Abstract and Applied Analysis

Abstract and Applied Analysis / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 523986 | https://doi.org/10.1155/2012/523986

Qingping Zeng, Huaijie Zhong, "A Note on Property and Perturbations", Abstract and Applied Analysis, vol. 2012, Article ID 523986, 10 pages, 2012. https://doi.org/10.1155/2012/523986

A Note on Property ( 𝑔 𝑏 ) and Perturbations

Academic Editor: Sergey V. Zelik
Received14 Apr 2012
Accepted26 Jul 2012
Published21 Aug 2012

Abstract

An operator 𝑇∈ℬ(𝑋) defined on a Banach space 𝑋 satisfies property (𝑔𝑏) if the complement in the approximate point spectrum ğœŽğ‘Ž(𝑇) of the upper semi-B-Weyl spectrum ğœŽğ‘†ğµğ¹âˆ’+(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, ğœŽ(𝑇) its spectrum, and ğœŽğ‘Ž(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 ğœŽğ‘Š(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆𝐼isnotaWeyloperator},(1.1) the upper semi-Weyl spectrum (in the literature called also Weyl essential approximate point spectrum) defined by ğœŽğ‘†ğ¹âˆ’+(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 ğœŽğµ(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 ğœŽğ‘ˆğµ(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆𝐼isnotanuppersemi-Browderoperator}.(1.4)

An operator 𝑇∈ℬ(𝑋) is called Riesz if its essential spectrum ğœŽğ‘’(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 ğœŽğ‘†ğ¹âˆ’+(𝑇+𝑅)=ğœŽğ‘†ğ¹âˆ’+ğœŽ(𝑇),𝑊(𝑇+𝑅)=ğœŽğ‘ŠğœŽ(𝑇),𝑈𝐵(𝑇+𝑅)=ğœŽğ‘ˆğµğœŽ(𝑇),𝐵(𝑇+𝑅)=ğœŽğµ(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: ğœŽğ¿ğ·(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆𝐼isnotaleftDrazininvertibleoperatorğœŽ};𝐷(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆𝐼isnotaDrazininvertibleoperatorğœŽ};𝑆𝐵𝐹−+(𝑇)∶={𝜆∈ℂ∶𝑇−𝜆𝐼isnotanuppersemi-B-Weyloperator}.(1.6)

Let Π(𝑇) denote the set of all poles of 𝑇. We say that ğœ†âˆˆğœŽğ‘Ž(𝑇) is a left pole of 𝑇 if 𝑇−𝜆𝐼 is left Drazin invertible. Let Î ğ‘Ž(𝑇) denote the set of all left poles of 𝑇. It is well know that Π(𝑇)=ğœŽ(𝑇)â§µğœŽğ·(𝑇)=isoğœŽ(𝑇)â§µğœŽğ·(𝑇) and Î ğ‘Ž(𝑇)=ğœŽğ‘Ž(𝑇)â§µğœŽğ¿ğ·(𝑇)=isoğœŽğ‘Ž(𝑇)â§µğœŽğ¿ğ·(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 isoğœŽğ‘Ž(𝑇)=∅ or every isolated point of ğœŽğ‘Ž(𝑇) is a left pole of 𝑇.

Following Harte and Lee [4], we say that 𝑇∈ℬ(𝑋) satisfies Browder's theorem if ğœŽğ‘Š(𝑇)=ğœŽğµ(𝑇), while, according to Djordjević and Han [5], we say that 𝑇 satisfies a-Browder's theorem if ğœŽSF−+(𝑇)=ğœŽğ‘ˆğµ(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 ğœŽğ‘Ž(𝑇)â§µğœŽğ‘†ğµğ¹âˆ’+(𝑇)=Π(𝑇),(1.7) while 𝑇∈ℬ(𝑋) is said to satisfy generalized a-Browder's theorem if ğœŽğ‘Ž(𝑇)â§µğœŽğ‘†ğµğ¹âˆ’+(𝑇)=Î ğ‘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 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∗)âŠ†ğœŽğ‘†ğµğ¹âˆ’+(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∗)âŠ†ğœŽğ‘†ğµğ¹âˆ’+(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)=Î ğ‘Ž(𝑇).

