# A Note on Property(gb) and Perturbations

Authors: Qingping Zeng; Huaijie Zhong
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

An operatorT∈ℬ(X) defined on a Banach space X satisfies property (gb) if the complement in the approximate point spectrum σa(T) of the upper semi-B-Weyl spectrum σSBF+-(T) coincides with the set Π(T) of all poles of the resolvent of T. In this paper, we continue to study property (gb) and the stability of it, for a bounded linear operator T acting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, and by quasinilpotent operators commuting with T. Two counterexamples show that property (gb) in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.


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## Body

## 1. Introduction

Throughout this paper, letℬ(X) denote the Banach algebra of all bounded linear operators acting on an infinite-dimensional complex Banach space X, and let ℱ(X) denote its ideal of finite rank operators on X. For an operator T∈ℬ(X), let T* denote its dual, 𝒩(T) its kernel, α(T) its nullity, ℛ(T) its range, β(T) its defect, σ(T) its spectrum, and σa(T) its approximate point spectrum. If the range ℛ(T) is closed and α(T)<∞ (resp., β(T)<∞), then T is said to be upper semi-Fredholm (resp., lower semi-Fredholm). If T∈ℬ(X) is both upper and lower semi-Fredholm, then T is said to be Fredholm. If T∈ℬ(X) is either upper or lower semi-Fredholm, then T is said to be semi-Fredholm, and its index is defined by ind(T)=α(T)-β(T). The upper semi-Weyl operators are defined as the class of upper semi-Fredholm operators with index less than or equal to zero, while Weyl operators are defined as the class of Fredholm operators of index zero. These classes of operators generate the following spectra: the Weyl spectrum defined by
(1.1)σW(T):={λ∈C:T-λIis not a Weyl operator},
the upper semi-Weyl spectrum (in the literature called also Weyl essential approximate point spectrum) defined by
(1.2)σSF+-(T):={λ∈C:T-λIis not an upper semi-Weyl operator}.Recall that thedescent and the ascent of T∈ℬ(X) are dsc(T)=inf{n∈ℕ:ℛ(Tn)=ℛ(Tn+1)} and asc(T)=inf{n∈ℕ:𝒩(Tn)=𝒩(Tn+1)}, respectively (the infimum of an empty set is defined to be ∞). If asc(T)<∞ and ℛ(Tasc(T)+1) is closed, then T is said to be left Drazin invertible. If dsc(T)<∞ and ℛ(Tdsc(T)) is closed, then T is said to be right Drazin invertible. If asc(T)=dsc(T)<∞, then T is said to be Drazin invertible. Clearly, T∈ℬ(X) is both left and right Drazin invertible if and only if T is Drazin invertible. An operator T∈ℬ(X) is called upper semi-Browder if it is an upper semi-Fredholm operator with finite ascent, while T is called Browder if it is a Fredholm operator of finite ascent and descent. The Browder spectrum of T∈ℬ(X) is defined by
(1.3)σB(T):={λ∈C:T-λIisnotaBrowderoperator},
the upper semi-Browder spectrum (in the literature called also Browder essential approximate point spectrum) is defined by
(1.4)σUB(T):={λ∈C:T-λIis not an upper semi-Browder operator}.An operatorT∈ℬ(X) is called Riesz if its essential spectrum σe(T):={λ∈ℂ:T-λIisnotFredholm}={0}.Suppose thatT∈ℬ(X) and that R∈ℬ(X) is a Riesz operator commuting with T. Then it follows from [1, Proposition 5] and [2, Theorem 1] that
(1.5)σSF+-(T+R)=σSF+-(T),σW(T+R)=σW(T),σUB(T+R)=σUB(T),σB(T+R)=σB(T).For each integern, define Tn to be the restriction of T to ℛ(Tn) viewed as the map from ℛ(Tn) into ℛ(Tn) (in particular T0=T). If there exists n∈ℕ such that ℛ(Tn) is closed and Tn is upper semi-Fredholm, then T is called upper semi-B-Fredholm. It follows from [3, Proposition 2.1] that if there exists n∈ℕ such that ℛ(Tn) is closed and Tn is upper semi-Fredholm, then ℛ(Tm) is closed, Tm is upper semi-Fredholm, and ind(Tm) = ind(Tn) for all m≥n. This enables us to define the index of an upper semi-B-Fredholm operator T as the index of the upper semi-Fredholm operator Tn, where n is an integer satisfying that ℛ(Tn) is closed and Tn is upper semi-Fredholm. An operator T∈ℬ(X) is called upper semi-B-Weyl if T is upper semi-B-Fredholm and ind(T)≤0.ForT∈ℬ(X), let us define the left Drazin spectrum, the Drazin spectrum, and the upper semi-B-Weyl spectrum of T as follows, respectively:
