Abstract

An operator 𝑇∈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.

1. Introduction

Throughout this paper, let ℬ(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 𝜎𝑊(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆𝐼isnotaWeyloperator},(1.1) the upper semi-Weyl spectrum (in the literature called also Weyl essential approximate point spectrum) defined by 𝜎𝑆𝐹−+(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆𝐼isnotanuppersemi-Weyloperator}.(1.2)

Recall that the descent 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 𝜎𝐵(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆𝐼isnotaBrowderoperator},(1.3) the upper semi-Browder spectrum (in the literature called also Browder essential approximate point spectrum) is defined by 𝜎𝑈𝐵(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆𝐼isnotanuppersemi-Browderoperator}.(1.4)

An operator 𝑇∈ℬ(𝑋) is called Riesz if its essential spectrum 𝜎𝑒(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆𝐼isnotFredholm}={0}.

Suppose that 𝑇∈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 𝜎𝑆𝐹−+(𝑇+𝑅)=𝜎𝑆𝐹−+𝜎(𝑇),𝑊(𝑇+𝑅)=𝜎𝑊𝜎(𝑇),𝑈𝐵(𝑇+𝑅)=𝜎𝑈𝐵𝜎(𝑇),𝐵(𝑇+𝑅)=𝜎𝐵(𝑇).(1.5)

For each integer 𝑛, 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.

For 𝑇∈ℬ(𝑋), let us define the left Drazin spectrum, the Drazin spectrum, and the upper semi-B-Weyl spectrum of 𝑇 as follows, respectively: 𝜎𝐿𝐷(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆𝐼isnotaleftDrazininvertibleoperator𝜎};𝐷(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆𝐼isnotaDrazininvertibleoperator𝜎};𝑆𝐵𝐹−+(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆𝐼isnotanuppersemi-B-Weyloperator}.(1.6)

Let Π(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 operator 𝑇∈ℬ(𝑋) is said to possess property (𝑔𝑏) if 𝜎𝑎(𝑇)⧵𝜎𝑆𝐵𝐹−+(𝑇)=Π(𝑇),(1.7) while 𝑇∈ℬ(𝑋) is said to satisfy generalized a-Browder's theorem if 𝜎𝑎(𝑇)⧵𝜎𝑆𝐵𝐹−+(𝑇)=Π𝑎(𝑇).(1.8)
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 operator 𝑇∈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.
Let 𝑆(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 operator 𝑇∈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 equality 𝜎𝑆𝐵𝐹−+(𝑇)=𝜎𝐷(𝑇) holds for 𝑇∈ℬ(𝑋), then 𝑇 possesses property (𝑔𝑏).

Proof. Suppose that 𝜎𝑆𝐵𝐹−+(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. If 𝑇∈ℬ(𝑋), then 𝜎𝑆𝐵𝐹−+(𝑇)∪𝑆(𝑇∗)=𝜎𝐷(𝑇).

Proof. Let 𝜆∉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. If 𝑇∈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 that 𝑆(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 Theorem 2.4 is not true.

Example 2.5. Let 𝑋 be the Hilbert space 𝑙2(ℕ), and let 𝑇∶𝑙2(ℕ)→𝑙2(ℕ) be the unilateral right shift operator defined by 𝑇𝑥1,𝑥2=,…0,𝑥1,𝑥2∀𝑥,…𝑛∈𝑙2(ℕ).(2.1) Then, 𝜎𝑎||𝜆||,𝜎(𝑇)=𝜆∈ℂ∶=1𝑆𝐵𝐹−+(||𝜆||,𝑇)=𝜆∈ℂ∶=1Π(𝑇)=∅.(2.2) 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. If 𝑇∈ℬ(𝑋) satisfies property (𝑔𝑏) and 𝑁 is a nilpotent operator that commutes with 𝑇, then 𝑇+𝑁 satisfies property (𝑔𝑏).

Proof. Suppose that 𝑇∈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)⧵𝜎𝐿𝐷Π(𝑇+𝑁)=𝑎(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. Let 𝑇∶𝑙2(ℕ)→𝑙2(ℕ) be the unilateral right shift operator defined by 𝑇𝑥1,𝑥2=,…0,𝑥1,𝑥2∀𝑥,…𝑛∈𝑙2(ℕ).(2.3) Let 𝑁∶𝑙2(ℕ)→𝑙2(ℕ) be a nilpotent operator with rank one defined by 𝑁𝑥1,𝑥2=,…0,−𝑥1∀𝑥,0,…𝑛∈𝑙2(ℕ).(2.4) Then 𝑇𝑁≠𝑁𝑇. Moreover, 𝜎||𝜆||,𝜎(𝑇)=𝜆∈ℂ∶0≤≤1𝑎(||𝜆||,𝜎||𝜆||,𝜎𝑇)=𝜆∈ℂ∶=1(𝑇+𝑁)=𝜆∈ℂ∶0≤≤1𝑎(||𝜆||𝑇+𝑁)=𝜆∈ℂ∶=1∪{0}.(2.5) 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 isomorphism 𝑋/𝒩(𝑇𝑑)≈ℛ(𝑇𝑑) and following [21], a topology on ℛ(𝑇𝑑) is defined as follows.