Lemma 2.2. If the equality ğœŽğ‘†ğµğ¹âˆ’+(𝑇)=ğœŽğ·(𝑇) holds for 𝑇∈ℬ(𝑋), then 𝑇 possesses property (𝑔𝑏).

Proof. Suppose that ğœŽğ‘†ğµğ¹âˆ’+(𝑇)=ğœŽğ·(𝑇). If ğœ†âˆˆğœŽğ‘Ž(𝑇)â§µğœŽğ‘†ğµğ¹âˆ’+(𝑇), then ğœ†âˆˆğœŽğ‘Ž(𝑇)â§µğœŽğ·(𝑇)⊆Π(𝑇). This implies that ğœŽğ‘Ž(𝑇)â§µğœŽğ‘†ğµğ¹âˆ’+(𝑇)=Π(𝑇). Since Π(𝑇)âŠ†ğœŽğ‘Ž(𝑇)â§µğœŽğ‘†ğµğ¹âˆ’+(𝑇) is always true, ğœŽğ‘Ž(𝑇)â§µğœŽğ‘†ğµğ¹âˆ’+(𝑇)=Π(𝑇), that is, 𝑇 possesses property (𝑔𝑏).

Lemma 2.3. If 𝑇∈ℬ(𝑋), then ğœŽğ‘†ğµğ¹âˆ’+(𝑇)∪𝑆(𝑇∗)=ğœŽğ·(𝑇).

Proof. Let ğœ†âˆ‰ğœŽğ‘†ğµğ¹âˆ’+(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, ğœ†âˆ‰ğœŽğ·(𝑇). Hence, ğœŽğ·(𝑇)âŠ†ğœŽğ‘†ğµğ¹âˆ’+(𝑇)∪𝑆(𝑇∗). Since the reverse inclusion obviously holds, we get ğœŽğ‘†ğµğ¹âˆ’+(𝑇)∪𝑆(𝑇∗)=ğœŽğ·(𝑇).

Theorem 2.4. If 𝑇∈ℬ(𝑋) satisfies 𝑆(𝑇∗)âŠ†ğœŽğ‘†ğµğ¹âˆ’+(𝑇), then 𝑇 possesses property (𝑔𝑏). In particular, if 𝑇∗ has SVEP, then 𝑇 possesses property (𝑔𝑏).

Proof. Suppose that 𝑆(𝑇∗)âŠ†ğœŽğ‘†ğµğ¹âˆ’+(𝑇). Then by Lemma 2.3, we get ğœŽğ‘†ğµğ¹âˆ’+(𝑇)=ğœŽğ·(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 𝑇𝑥1,𝑥2=,…0,𝑥1,𝑥2∀𝑥,…𝑛∈𝑙2(ℕ).(2.1) Then, ğœŽğ‘Žî€½||𝜆||,ğœŽ(𝑇)=𝜆∈ℂ∶=1𝑆𝐵𝐹−+(||𝜆||,𝑇)=𝜆∈ℂ∶=1Π(𝑇)=∅.(2.2) Hence ğœŽğ‘Ž(𝑇)â§µğœŽğ‘†ğµğ¹âˆ’+(𝑇)=Π(𝑇), that is, 𝑇 possesses property (𝑔𝑏), but 𝑆(𝑇∗̸)={𝜆∈ℂ∶0≤|𝜆|<1}⊆{𝜆∈ℂ∶|𝜆|=1}=ğœŽğ‘†ğµğ¹âˆ’+(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)=Î ğ‘Ž(𝑇). Hence, 𝑇+𝑁 satisfies generalized a-Browder's theorem. By [18], ğœŽ(𝑇+𝑁)=ğœŽ(𝑇) and ğœŽğ‘Ž(𝑇+𝑁)=ğœŽğ‘Ž(𝑇). Hence, by [19, Theorem 2.2] and [20, Theorem 3.2], we have that Π(𝑇+𝑁)=ğœŽ(𝑇+𝑁)â§µğœŽğ·(𝑇+𝑁)=ğœŽ(𝑇)â§µğœŽğ·(𝑇)=Π(𝑇)=Î ğ‘Ž(𝑇)=ğœŽğ‘Ž(𝑇)â§µğœŽğ¿ğ·(𝑇)=ğœŽğ‘Ž(𝑇+𝑁)â§µğœŽğ¿ğ·Î (𝑇+𝑁)=ğ‘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 𝑇∶𝑙2(ℕ)→𝑙2(ℕ) be the unilateral right shift operator defined by 𝑇𝑥1,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 𝑁𝑥1,𝑥2=,…0,−𝑥1∀𝑥,0,…𝑛∈𝑙2(ℕ).(2.4) Then 𝑇𝑁≠𝑁𝑇. Moreover, ğœŽî€½||𝜆||,ğœŽ(𝑇)=𝜆∈ℂ∶0≤≤1ğ‘Ž(||𝜆||,ğœŽî€½||𝜆||,ğœŽğ‘‡)=𝜆∈ℂ∶=1(𝑇+𝑁)=𝜆∈ℂ∶0≤≤1ğ‘Ž(||𝜆||𝑇+𝑁)=𝜆∈ℂ∶=1∪{0}.(2.5) It follows that Î ğ‘Ž(𝑇)=Π(𝑇)=∅ and {0}=Î ğ‘Ž(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)ğœŽSBF−+(𝑇+𝐹)=ğœŽSBF−+(𝑇),(2)ğœŽğ·(𝑇+𝐹)=ğœŽğ·(𝑇).