(1.6)σLD(T):={λ∈C:T-λIisnotaleftDrazininvertibleoperator};σD(T):={λ∈C:T-λIisnotaDrazininvertibleoperator};σSBF+-(T):={λ∈C:T-λIisnotanuppersemi-B-Weyloperator}.LetΠ(T) denote the set of all poles of T. We say that λ∈σa(T) is a left pole of T if T-λI is left Drazin invertible. Let Πa(T) denote the set of all left poles of T. It is well know that Π(T)=σ(T)∖σD(T)=isoσ(T)∖σD(T) and Πa(T)=σa(T)∖σLD(T)=isoσa(T)∖σLD(T). Here and henceforth, for A⊆ℂ, isoA is the set of isolated points of A. An operator T∈ℬ(X) is called a-polaroid if isoσa(T)=∅ or every isolated point of σa(T) is a left pole of T.Following Harte and Lee [4], we say that T∈ℬ(X) satisfies Browder's theorem if σW(T)=σB(T), while, according to Djordjević and Han [5], we say that T satisfies a-Browder's theorem if σSF+-(T)=σUB(T).The following two variants of Browder's theorem have been introduced by Berkani and Zariouh [6] and Berkani and Koliha [7], respectively.Definition 1.1.
An operatorT∈ℬ(X) is said to possess property (gb) if
(1.7)σa(T)∖σSBF+-(T)=Π(T),
while T∈ℬ(X) is said to satisfy generalized a-Browder's theorem if
(1.8)σa(T)∖σSBF+-(T)=Πa(T).
From formulas (1.5), it follows immediately that Browder's theorem and a-Browder's theorem are preserved under commuting Riesz perturbations. It is proved in [8, Theorem 2.2] that generalized a-Browder's theorem is equivalent to a-Browder's theorem. Hence, generalized a-Browder's theorem is stable under commuting Riesz perturbations. That is, if T∈ℬ(X) satisfies generalized a-Browder's theorem and R is a Riesz operator commuting with T, then T+R satisfies generalized a-Browder's theorem.
The single-valued extension property was introduced by Dunford in [9, 10] and has an important role in local spectral theory and Fredholm theory, see the recent monographs [11] by Aiena and [12] by Laursen and Neumann.Definition 1.2.
An operatorT∈ℬ(X) is said to have the single-valued extension property at λ0∈ℂ (SVEP at λ0 for brevity) if for every open neighborhood U of λ0 the only analytic function f:U→X which satisfies the equation (λI-T)f(λ)=0 for all λ∈U is the function f(λ)≡0.
LetS(T):={λ∈ℂ:TdoesnothavetheSVEPatλ}. An operator T∈ℬ(X) is said to have SVEP if S(T)=∅.
In this paper, we continue the study of property(gb) which is studied in some recent papers [6, 13–15]. We show that property (gb) is satisfied by an operator T satisfying S(T*)⊆σSBF+-(T). We give a revised proof of [15, Theorem 3.10] to prove that property (gb) is preserved under commuting nilpotent perturbations. We show also that if T∈ℬ(X) satisfies S(T*)⊆σSBF+-(T) and F is a finite rank operator commuting with T, then T+F satisfies property (gb). We show that if T∈ℬ(X) is an a-polaroid operator satisfying property (gb) and Q is a quasinilpotent operator commuting with T, then T+Q satisfies property (gb). Two counterexamples are also given to show that property (gb) in general is not preserved under commuting quasinilpotent perturbations or commuting finite rank perturbations. These results improve and revise some recent results of Rashid in [15].


## 2. Main Results

We begin with the following lemmas.Lemma 2.1 (See [6], Corollary 2.9).
An operatorT∈ℬ(X) possesses property (gb) if and only if T satisfies generalized a-Browder's theorem and Π(T)=Πa(T).Lemma 2.2.
If the equalityσSBF+-(T)=σD(T) holds for T∈ℬ(X), then T possesses property (gb).Proof.
Suppose thatσSBF+-(T)=σD(T). If λ∈σa(T)∖σSBF+-(T), then λ∈σa(T)∖σD(T)⊆Π(T). This implies that σa(T)∖σSBF+-(T)=Π(T). Since Π(T)⊆σa(T)∖σSBF+-(T) is always true, σa(T)∖σSBF+-(T)=Π(T), that is, T possesses property (gb).Lemma 2.3.
