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Abstract and Applied Analysis
VolumeΒ 2012, Article IDΒ 523986, 10 pages
http://dx.doi.org/10.1155/2012/523986
Research Article

A Note on Property (𝑔𝑏) and Perturbations

School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007, China

Received 14 April 2012; Accepted 26 July 2012

Academic Editor: Sergey V.Β Zelik

Copyright Β© 2012 Qingping Zeng and Huaijie Zhong. 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.

Abstract

An operator π‘‡βˆˆβ„¬(𝑋) defined on a Banach space 𝑋 satisfies property (𝑔𝑏) if the complement in the approximate point spectrum πœŽπ‘Ž(𝑇) of the upper semi-B-Weyl spectrum πœŽπ‘†π΅πΉβˆ’+(𝑇) coincides with the set Ξ (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 β„±(𝑋) denote its ideal of finite rank operators on 𝑋. For an operator π‘‡βˆˆβ„¬(𝑋), let π‘‡βˆ— denote its dual, 𝒩(𝑇) its kernel, 𝛼(𝑇) its nullity, β„›(𝑇) its range, 𝛽(𝑇) its defect, 𝜎(𝑇) its spectrum, and πœŽπ‘Ž(𝑇) its approximate point spectrum. If the range β„›(𝑇) is closed and 𝛼(𝑇)<∞ (resp., 𝛽(𝑇)<∞), then 𝑇 is said to be upper semi-Fredholm (resp., lower semi-Fredholm). If π‘‡βˆˆβ„¬(𝑋) is both upper and lower semi-Fredholm, then 𝑇 is said to be Fredholm. If π‘‡βˆˆβ„¬(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)βˆ’π›½(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 πœŽπ‘Š(𝑇)∢={πœ†βˆˆβ„‚βˆΆπ‘‡βˆ’πœ†πΌisnotaWeyloperator},(1.1) the upper semi-Weyl spectrum (in the literature called also Weyl essential approximate point spectrum) defined by πœŽπ‘†πΉβˆ’+(𝑇)∢={πœ†βˆˆβ„‚βˆΆπ‘‡βˆ’πœ†πΌisnotanuppersemi-Weyloperator}.(1.2)

Recall that the descent and the ascent of π‘‡βˆˆβ„¬(𝑋) are dsc(𝑇)=inf{π‘›βˆˆβ„•βˆΆβ„›(𝑇𝑛)=β„›(𝑇𝑛+1)} and asc(𝑇)=inf{π‘›βˆˆβ„•βˆΆπ’©(𝑇𝑛)=𝒩(𝑇𝑛+1)}, respectively (the infimum of an empty set is defined to be ∞). If asc(𝑇)<∞ and β„›(𝑇asc(𝑇)+1) is closed, then 𝑇 is said to be left Drazin invertible. If dsc(𝑇)<∞ and β„›(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, π‘‡βˆˆβ„¬(𝑋) is both left and right Drazin invertible if and only if 𝑇 is Drazin invertible. An operator π‘‡βˆˆβ„¬(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 π‘‡βˆˆβ„¬(𝑋) is defined by 𝜎𝐡(𝑇)∢={πœ†βˆˆβ„‚βˆΆπ‘‡βˆ’πœ†πΌisnotaBrowderoperator},(1.3) the upper semi-Browder spectrum (in the literature called also Browder essential approximate point spectrum) is defined by πœŽπ‘ˆπ΅(𝑇)∢={πœ†βˆˆβ„‚βˆΆπ‘‡βˆ’πœ†πΌisnotanuppersemi-Browderoperator}.(1.4)

An operator π‘‡βˆˆβ„¬(𝑋) is called Riesz if its essential spectrum πœŽπ‘’(𝑇)∢={πœ†βˆˆβ„‚βˆΆπ‘‡βˆ’πœ†πΌisnotFredholm}={0}.