Definition 2.8. Let 𝑇∈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), ‖𝑦‖ℛ(𝑇𝑑)∶=inf‖𝑥‖∶𝑥∈𝑋,𝑦=𝑇𝑑𝑥.(2.6)
For a detailed discussion of operator ranges and their topologies, we refer the reader to [22, 23].

Definition 2.9. Let 𝑇∈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 for 𝑛≥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. If 𝑇∈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 that 0∉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. 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).

Proof. Since 𝐹 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. Let 𝑈∶𝑙2(ℕ)→𝑙2(ℕ) be the unilateral right shift operator defined by 𝑈𝑥1,𝑥2=,…0,𝑥1,𝑥2∀𝑥,…𝑛∈𝑙2(ℕ).(2.7) For fixed 0<𝜀<1, let 𝐹𝜀∶𝑙2(ℕ)→𝑙2(ℕ) be a finite rank operator defined by 𝐹𝜀𝑥1,𝑥2=,…−𝜀𝑥1∀𝑥,0,0,…𝑛∈𝑙2(ℕ).(2.8) 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, 𝜎||𝜆||,𝜎(𝑇)=𝜎(𝑈)∪𝜎(𝐼)=𝜆∈ℂ∶0≤≤1𝑎(𝑇)=𝜎𝑎(𝑈)∪𝜎𝑎(||𝜆||,𝜎𝐼)=𝜆∈ℂ∶=1(𝑇+𝐹)=𝜎(𝑈)∪𝜎𝐼+𝐹𝜀=||𝜆||,𝜎𝜆∈ℂ∶0≤≤1𝑎(𝑇+𝐹)=𝜎𝑎(𝑈)∪𝜎𝑎𝐼+𝐹𝜀=||𝜆||𝜆∈ℂ∶=1∪{1−𝜀}.(2.9) 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 𝜎𝑆𝐵𝐹−+(𝑇+𝑄)=𝜎𝑆𝐵𝐹−+(𝑇).(2.10) The next example shows that this equality does not hold in general.

Example 2.13. Let 𝑄 denote the Volterra operator on the Banach space 𝐶[0,1] defined by (𝑄𝑓)(𝑡)=𝑡0𝑓(𝑠)d[][].𝑠∀𝑓∈𝐶0,1𝑡∈0,1(2.11)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. Let 𝑈∶𝑙2(ℕ)→𝑙2(ℕ) be the unilateral right shift operator defined by 𝑈𝑥1,𝑥2=,…0,𝑥1,𝑥2∀𝑥,…𝑛∈𝑙2(ℕ).(2.12)
Let 𝑉∶𝑙2(ℕ)→𝑙2(ℕ) be a quasinilpotent operator defined by 𝑉𝑥1,𝑥2=,…0,𝑥1𝑥,0,33,𝑥44∀𝑥,…𝑛∈𝑙2(ℕ).(2.13)
Let 𝑁∶𝑙2(ℕ)→𝑙2(ℕ) be a quasinilpotent operator defined by 𝑁𝑥1,𝑥2=𝑥,⋯0,0,0,−33𝑥,−44∀𝑥,…𝑛∈𝑙2(ℕ).(2.14)
It is easy to verify that 𝑉𝑁=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, 𝜎||𝜆||,𝜎(𝑇)=𝜎(𝑈)∪𝜎(𝑉)=𝜆∈ℂ∶0≤≤1𝑎(𝑇)=𝜎𝑎(𝑈)∪𝜎𝑎(||𝜆||𝜎||𝜆||,𝜎𝑉)=𝜆∈ℂ∶=1∪{0},(𝑇+𝑄)=𝜎(𝑈)∪𝜎(𝑉+𝑁)=𝜆∈ℂ∶0≤≤1𝑎(𝑇+𝑄)=𝜎𝑎(𝑈)∪𝜎𝑎(||𝜆||𝑉+𝑁)=𝜆∈ℂ∶=1∪{0}.(2.15)
It follows that Π𝑎(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 that 𝑇∈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. Since 𝑇 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).
Let 𝜆∈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.

Acknowledgments

This work has been supported by National Natural Science Foundation of China (11171066), Specialized Research Fund for the Doctoral Program of Higher Education (2010350311001, 20113503120003), Natural Science Foundation of Fujian Province (2009J01005, 2011J05002), and Foundation of the Education Department of Fujian Province, (JB10042).