IfT∈ℬ(X), then σSBF+-(T)∪S(T*)=σD(T).Proof.
Letλ∉σSBF+-(T)∪S(T*). Then T-λ is an upper semi-Weyl operator and T* has SVEP at λ. Thus, T-λ is an upper semi-B-Fredholm operator and ind(T-λ)≤0. Hence, there exists n∈ℕ such that ℛ((T-λ)n) is closed, (T-λ)n is an upper semi-Fredholm operator, and ind(T-λ)n≤0. By [16, Theorem 2.11], dsc(T-λ)<∞. Thus, dsc(T-λ)n<∞, by [11, Theorem 3.4(ii)], ind(T-λ)n≥0. By [11, Theorem 3.4(iv)], asc(T-λ)n=dsc(T-λ)n<∞. Consequently, (T-λ)n is a Browder operator. Thus, by [17, Theorem 2.9], we then conclude that T-λ is Drazin invertible, that is, λ∉σD(T). Hence, σD(T)⊆σSBF+-(T)∪S(T*). Since the reverse inclusion obviously holds, we get σSBF+-(T)∪S(T*)=σD(T).Theorem 2.4.
IfT∈ℬ(X) satisfies S(T*)⊆σSBF+-(T), then T possesses property (gb). In particular, if T* has SVEP, then T possesses property (gb).Proof.
Suppose thatS(T*)⊆σSBF+-(T). Then by Lemma 2.3, we get σSBF+-(T)=σD(T). Consequently, by Lemma 2.2, T possesses property (gb). If T* has SVEP, then S(T*)=∅; the conclusion follows immediately.The following example shows that the converse of Theorem2.4 is not true.Example 2.5.
LetX be the Hilbert space l2(ℕ), and let T:l2(ℕ)→l2(ℕ) be the unilateral right shift operator defined by
(2.1)T(x1,x2,…)=(0,x1,x2,…)∀(xn)∈l2(N).
Then,
(2.2)σa(T)={λ∈C:|λ|=1},σSBF+-(T)={λ∈C:|λ|=1},Π(T)=∅.
Hence σa(T)∖σSBF+-(T)=Π(T), that is, T possesses property (gb), but S(T*)={λ∈ℂ:0≤|λ|<1}⊈{λ∈ℂ:|λ|=1}=σSBF+-(T).
The next theorem improves a recent result of Berkani and Zariouh [14, Theorem 2.5] by removing the extra assumption that T is an a-polaroid operator. It also improves [14, Theorem 2.7]. We mention that it had been established in [15, Theorem 3.10], but its proof was not so clear. Hence, we give a revised proof of it.Theorem 2.6.
IfT∈ℬ(X) satisfies property (gb) and N is a nilpotent operator that commutes with T, then T+N satisfies property (gb).Proof.
Suppose thatT∈ℬ(X) satisfies property (gb) and N is a nilpotent operator that commutes with T. By Lemma 2.1, T satisfies generalized a-Browder's theorem and Π(T)=Πa(T). Hence, T+N satisfies generalized a-Browder's theorem. By [18], σ(T+N)=σ(T) and σa(T+N)=σa(T). Hence, by [19, Theorem 2.2] and [20, Theorem 3.2], we have that Π(T+N)=σ(T+N)∖σD(T+N)=σ(T)∖σD(T)=Π(T)=Πa(T)=σa(T)∖σLD(T)=σa(T+N)∖σLD(T+N)=Πa(T+N). By Lemma 2.1 again, T+N satisfies property (gb).The following example, which is a revised version of [15, Example 3.11], shows that the hypothesis of commutativity in Theorem 2.6 is crucial.Example 2.7.
LetT:l2(ℕ)→l2(ℕ) be the unilateral right shift operator defined by
(2.3)T(x1,x2,…)=(0,x1,x2,…)∀(xn)∈l2(N).
Let N:l2(ℕ)→l2(ℕ) be a nilpotent operator with rank one defined by
(2.4)N(x1,x2,…)=(0,-x1,0,…)∀(xn)∈l2(N).
Then TN≠NT. Moreover,
(2.5)σ(T)={λ∈C:0≤|λ|≤1},σa(T)={λ∈C:|λ|=1},σ(T+N)={λ∈ℂ:0≤|λ|≤1},σa(T+N)={λ∈ℂ:|λ|=1}∪{0}.