Suppose that π‘‡βˆˆβ„¬(𝑋) and that π‘…βˆˆβ„¬(𝑋) is a Riesz operator commuting with 𝑇. Then it follows from [1, Proposition 5] and [2, Theorem 1] that πœŽπ‘†πΉβˆ’+(𝑇+𝑅)=πœŽπ‘†πΉβˆ’+𝜎(𝑇),π‘Š(𝑇+𝑅)=πœŽπ‘ŠπœŽ(𝑇),π‘ˆπ΅(𝑇+𝑅)=πœŽπ‘ˆπ΅πœŽ(𝑇),𝐡(𝑇+𝑅)=𝜎𝐡(𝑇).(1.5)

For each integer 𝑛, define 𝑇𝑛 to be the restriction of 𝑇 to β„›(𝑇𝑛) viewed as the map from β„›(𝑇𝑛) into β„›(𝑇𝑛) (in particular 𝑇0=𝑇). If there exists π‘›βˆˆβ„• such that β„›(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 π‘›βˆˆβ„• such that β„›(𝑇𝑛) is closed and 𝑇𝑛 is upper semi-Fredholm, then β„›(π‘‡π‘š) is closed, π‘‡π‘š is upper semi-Fredholm, and ind(π‘‡π‘š) = ind(𝑇𝑛) for all π‘šβ‰₯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 β„›(𝑇𝑛) is closed and 𝑇𝑛 is upper semi-Fredholm. An operator π‘‡βˆˆβ„¬(𝑋) is called upper semi-B-Weyl if 𝑇 is upper semi-B-Fredholm and ind(𝑇)≀0.

For π‘‡βˆˆβ„¬(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)∢={πœ†βˆˆβ„‚βˆΆπ‘‡βˆ’πœ†πΌisnotaleftDrazininvertibleoperator𝜎};𝐷(𝑇)∢={πœ†βˆˆβ„‚βˆΆπ‘‡βˆ’πœ†πΌisnotaDrazininvertibleoperator𝜎};π‘†π΅πΉβˆ’+(𝑇)∢={πœ†βˆˆβ„‚βˆΆπ‘‡βˆ’πœ†πΌisnotanuppersemi-B-Weyloperator}.(1.6)

Let Ξ (𝑇) denote the set of all poles of 𝑇. We say that πœ†βˆˆπœŽπ‘Ž(𝑇) is a left pole of 𝑇 if π‘‡βˆ’πœ†πΌ is left Drazin invertible. Let Ξ π‘Ž(𝑇) denote the set of all left poles of 𝑇. It is well know that Ξ (𝑇)=𝜎(𝑇)⧡𝜎𝐷(𝑇)=iso𝜎(𝑇)⧡𝜎𝐷(𝑇) and Ξ π‘Ž(𝑇)=πœŽπ‘Ž(𝑇)⧡𝜎𝐿𝐷(𝑇)=isoπœŽπ‘Ž(𝑇)⧡𝜎𝐿𝐷(𝑇). Here and henceforth, for π΄βŠ†β„‚, iso𝐴 is the set of isolated points of 𝐴. An operator π‘‡βˆˆβ„¬(𝑋) is called a-polaroid if isoπœŽπ‘Ž(𝑇)=βˆ… or every isolated point of πœŽπ‘Ž(𝑇) is a left pole of 𝑇.