Proof. (1) Without loss of generality, we need only to show that 0âˆ‰ğœŽğ‘†ğµğ¹âˆ’+(𝑇+𝐹) if and only if 0âˆ‰ğœŽğ‘†ğµğ¹âˆ’+(𝑇). By symmetry, it suffices to prove that 0âˆ‰ğœŽğ‘†ğµğ¹âˆ’+(𝑇+𝐹) if 0âˆ‰ğœŽğ‘†ğµğ¹âˆ’+(𝑇).
Suppose that 0âˆ‰ğœŽğ‘†ğµğ¹âˆ’+(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, 0âˆ‰ğœŽğ‘†ğµğ¹âˆ’+(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∗)âŠ†ğœŽSBF−+(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, ğœŽğ‘†ğµğ¹âˆ’+(𝑇+𝐹)=ğœŽğ‘†ğµğ¹âˆ’+(𝑇) and ğœŽğ·(𝑇+𝐹)=ğœŽğ·(𝑇). Since 𝑆(𝑇∗)âŠ†ğœŽğ‘†ğµğ¹âˆ’+(𝑇), by Lemma 2.3, ğœŽğ‘†ğµğ¹âˆ’+(𝑇)=ğœŽğ·(𝑇). Thus, ğœŽğ‘†ğµğ¹âˆ’+(𝑇+𝐹)=ğœŽğ·(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 𝑈𝑥1,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 𝐹𝜀𝑥1,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, ğœŽî€½||𝜆||,ğœŽ(𝑇)=ğœŽ(𝑈)âˆªğœŽ(𝐼)=𝜆∈ℂ∶0≤≤1ğ‘Ž(𝑇)=ğœŽğ‘Ž(𝑈)âˆªğœŽğ‘Ž(||𝜆||,ğœŽî€·ğ¼)=𝜆∈ℂ∶=1(𝑇+𝐹)=ğœŽ(𝑈)âˆªğœŽğ¼+𝐹𝜀=||𝜆||,ğœŽğœ†âˆˆâ„‚âˆ¶0≤≤1ğ‘Ž(𝑇+𝐹)=ğœŽğ‘Ž(𝑈)âˆªğœŽğ‘Žî€·ğ¼+𝐹𝜀=||𝜆||𝜆∈ℂ∶=1∪{1−𝜀}.(2.9) It follows that Î ğ‘Ž(𝑇)=Π(𝑇)=∅ and {1−𝜀}=Î ğ‘Ž(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 ğœŽğ‘†ğµğ¹âˆ’+(𝑇+𝑄)=ğœŽğ‘†ğµğ¹âˆ’+(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 