It follows that Πa(T)=Π(T)=∅ and {0}=Πa(T+N)≠Π(T+N)=∅. Hence, by Lemma 2.1, T+N does not satisfy property (gb). But since T has SVEP, T satisfies a-Browder's theorem or equivalently, by [8, Theorem 2.2], T satisfies generalized a-Browder's theorem. Therefore, by Lemma 2.1 again, T satisfies property (gb).
To continue the discussion of this paper, we recall some classical definitions. Using the isomorphismX/𝒩(Td)≈ℛ(Td) and following [21], a topology on ℛ(Td) is defined as follows.Definition 2.8.
LetT∈ℬ(X). For every d∈ℕ, the operator range topological on ℛ(Td) is defined by the norm ||·||ℛ(Td) such that for all y∈ℛ(Td),
(2.6)‖y‖R(Td):=inf⁡{‖x‖:x∈X,y=Tdx}.
For a detailed discussion of operator ranges and their topologies, we refer the reader to [22, 23].Definition 2.9.
LetT∈ℬ(X) and let d∈ℕ. Then T has uniformdescent for n≥d if kn(T)=0 for all n≥d. If in addition ℛ(Tn) is closed in the operator range topology of ℛ(Td) for all n≥d, then we say that T has eventual topological uniform descent, and, more precisely, that T has topological uniform descent forn≥d.
Operators with eventual topological uniform descent are introduced by Grabiner in [21]. It includes many classes of operators introduced in the introduction of this paper, such as upper semi-B-Fredholm operators, left Drazin invertible operators, and Drazin invertible operators. It also includes many other classes of operators such as operators of Kato type, quasi-Fredholm operators, operators with finite descent, and operators with finite essential descent. A very detailed and far-reaching account of these notations can be seen in [11, 18, 24]. Especially, operators which have topological uniform descent for n≥0 are precisely the semi-regular operators studied by Mbekhta in [25]. Discussions of operators with eventual topological uniform descent may be found in [21, 26–29].Lemma 2.10.
IfT∈ℬ(X) and F is a finite rank operator commuting with T, then(1)
σSBF+-(T+F)=σSBF+-(T),(2)
σD(T+F)=σD(T).Proof.
(
1
) Without loss of generality, we need only to show that 0∉σSBF+-(T+F) if and only if 0∉σSBF+-(T). By symmetry, it suffices to prove that 0∉σSBF+-(T+F) if 0∉σSBF+-(T).
Suppose that0∉σSBF+-(T). Then T is an upper semi-B-Fredholm operator and ind(T)≤0. Hence, it follows from [24, Theorem 3.6] and [20, Theorem 3.2] that T+F is also an upper semi-B-Fredholm operator. Thus, by [21, Theorem 5.8], ind(T+F)=ind(T)≤0. Consequently, T+F is an upper semi-B-Weyl operator, that is, 0∉σSBF+-(T), and this completes the proof of (1).
(
2
) Noting that an operator is Drazin invertible if and only if it is of finite ascent and finite descent, the conclusion follows from [19, Theorem 2.2].Theorem 2.11.
IfT∈ℬ(X) satisfies S(T*)⊆σSBF+-(T) and F is a finite rank operator commuting with T, then T+F satisfies property (gb).Proof.
SinceF is a finite rank operator commuting with T, by Lemma 2.10, σSBF+-(T+F)=σSBF+-(T) and σD(T+F)=σD(T). Since S(T*)⊆σSBF+-(T), by Lemma 2.3, σSBF+-(T)=σD(T). Thus, σSBF+-(T+F)=σD(T+F). By Lemma 2.2, T+F satisfies property (gb).The following example illustrates that property(gb) in general is not preserved under commuting finite rank perturbations.Example 2.12.
LetU:l2(ℕ)→l2(ℕ) be the unilateral right shift operator defined by
(2.7)U(x1,x2,…)=(0,x1,x2,…)∀(xn)∈l2(N).
For fixed 0<ε<1, let Fε:l2(ℕ)→l2(ℕ) be a finite rank operator defined by
(2.8)Fε(x1,x2,…)=(-εx1,0,0,…)∀(xn)∈l2(N).
We consider the operators T and F defined by T=U⊕I and F=0⊕Fε, respectively. Then F is a finite rank operator and TF=FT. Moreover,
(2.9)σ(T)=σ(U)∪σ(I)={λ∈C:0≤|λ|≤1},σa(T)=σa(U)∪σa(I)={λ∈C:|λ|=1},σ(T+F)=σ(U)∪σ(I+Fε)={λ∈C:0≤|λ|≤1},σa(T+F)=σa(U)∪σa(I+Fε)={λ∈C:|λ|=1}∪{1-ε}.