Following Harte and Lee [4], we say that π‘‡βˆˆβ„¬(𝑋) satisfies Browder's theorem if πœŽπ‘Š(𝑇)=𝜎𝐡(𝑇), while, according to DjordjeviΔ‡ and Han [5], we say that 𝑇 satisfies a-Browder's theorem if 𝜎SFβˆ’+(𝑇)=πœŽπ‘ˆπ΅(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 π‘‡βˆˆβ„¬(𝑋) is said to possess property (𝑔𝑏) if πœŽπ‘Ž(𝑇)β§΅πœŽπ‘†π΅πΉβˆ’+(𝑇)=Ξ (𝑇),(1.7) while π‘‡βˆˆβ„¬(𝑋) is said to satisfy generalized a-Browder's theorem if πœŽπ‘Ž(𝑇)β§΅πœŽπ‘†π΅πΉβˆ’+(𝑇)=Ξ π‘Ž(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 π‘‡βˆˆβ„¬(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 π‘‡βˆˆβ„¬(𝑋) is said to have the single-valued extension property at πœ†0βˆˆβ„‚ (SVEP at πœ†0 for brevity) if for every open neighborhood π‘ˆ of πœ†0 the only analytic function π‘“βˆΆπ‘ˆβ†’π‘‹ which satisfies the equation (πœ†πΌβˆ’π‘‡)𝑓(πœ†)=0 for all πœ†βˆˆπ‘ˆ is the function 𝑓(πœ†)≑0.
Let 𝑆(𝑇)∢={πœ†βˆˆβ„‚βˆΆπ‘‡doesnothavetheSVEPatπœ†}. An operator π‘‡βˆˆβ„¬(𝑋) is said to have SVEP if 𝑆(𝑇)=βˆ….
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(π‘‡βˆ—)βŠ†πœŽπ‘†π΅πΉβˆ’+(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 π‘‡βˆˆβ„¬(𝑋) satisfies 𝑆(π‘‡βˆ—)βŠ†πœŽπ‘†π΅πΉβˆ’+(𝑇) and 𝐹 is a finite rank operator commuting with 𝑇, then 𝑇+𝐹 satisfies property (𝑔𝑏). We show that if π‘‡βˆˆβ„¬(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 π‘‡βˆˆβ„¬(𝑋) possesses property (𝑔𝑏) if and only if 𝑇 satisfies generalized a-Browder's theorem and Ξ (𝑇)=Ξ π‘Ž(𝑇).

Lemma 2.2. If the equality πœŽπ‘†π΅πΉβˆ’+(𝑇)=𝜎𝐷(𝑇) holds for π‘‡βˆˆβ„¬(𝑋), then 𝑇 possesses property (𝑔𝑏).

Proof. Suppose that πœŽπ‘†π΅πΉβˆ’+(𝑇)=𝜎𝐷(𝑇). If πœ†βˆˆπœŽπ‘Ž(𝑇)β§΅πœŽπ‘†π΅πΉβˆ’+(𝑇), then πœ†βˆˆπœŽπ‘Ž(𝑇)⧡𝜎𝐷(𝑇)βŠ†Ξ (𝑇). This implies that πœŽπ‘Ž(𝑇)β§΅πœŽπ‘†π΅πΉβˆ’+(𝑇)=Ξ (𝑇). Since Ξ (𝑇)βŠ†πœŽπ‘Ž(𝑇)β§΅πœŽπ‘†π΅πΉβˆ’+(𝑇) is always true, πœŽπ‘Ž(𝑇)β§΅πœŽπ‘†π΅πΉβˆ’+(𝑇)=Ξ (𝑇), that is, 𝑇 possesses property (𝑔𝑏).

Lemma 2.3. If π‘‡βˆˆβ„¬(𝑋), then πœŽπ‘†π΅πΉβˆ’+(𝑇)βˆͺ𝑆(π‘‡βˆ—)=𝜎𝐷(𝑇).

Proof. Let πœ†βˆ‰πœŽπ‘†π΅πΉβˆ’+(𝑇)βˆͺ𝑆(π‘‡βˆ—). Then π‘‡βˆ’πœ† is an upper semi-Weyl operator and π‘‡βˆ— has SVEP at πœ†. Thus, π‘‡βˆ’πœ† is an upper semi-B-Fredholm operator and ind(π‘‡βˆ’πœ†)≀0. Hence, there exists π‘›βˆˆβ„• such that β„›((π‘‡βˆ’πœ†)𝑛) is closed, (π‘‡βˆ’πœ†)𝑛 is an upper semi-Fredholm operator, and ind(π‘‡βˆ’πœ†)𝑛≀0. By [16, Theorem 2.11], dsc(π‘‡βˆ’πœ†)<∞. Thus, dsc(π‘‡βˆ’πœ†)𝑛<∞, by [11, Theorem 3.4(ii)], ind(π‘‡βˆ’πœ†)𝑛β‰₯0. By [11, Theorem 3.4(iv)], asc(π‘‡βˆ’πœ†)𝑛=dsc(π‘‡βˆ’πœ†)𝑛<∞. Consequently, (π‘‡βˆ’πœ†)𝑛 is a Browder operator. Thus, by [17, Theorem 2.9], we then conclude that π‘‡βˆ’πœ† is Drazin invertible, that is, πœ†βˆ‰πœŽπ·(𝑇). Hence, 𝜎𝐷(𝑇)βŠ†πœŽπ‘†π΅πΉβˆ’+(𝑇)βˆͺ𝑆(π‘‡βˆ—). Since the reverse inclusion obviously holds, we get πœŽπ‘†π΅πΉβˆ’+(𝑇)βˆͺ𝑆(π‘‡βˆ—)=𝜎𝐷(𝑇).

