1. Then [M1,M2]=[N1,N2]=0. Defineφ∈B(B(H)) byφ(X)=M1XN1+M2XN2. Then φ−1(0)⊥kφ(B(H)) (see [14] or [8]), which implies
that asc(φ)=asc(Ᏹ)≤1. The following corollary, which generalizes [14, Theorem 2], is now obvious.
numberλis as inTheorem 2.3, thenasc(Ᏹµ)≤1, andᏱ−1µ (0)⊥kᏱµ(B(H))forµ=0,λ.
Fur-thermore, ifAandBare normally constituted, thenᏱ−1
µ (0)=Ᏹ−∗1µ(0)forµ=0,λ.
Perturbation by quasinilpotents. Recall that everyspectral operatorT∈B(ᐄ) is the sum T=S+Qof a scalar type operatorSand a quasinilpotent operatorQsuch that [S,Q]=0 [11]. LetA=(J1,J2) andB=(K1,K2) be tuples of operators inB(H) such thatJi=Ai+Qi
and Ki=Bi+Ri, i=1, 2, for some scalar operators Ai, Bi and quasinilpotent
opera-torsQi,Ri. If we define E∈B(B(H)) by E(X)=J1XK1+J2XK2, then E(X)=Ᏹ(X) + φ(X), whereᏱ(X) is defined as inCorollary 2.5andφ(X)=A1XR1+A2XR2+Q1XB1+ Q2XB2+Q1XR1+Q2XR2. Recall that the sum of two commuting quasinilpotent oper-ators, as well as the product of two commuting operators one of which is quasinilpo-tent, is quasinilpotent [5, Lemma 3.8, Chapter 4]. Representing the operatorX→SXTby X→LSRT(X), where (S,T) denotes any of the operator pairs (Ai,Ri), (Qi,Bi), or (Qi,Ri),
i=1, 2, and assuming that the operators in the sets{A1,A2,Q1,Q2}and{R1,R2,B1,B2} mutually commute, it follows that the operatorφis quasinilpotent.
Theorem2.6. Let the operatorEbe defined as above. If the operators in the sets{A1,A2, Q1,Q2}and{R1,R2,B1,B2}mutually commute, thenX∈E−1(0)⇒X∈Ᏹ−1(0).
Proof. LetX∈E−1(0). The hypothesis that the operators in the sets{A1,A2,Q1,Q2}and {R1,R2,B1,B2}mutually commute then implies that
−φ(X)=E(X)=T1
M1
T1−1XT2
N1+M2
T1−1XT2
N2
T2−1, (2.11)
where the operatorφis quasinilpotent, and where the normal operators Mi,Ni, [M1, M2]=0=[N1,N2], and the invertible operatorsTi,i=1, 2, are defined as in the proof of
Theorem 2.1. DefineΦ∈B(B(H)) byΦ(Y)=M1YN1+M2YN2. Since the operatorφis quasinilpotent,
lim
n→∞Φn
T1−1XT2 1/n≤
T1−1T2lim
n→∞φn(X) 1/n=
0. (2.12)
As earlier remarked upon,H0(Φ)=Φ−p(0) for some integerp≥1. Since asc(Φ)≤1 (by Corollary 2.5), it follows thatΦ(T1−1XT2)=0. HenceX∈Ᏹ−1(0).
3. Weyl’s theorem
IfA,B∈B(ᐄ) are generalized scalar operators, thenLA,RB∈B(B(ᐄ)) are commuting
generalized scalar operators with two commuting spectral distributions, which implies thatLARB andLA+RB are generalized scalar operators (see [5, Theorem 3.3,
Proposi-tion 4.2, Theorem 4.3, Chapter 4]). LetA=(A1,A2,...,An) andB=(B1,B2,...,Bn) be
scalar operator. A finitely repeated application of this argument implies thatEλis a
gen-eralized scalar operator for allλ∈C. Thus
H0
Eλ
=E−λp(0) (3.1)
for some integer p≥1 and all λ∈C see [5, Theorem 4.5, Chapter 4]. In particular, asc(Eλ)≤p <∞for allλ∈CandE(=E0) has SVEP.
The following proposition will be required in the proof of our main result. Proposition3.1. (a)The following conditions are equivalent:
(i)λ∈isoσ(E);
(ii)λis a pole of orderpof the resolvent ofE; (iii) dsc(Eλ)<∞;
(iv)Eλis Kato type and (in the definition of Kato type) the subspaceᏺ⊆Eλ(B(ᐄ)).
(b)IfE∗denotes the conjugate operator ofE, thenσw(E∗)=σw(E),π00(E∗)=π00(E)= π0(E)=π0(E∗), andλ∈π00(E)⇒Eλ∈Φ(B(ᐄ)), andind(Eλ)=0.