0âˆ‰ğœŽğ‘†ğµğ¹âˆ’+(0)=ğœŽğ‘†ğµğ¹âˆ’+(𝑇), but 0âˆˆğœŽğ‘†ğµğ¹âˆ’+(𝑄)=ğœŽğ‘†ğµğ¹âˆ’+(0+𝑄)=ğœŽğ‘†ğµğ¹âˆ’+(𝑇+𝑄). Hence, ğœŽğ‘†ğµğ¹âˆ’+(𝑇+𝑄)â‰ ğœŽğ‘†ğµğ¹âˆ’+(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 𝑈𝑥1,𝑥2=,…0,𝑥1,𝑥2∀𝑥,…𝑛∈𝑙2(ℕ).(2.12)
Let 𝑉∶𝑙2(ℕ)→𝑙2(ℕ) be a quasinilpotent operator defined by 𝑉𝑥1,𝑥2=,…0,𝑥1𝑥,0,33,𝑥44∀𝑥,…𝑛∈𝑙2(ℕ).(2.13)
Let 𝑁∶𝑙2(ℕ)→𝑙2(ℕ) be a quasinilpotent operator defined by 𝑁𝑥1,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, ğœŽî€½||𝜆||,ğœŽ(𝑇)=ğœŽ(𝑈)âˆªğœŽ(𝑉)=𝜆∈ℂ∶0≤≤1ğ‘Ž(𝑇)=ğœŽğ‘Ž(𝑈)âˆªğœŽğ‘Ž(||𝜆||î€¾ğœŽî€½||𝜆||,ğœŽğ‘‰)=𝜆∈ℂ∶=1∪{0},(𝑇+𝑄)=ğœŽ(𝑈)âˆªğœŽ(𝑉+𝑁)=𝜆∈ℂ∶0≤≤1ğ‘Ž(𝑇+𝑄)=ğœŽğ‘Ž(𝑈)âˆªğœŽğ‘Ž(||𝜆||𝑉+𝑁)=𝜆∈ℂ∶=1∪{0}.(2.15)
It follows that Î ğ‘Ž(𝑇)=Π(𝑇)=∅ and {0}=Î ğ‘Ž(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)=Î ğ‘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 Π(𝑇+𝑄)=Î ğ‘Ž(𝑇+𝑄). Since Π(𝑇+𝑄)âŠ†Î ğ‘Ž(𝑇+𝑄) is always true, one needs only to show that Î ğ‘Ž(𝑇+𝑄)⊆Π(𝑇+𝑄).
Let ğœ†âˆˆÎ ğ‘Ž(𝑇+𝑄)=ğœŽğ‘Ž(𝑇+𝑄)â§µğœŽğ¿ğ·(𝑇+𝑄)=isoğœŽğ‘Ž(𝑇+𝑄)â§µğœŽğ¿ğ·(𝑇+𝑄). Then by [18], 𝜆∈isoğœŽğ‘Ž(𝑇). Since 𝑇 is a-polaroid, ğœ†âˆˆÎ ğ‘Ž(𝑇)=Π(𝑇). Thus by [29, Theorem 3.12], 𝜆∈Π(𝑇+𝑄). Therefore, Î ğ‘Ž(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).