Theorem 2.4. If π‘‡βˆˆβ„¬(𝑋) satisfies 𝑆(π‘‡βˆ—)βŠ†πœŽπ‘†π΅πΉβˆ’+(𝑇), then 𝑇 possesses property (𝑔𝑏). In particular, if π‘‡βˆ— has SVEP, then 𝑇 possesses property (𝑔𝑏).

Proof. Suppose that 𝑆(π‘‡βˆ—)βŠ†πœŽπ‘†π΅πΉβˆ’+(𝑇). Then by Lemma 2.3, we get πœŽπ‘†π΅πΉβˆ’+(𝑇)=𝜎𝐷(𝑇). Consequently, by Lemma 2.2, 𝑇 possesses property (𝑔𝑏). If π‘‡βˆ— has SVEP, then 𝑆(π‘‡βˆ—)=βˆ…; 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(β„•), and let π‘‡βˆΆπ‘™2(β„•)→𝑙2(β„•) be the unilateral right shift operator defined by 𝑇π‘₯1,π‘₯2ξ€Έ=ξ€·,…0,π‘₯1,π‘₯2ξ€Έβˆ€ξ€·π‘₯,β€¦π‘›ξ€Έβˆˆπ‘™2(β„•).(2.1) Then, πœŽπ‘Žξ€½||πœ†||ξ€Ύ,𝜎(𝑇)=πœ†βˆˆβ„‚βˆΆ=1π‘†π΅πΉβˆ’+(ξ€½||πœ†||ξ€Ύ,𝑇)=πœ†βˆˆβ„‚βˆΆ=1Ξ (𝑇)=βˆ….(2.2) Hence πœŽπ‘Ž(𝑇)β§΅πœŽπ‘†π΅πΉβˆ’+(𝑇)=Ξ (𝑇), that is, 𝑇 possesses property (𝑔𝑏), but 𝑆(π‘‡βˆ—ΜΈ)={πœ†βˆˆβ„‚βˆΆ0≀|πœ†|<1}βŠ†{πœ†βˆˆβ„‚βˆΆ|πœ†|=1}=πœŽπ‘†π΅πΉβˆ’+(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 π‘‡βˆˆβ„¬(𝑋) satisfies property (𝑔𝑏) and 𝑁 is a nilpotent operator that commutes with 𝑇, then 𝑇+𝑁 satisfies property (𝑔𝑏).