Proof. (a) (i)⇒(ii). Ifλ∈isoσ(E), thenB(ᐄ)=H0(Eλ)⊕K(Eλ)=Eλ−p(0)⊕K(Eλ) for
some integerp≥1. But thenE−λp(0) is complemented by the closed subspaceK(Eλ)⊆ Eλ(B(ᐄ))⇒K(Eλ)=Eλp(B(ᐄ)) [15, Theorem 3.4]. Henceλis a pole of the resolvent ofE.
(ii)⇒(iii). The implication is obvious.
(iii)⇒(iv). If dsc(Eλ)<∞, then we have the following implications:
H0
Eλ
=E−λp(0), ∀λ∈C, =⇒ascEλ
=dscEλ
≤p <∞, [16, Proposition 4.10.6], =⇒B(ᐄ)=E−λp(0)⊕EλpB(ᐄ)=ᏹ⊕ᏺ
=⇒Eλis Kato type, ᏺ⊆EλB(ᐄ).
(3.2)
(iv)⇒(i). IfEλ is Kato type, then B(ᐄ)=ᏹ⊕ᏺ, whereEλ|ᏹis nilpotent andEλ|ᏺ is
semiregular. SinceE−λn(0)⊆ᏹ⊆H0(Eλ)=E−λp(0) for all nonnegative integersn, and the
closed subspaceᏺ⊆Eλ(B(ᐄ)),λ∈iso(E) [15, Theorem 3.2].
(b) The following implications hold:
λ /∈σw
E∗⇐⇒E∗λ ∈ΦB(ᐄ)∗, indE∗λ=0, ⇐⇒Eλ∈ΦB(ᐄ), indEλ)=0, ⇐⇒λ /∈σw(E.
(3.3)
Henceσw(E)=σw(E∗). Again,
λ∈isoσE∗⇐⇒λ∈isoσ(E)
⇐⇒B(ᐄ)=E−λp(0)⊕EλpB(ᐄ)⇐⇒λ∈π0(E) ⇐⇒B(ᐄ)∗=E∗λ−p(0)⊕E∗λpB(ᐄ)∗
⇐⇒λ∈π0
E∗.
Recall that if the ascent and the descent of an operatorTare finite, and either 0< α(T)< ∞or 0< β(T)<∞, then asc(T)=dsc(T)<∞and 0< α(T)=β(T)<∞[12, Proposition 38.6]. Henceπ00(E∗)=π00(E)=π0(E)=π0(E∗), andλ∈π00(E)⇒Eλ∈Φ(B(ᐄ)) with
ind(Eλ)=0.
It is evident fromProposition 3.1(a) that a sufficient condition forEλto have closed
range is thatλ∈isoσ(E).Proposition 3.1(b) implies that bothEandE∗ satisfy Weyl’s theorem: more is true. LetH(σ(E)) denote the set of functions f which are defined and analytic on an open neighborhood ofσ(E).
Theorem3.2. (a)f(E)and f(E∗)satisfy Weyl’s theorem for every f ∈H(σ(E)). (b)E∗satisfiesa-Weyl’s theorem.
Proof. (a) A proof follows from [19, Theorem 3.1]. Alternatively, one argues as follows. If we letEdenote either ofEorE∗, thenσ(f(E))=σ(f(E)) andσw(f(E))=σw(f(E)).
SinceEis isoloid (i.e., isolated points ofEare eigenvalues ofE) and Weyl’s theorem holds forE(byProposition 3.1), f(σw(E))= f(σ(E)\π00(E))=σ(f(E))\π00(f(E)) [17, lemma] and f(σw(E))=σw(f(E)) [6, Corollary 2.6]. (We note here that although
[17, lemma] is stated for a Hilbert space, it equally holds in the setting of a Banach space.) Hence, since f(E) satisfies propertyH(p), then [19, Theorem 3.4] implies (by Proposition 3.1) that Weyl’s theorem holds for f(E),σ(f(E))\σw(f(E))=π00(f(E)). (b) The operator Ehas SVEP and the operator E∗ satisfies Weyl’s theorem; hence σ(E∗)=σa(E∗) [16, page 35] andσa(E∗)\σw(E∗)=πa0(E∗). We prove that σea(E∗)⊇
σw(E∗): sinceσea(E∗)⊆σw(E∗) always, this would complete the proof. If λ /∈σea(E∗),
then E∗λ ∈Φ+(B(ᐄ)∗) and ind(E∗λ)≤0⇔Eλ ∈Φ−(B(ᐄ)) and ind(Eλ)≥0, where Φ−(B(ᐄ))= {T∈B(B(ᐄ)) :β(T)<∞}. Since asc(Eλ)<∞, ind(Eλ)≤0. Henceα(Eλ)=
β(Eλ)<∞ and asc(Eλ)=dsc(Eλ)<∞ [12, Proposition 38.6], which implies that λ /∈
σw(E∗).
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B. P. Duggal: Department of Mathematics and Computer Science, Faculty of Science, United Arab Emirates University, P.O. Box 17551, Al-Ain, United Arab Emirates