It follows that Πa(T)=Π(T)=∅ and {1-ε}=Πa(T+F)≠Π(T+F)=∅. Hence, by Lemma 2.1, T+F does not satisfy property (gb). But since T has SVEP, T satisfies a-Browder's theorem or equivalently, by [8, Theorem 2.2], T satisfies generalized a-Browder's theorem. Therefore by Lemma 2.1 again, T satisfies property (gb).
Rashid gives in [15, Theorem 3.15] that if T∈ℬ(X) and Q is a quasinilpotent operator that commute with T, then
(2.10)σSBF+-(T+Q)=σSBF+-(T).
The next example shows that this equality does not hold in general.Example 2.13.
LetQ denote the Volterra operator on the Banach space C[0,1] defined by
(2.11)(Qf)(t)=∫0t‍f(s)ds∀f∈C[0,1]t∈[0,1].Q is injective and quasinilpotent. Hence, it is easy to see that ℛ(Qn) is not closed for every n∈ℕ. Let T=0∈ℬ(C[0,1]). It is easy to see that TQ=0=QT and 0∉σSBF+-(0)=σSBF+-(T), but 0∈σSBF+-(Q)=σSBF+-(0+Q)=σSBF+-(T+Q). Hence, σSBF+-(T+Q)≠σSBF+-(T).
Rashid claims in [15, Theorem 3.16] that property (gb) is stable under commuting quasinilpotent perturbations, but its proof relies on [15, Theorem 3.15] which, by Example 2.13, is not always true. The following example shows that property (gb) in general is not preserved under commuting quasinilpotent perturbations.Example 2.14.
LetU:l2(ℕ)→l2(ℕ) be the unilateral right shift operator defined by
(2.12)U(x1,x2,…)=(0,x1,x2,…)∀(xn)∈l2(N).
LetV:l2(ℕ)→l2(ℕ) be a quasinilpotent operator defined by
(2.13)V(x1,x2,…)=(0,x1,0,x33,x44,…)∀(xn)∈l2(N).
LetN:l2(ℕ)→l2(ℕ) be a quasinilpotent operator defined by
(2.14)N(x1,x2,⋯)=(0,0,0,-x33,-x44,…)∀(xn)∈l2(N).
It is easy to verify thatVN=NV. We consider the operators T and Q defined by T=U⊕V and Q=0⊕N, respectively. Then Q is quasinilpotent and TQ=QT. Moreover,
(2.15)σ(T)=σ(U)∪σ(V)={λ∈C:0≤|λ|≤1},σa(T)=σa(U)∪σa(V)={λ∈C:|λ|=1}∪{0},σ(T+Q)=σ(U)∪σ(V+N)={λ∈C:0≤|λ|≤1},σa(T+Q)=σa(U)∪σa(V+N)={λ∈C:|λ|=1}∪{0}.
It follows thatΠa(T)=Π(T)=∅ and {0}=Πa(T+Q)≠Π(T+Q)=∅. Hence, by Lemma 2.1, T+Q does not satisfy property (gb). But since T has SVEP, T satisfies a-Browder's theorem or equivalently, by [8, Theorem 2.2], T satisfies generalized a-Browder's theorem. Therefore, by Lemma 2.1 again, T satisfies property (gb).
Our last result, which also improves [14, Theorem 2.5] from a different standpoint, gives the correct version of [15, Theorem 3.16].Theorem 2.15.
Suppose thatT∈ℬ(X) obeys property (gb) and that Q∈ℬ(X) is a quasinilpotent operator commuting with T. If T is a-polaroid, then T+Q obeys (gb).Proof.
SinceT satisfies property (gb), by Lemma 2.1, T satisfies generalized a-Browder's theorem and Π(T)=Πa(T). Hence, T+Q satisfies generalized a-Browder's theorem. In order to show that T+Q satisfies property (gb), by Lemma 2.1 again, it suffices to show that Π(T+Q)=Πa(T+Q). Since Π(T+Q)⊆Πa(T+Q) is always true, one needs only to show that Πa(T+Q)⊆Π(T+Q).
Letλ∈Πa(T+Q)=σa(T+Q)∖σLD(T+Q)=isoσa(T+Q)∖σLD(T+Q). Then by [18], λ∈isoσa(T). Since T is a-polaroid, λ∈Πa(T)=Π(T). Thus by [29, Theorem 3.12], λ∈Π(T+Q). Therefore, Πa(T+Q)⊆Π(T+Q), and this completes the proof.



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