Proof. Suppose that π‘‡βˆˆβ„¬(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 Ξ (𝑇)=Ξ π‘Ž(𝑇). Hence, 𝑇+𝑁 satisfies generalized a-Browder's theorem. By [18], 𝜎(𝑇+𝑁)=𝜎(𝑇) and πœŽπ‘Ž(𝑇+𝑁)=πœŽπ‘Ž(𝑇). Hence, by [19, Theorem 2.2] and [20, Theorem 3.2], we have that Ξ (𝑇+𝑁)=𝜎(𝑇+𝑁)⧡𝜎𝐷(𝑇+𝑁)=𝜎(𝑇)⧡𝜎𝐷(𝑇)=Ξ (𝑇)=Ξ π‘Ž(𝑇)=πœŽπ‘Ž(𝑇)⧡𝜎𝐿𝐷(𝑇)=πœŽπ‘Ž(𝑇+𝑁)⧡𝜎𝐿𝐷Π(𝑇+𝑁)=π‘Ž(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 π‘‡βˆΆπ‘™2(β„•)→𝑙2(β„•) be the unilateral right shift operator defined by 𝑇π‘₯1,π‘₯2ξ€Έ=ξ€·,…0,π‘₯1,π‘₯2ξ€Έβˆ€ξ€·π‘₯,β€¦π‘›ξ€Έβˆˆπ‘™2(β„•).(2.3) Let π‘βˆΆπ‘™2(β„•)→𝑙2(β„•) be a nilpotent operator with rank one defined by 𝑁π‘₯1,π‘₯2ξ€Έ=ξ€·,…0,βˆ’π‘₯1ξ€Έβˆ€ξ€·π‘₯,0,β€¦π‘›ξ€Έβˆˆπ‘™2(β„•).(2.4) Then 𝑇𝑁≠𝑁𝑇. Moreover, πœŽξ€½||πœ†||ξ€Ύ,𝜎(𝑇)=πœ†βˆˆβ„‚βˆΆ0≀≀1π‘Ž(ξ€½||πœ†||ξ€Ύ,πœŽξ€½||πœ†||ξ€Ύ,πœŽπ‘‡)=πœ†βˆˆβ„‚βˆΆ=1(𝑇+𝑁)=πœ†βˆˆβ„‚βˆΆ0≀≀1π‘Ž(ξ€½||πœ†||𝑇+𝑁)=πœ†βˆˆβ„‚βˆΆ=1βˆͺ{0}.(2.5) It follows that Ξ π‘Ž(𝑇)=Ξ (𝑇)=βˆ… and {0}=Ξ π‘Ž(𝑇+𝑁)β‰ Ξ (𝑇+𝑁)=βˆ…. 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)β‰ˆβ„›(𝑇𝑑) and following [21], a topology on β„›(𝑇𝑑) is defined as follows.

Definition 2.8. Let π‘‡βˆˆβ„¬(𝑋). For every π‘‘βˆˆβ„•, the operator range topological on β„›(𝑇𝑑) is defined by the norm ||β‹…||β„›(𝑇𝑑) such that for all π‘¦βˆˆβ„›(𝑇𝑑), ‖𝑦‖ℛ(𝑇𝑑)ξ€½βˆΆ=infβ€–π‘₯β€–βˆΆπ‘₯βˆˆπ‘‹,𝑦=𝑇𝑑π‘₯ξ€Ύ.(2.6)
For a detailed discussion of operator ranges and their topologies, we refer the reader to [22, 23].

Definition 2.9. Let π‘‡βˆˆβ„¬(𝑋) and let π‘‘βˆˆβ„•. Then 𝑇 has π‘’π‘›π‘–π‘“π‘œπ‘Ÿπ‘šπ‘‘π‘’π‘ π‘π‘’π‘›π‘‘ for 𝑛β‰₯𝑑 if π‘˜π‘›(𝑇)=0 for all 𝑛β‰₯𝑑. If in addition β„›(𝑇𝑛) is closed in the operator range topology of β„›(𝑇𝑑) for all 𝑛β‰₯𝑑, then we say that 𝑇 has eventual topological uniform descent, and, more precisely, that 𝑇 has topological uniform descent for 𝑛β‰₯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 𝑛β‰₯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 π‘‡βˆˆβ„¬(𝑋) and 𝐹 is a finite rank operator commuting with 𝑇, then(1)𝜎SBFβˆ’+(𝑇+𝐹)=𝜎SBFβˆ’+(𝑇),(2)𝜎𝐷(𝑇+𝐹)=𝜎𝐷(𝑇).