References

  1. H.-O. Tylli, “On the asymptotic behaviour of some quantities related to semi-Fredholm operators,” Journal of the London Mathematical Society, vol. 31, no. 2, pp. 340–348, 1985. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. V. Rakočević, “Semi-Browder operators and perturbations,” Studia Mathematica, vol. 122, no. 2, pp. 131–137, 1997. View at: Google Scholar | Zentralblatt MATH
  3. M. Berkani and M. Sarih, “On semi B-Fredholm operators,” Glasgow Mathematical Journal, vol. 43, no. 3, pp. 457–465, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. R. Harte and W. Y. Lee, “Another note on Weyl's theorem,” Transactions of the American Mathematical Society, vol. 349, no. 5, pp. 2115–2124, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. S. V. Djordjević and Y. M. Han, “Browder's theorems and spectral continuity,” Glasgow Mathematical Journal, vol. 42, no. 3, pp. 479–486, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. M. Berkani and H. Zariouh, “Extended Weyl type theorems,” Mathematica Bohemica, vol. 134, no. 4, pp. 369–378, 2009. View at: Google Scholar | Zentralblatt MATH
  7. M. Berkani and J. J. Koliha, “Weyl type theorems for bounded linear operators,” Acta Scientiarum Mathematicarum, vol. 69, no. 1-2, pp. 359–376, 2003. View at: Google Scholar | Zentralblatt MATH
  8. M. Amouch and H. Zguitti, “On the equivalence of Browder's and generalized Browder's theorem,” Glasgow Mathematical Journal, vol. 48, no. 1, pp. 179–185, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. N. Dunford, “Spectral theory. II. Resolutions of the identity,” Pacific Journal of Mathematics, vol. 2, pp. 559–614, 1952. View at: Google Scholar | Zentralblatt MATH
  10. N. Dunford, “Spectral operators,” Pacific Journal of Mathematics, vol. 4, pp. 321–354, 1954. View at: Google Scholar | Zentralblatt MATH
  11. P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2004. View at: Zentralblatt MATH
  12. K. B. Laursen and M. M. Neumann, An Introduction to Local Spectral Theory, vol. 20, Oxford University Press, New York, NY, USA, 2000.
  13. M. Berkani and H. Zariouh, “New extended Weyl type theorems,” Matematichki Vesnik, vol. 62, no. 2, pp. 145–154, 2010. View at: Google Scholar
  14. M. Berkani and H. Zariouh, “Extended Weyl type theorems and perturbations,” Mathematical Proceedings of the Royal Irish Academy, vol. 110A, no. 1, pp. 73–82, 2010. View at: Publisher Site | Google Scholar
  15. M. H. M. Rashid, “Property (gb) and perturbations,” Journal of Mathematical Analysis and Applications, vol. 383, no. 1, pp. 82–94, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  16. P. Aiena, “Quasi-Fredholm operators and localized SVEP,” Acta Scientiarum Mathematicarum, vol. 73, no. 1-2, pp. 251–263, 2007. View at: Google Scholar | Zentralblatt MATH
  17. P. Aiena, M. T. Biondi, and C. Carpintero, “On Drazin invertibility,” Proceedings of the American Mathematical Society, vol. 136, no. 8, pp. 2839–2848, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  18. M. Mbekhta and V. Müller, “On the axiomatic theory of spectrum. II,” Studia Mathematica, vol. 119, no. 2, pp. 129–147, 1996. View at: Google Scholar | Zentralblatt MATH
  19. M. A. Kaashoek and D. C. Lay, “Ascent, descent, and commuting perturbations,” Transactions of the American Mathematical Society, vol. 169, pp. 35–47, 1972. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  20. O. Bel Hadj Fredj, M. Burgos, and M. Oudghiri, “Ascent spectrum and essential ascent spectrum,” Studia Mathematica, vol. 187, no. 1, pp. 59–73, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  21. S. Grabiner, “Uniform ascent and descent of bounded operators,” Journal of the Mathematical Society of Japan, vol. 34, no. 2, pp. 317–337, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  22. P. A. Fillmore and J. P. Williams, “On operator ranges,” Advances in Mathematics, vol. 7, pp. 254–281, 1971. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  23. S. Grabiner, “Ranges of products of operators,” Canadian Journal of Mathematics, vol. 26, pp. 1430–1441, 1974. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  24. M. Berkani, “Restriction of an operator to the range of its powers,” Studia Mathematica, vol. 140, no. 2, pp. 163–175, 2000. View at: Google Scholar | Zentralblatt MATH
  25. M. Mbekhta, “Résolvant Généralisé et Théorie Spectrale,” Journal of Operator Theory, vol. 21, no. 1, pp. 69–105, 1989. View at: Google Scholar | Zentralblatt MATH
  26. M. Berkani, N. Castro, and S. V. Djordjević, “Single valued extension property and generalized Weyl's theorem,” Mathematica Bohemica, vol. 131, no. 1, pp. 29–38, 2006. View at: Google Scholar | Zentralblatt MATH
  27. X. Cao, “Topological uniform descent and Weyl type theorem,” Linear Algebra and its Applications, vol. 420, no. 1, pp. 175–182, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  28. Q. Jiang, H. Zhong, and Q. Zeng, “Topological uniform descent and localized SVEP,” Journal of Mathematical Analysis and Applications, vol. 390, no. 1, pp. 355–361, 2012. View at: Publisher Site | Google Scholar
  29. Q. Zeng, H. Zhong, and Z. Wu, “Small essential spectral radius perturbations of operators with topological uniform descent,” Bulletin of the Australian Mathematical Society, vol. 85, no. 1, pp. 26–45, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH

Copyright © 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.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views683
Downloads556
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.