Proof. (1) Without loss of generality, we need only to show that 0βˆ‰πœŽπ‘†π΅πΉβˆ’+(𝑇+𝐹) if and only if 0βˆ‰πœŽπ‘†π΅πΉβˆ’+(𝑇). By symmetry, it suffices to prove that 0βˆ‰πœŽπ‘†π΅πΉβˆ’+(𝑇+𝐹) if 0βˆ‰πœŽπ‘†π΅πΉβˆ’+(𝑇).
Suppose that 0βˆ‰πœŽπ‘†π΅πΉβˆ’+(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, 0βˆ‰πœŽπ‘†π΅πΉβˆ’+(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 π‘‡βˆˆβ„¬(𝑋) satisfies 𝑆(π‘‡βˆ—)βŠ†πœŽSBFβˆ’+(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, πœŽπ‘†π΅πΉβˆ’+(𝑇+𝐹)=πœŽπ‘†π΅πΉβˆ’+(𝑇) and 𝜎𝐷(𝑇+𝐹)=𝜎𝐷(𝑇). Since 𝑆(π‘‡βˆ—)βŠ†πœŽπ‘†π΅πΉβˆ’+(𝑇), by Lemma 2.3, πœŽπ‘†π΅πΉβˆ’+(𝑇)=𝜎𝐷(𝑇). Thus, πœŽπ‘†π΅πΉβˆ’+(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 π‘ˆβˆΆπ‘™2(β„•)→𝑙2(β„•) be the unilateral right shift operator defined by π‘ˆξ€·π‘₯1,π‘₯2ξ€Έ=ξ€·,…0,π‘₯1,π‘₯2ξ€Έβˆ€ξ€·π‘₯,β€¦π‘›ξ€Έβˆˆπ‘™2(β„•).(2.7) For fixed 0<πœ€<1, let πΉπœ€βˆΆπ‘™2(β„•)→𝑙2(β„•) be a finite rank operator defined by πΉπœ€ξ€·π‘₯1,π‘₯2ξ€Έ=ξ€·,β€¦βˆ’πœ€π‘₯1ξ€Έβˆ€ξ€·π‘₯,0,0,β€¦π‘›ξ€Έβˆˆπ‘™2(β„•).(2.8) We consider the operators 𝑇 and 𝐹 defined by 𝑇=π‘ˆβŠ•πΌ and 𝐹=0βŠ•πΉπœ€, respectively. Then 𝐹 is a finite rank operator and 𝑇𝐹=𝐹𝑇. Moreover, πœŽξ€½||πœ†||ξ€Ύ,𝜎(𝑇)=𝜎(π‘ˆ)βˆͺ𝜎(𝐼)=πœ†βˆˆβ„‚βˆΆ0≀≀1π‘Ž(𝑇)=πœŽπ‘Ž(π‘ˆ)βˆͺπœŽπ‘Ž(ξ€½||πœ†||ξ€Ύ,πœŽξ€·πΌ)=πœ†βˆˆβ„‚βˆΆ=1(𝑇+𝐹)=𝜎(π‘ˆ)βˆͺ𝜎𝐼+πΉπœ€ξ€Έ=ξ€½||πœ†||ξ€Ύ,πœŽπœ†βˆˆβ„‚βˆΆ0≀≀1π‘Ž(𝑇+𝐹)=πœŽπ‘Ž(π‘ˆ)βˆͺπœŽπ‘Žξ€·πΌ+πΉπœ€ξ€Έ=ξ€½||πœ†||ξ€Ύπœ†βˆˆβ„‚βˆΆ=1βˆͺ{1βˆ’πœ€}.(2.9) It follows that Ξ π‘Ž(𝑇)=Ξ (𝑇)=βˆ… and {1βˆ’πœ€}=Ξ π‘Ž(𝑇+𝐹)β‰ Ξ (𝑇+𝐹)=βˆ…. 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 π‘‡βˆˆβ„¬(𝑋) and 𝑄 is a quasinilpotent operator that commute with 𝑇, then πœŽπ‘†π΅πΉβˆ’+(𝑇+𝑄)=πœŽπ‘†π΅πΉβˆ’+(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 𝐢[0,1] defined by (ξ€œπ‘„π‘“)(𝑑)=𝑑0𝑓(𝑠)d[][].π‘ βˆ€π‘“βˆˆπΆ0,1π‘‘βˆˆ0,1(2.11)𝑄 is injective and quasinilpotent. Hence, it is easy to see that β„›(𝑄𝑛) is not closed for every π‘›βˆˆβ„•. Let 𝑇=0βˆˆβ„¬(𝐢[0,1]). It is easy to see that 𝑇𝑄=0=𝑄𝑇 and 0βˆ‰πœŽπ‘†π΅πΉβˆ’+(0)=πœŽπ‘†π΅πΉβˆ’+(𝑇), but 0βˆˆπœŽπ‘†π΅πΉβˆ’+(𝑄)=πœŽπ‘†π΅πΉβˆ’+(0+𝑄)=πœŽπ‘†π΅πΉβˆ’+(𝑇+𝑄). Hence, πœŽπ‘†π΅πΉβˆ’+(𝑇+𝑄)β‰ πœŽπ‘†π΅πΉβˆ’+(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 π‘ˆβˆΆπ‘™2(β„•)→𝑙2(β„•) be the unilateral right shift operator defined by π‘ˆξ€·π‘₯1,π‘₯2ξ€Έ=ξ€·,…0,π‘₯1,π‘₯2ξ€Έβˆ€ξ€·π‘₯,β€¦π‘›ξ€Έβˆˆπ‘™2(β„•).(2.12)
Let π‘‰βˆΆπ‘™2(β„•)→𝑙2(β„•) be a quasinilpotent operator defined by 𝑉π‘₯1,π‘₯2ξ€Έ=ξ‚€,…0,π‘₯1π‘₯,0,33,π‘₯44ξ‚βˆ€ξ€·π‘₯,β€¦π‘›ξ€Έβˆˆπ‘™2(β„•).(2.13)
Let π‘βˆΆπ‘™2(β„•)→𝑙2(β„•) be a quasinilpotent operator defined by 𝑁π‘₯1,π‘₯2ξ€Έ=ξ‚€π‘₯,β‹―0,0,0,βˆ’33π‘₯,βˆ’44ξ‚βˆ€ξ€·π‘₯,β€¦π‘›ξ€Έβˆˆπ‘™2(β„•).(2.14)
It is easy to verify that 𝑉𝑁=𝑁𝑉. We consider the operators 𝑇 and 𝑄 defined by 𝑇=π‘ˆβŠ•π‘‰ and 𝑄=0βŠ•π‘, respectively. Then 𝑄 is quasinilpotent and 𝑇𝑄=𝑄𝑇. Moreover, πœŽξ€½||πœ†||ξ€Ύ,𝜎(𝑇)=𝜎(π‘ˆ)βˆͺ𝜎(𝑉)=πœ†βˆˆβ„‚βˆΆ0≀≀1π‘Ž(𝑇)=πœŽπ‘Ž(π‘ˆ)βˆͺπœŽπ‘Ž(ξ€½||πœ†||ξ€ΎπœŽξ€½||πœ†||ξ€Ύ,πœŽπ‘‰)=πœ†βˆˆβ„‚βˆΆ=1βˆͺ{0},(𝑇+𝑄)=𝜎(π‘ˆ)βˆͺ𝜎(𝑉+𝑁)=πœ†βˆˆβ„‚βˆΆ0≀≀1π‘Ž(𝑇+𝑄)=πœŽπ‘Ž(π‘ˆ)βˆͺπœŽπ‘Ž(ξ€½||πœ†||𝑉+𝑁)=πœ†βˆˆβ„‚βˆΆ=1βˆͺ{0}.(2.15)
It follows that Ξ π‘Ž(𝑇)=Ξ (𝑇)=βˆ… and {0}=Ξ π‘Ž(𝑇+𝑄)β‰ Ξ (𝑇+𝑄)=βˆ…. 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 π‘‡βˆˆβ„¬(𝑋) obeys property (𝑔𝑏) and that π‘„βˆˆβ„¬(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 Ξ (𝑇)=Ξ π‘Ž(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 Ξ (𝑇+𝑄)=Ξ π‘Ž(𝑇+𝑄). Since Ξ (𝑇+𝑄)βŠ†Ξ π‘Ž(𝑇+𝑄) is always true, one needs only to show that Ξ π‘Ž(𝑇+𝑄)βŠ†Ξ (𝑇+𝑄).
Let πœ†βˆˆΞ π‘Ž(𝑇+𝑄)=πœŽπ‘Ž(𝑇+𝑄)⧡𝜎𝐿𝐷(𝑇+𝑄)=isoπœŽπ‘Ž(𝑇+𝑄)⧡𝜎𝐿𝐷(𝑇+𝑄). Then by [18], πœ†βˆˆisoπœŽπ‘Ž(𝑇). Since 𝑇 is a-polaroid, πœ†βˆˆΞ π‘Ž(𝑇)=Ξ (𝑇). Thus by [29, Theorem 3.12], πœ†βˆˆΞ (𝑇+𝑄). Therefore, Ξ π‘Ž(𝑇+𝑄)βŠ†Ξ (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).

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