Abstract and Applied Analysis

Volume 2012 (2012), Article ID 523986, 10 pages

doi:10.1155/2012/523986

Research Article

A Note on Property and Perturbations

Qingping Zeng and Huaijie Zhong

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

Received 14 April 2012; Accepted 26 July 2012

Academic Editor: Sergey V. Zelik

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.

Abstract.

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

1. Introduction

Throughout this paper, let denote the Banach algebra of all bounded linear operators acting on an infinite-dimensional complex Banach space , and let denote its ideal of finite rank operators on . For an operator , let denote its dual, its kernel, its nullity, its range, its defect, its spectrum, and its approximate point spectrum. If the range is closed and (resp., ), then is said to be upper semi-Fredholm (resp., lower semi-Fredholm). If is both upper and lower semi-Fredholm, then is said to be Fredholm. If is either upper or lower semi-Fredholm, then is said to be semi-Fredholm, and its index is defined by . 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

the upper semi-Weyl spectrum (in the literature called also Weyl essential approximate point spectrum) defined by

Recall that the descent and the ascent of are and , respectively (the infimum of an empty set is defined to be ). If and is closed, then is said to be left Drazin invertible. If and is closed, then is said to be right Drazin invertible. If , then is said to be Drazin invertible. Clearly, is both left and right Drazin invertible if and only if is Drazin invertible. An operator is called upper semi-Browder if it is an upper semi-Fredholm operator with finite ascent, while is called Browder if it is a Fredholm operator of finite ascent and descent. The Browder spectrum of is defined by

the upper semi-Browder spectrum (in the literature called also Browder essential approximate point spectrum) is defined by

An operator is called Riesz if its essential spectrum .

Suppose that and that is a Riesz operator commuting with . Then it follows from [1, Proposition 5] and [2, Theorem 1] that

For each integer , define to be the restriction of to viewed as the map from into (in particular ). If there exists such that is closed and is upper semi-Fredholm, then is called upper semi-B-Fredholm. It follows from [3, Proposition 2.1] that if there exists such that is closed and is upper semi-Fredholm, then is closed, is upper semi-Fredholm, and = for all . This enables us to define the index of an upper semi-B-Fredholm operator as the index of the upper semi-Fredholm operator , where is an integer satisfying that is closed and is upper semi-Fredholm. An operator is called upper semi-B-Weyl if is upper semi-B-Fredholm and .

For , let us define the left Drazin spectrum, the Drazin spectrum, and the upper semi-B-Weyl spectrum of as follows, respectively:

Let denote the set of all poles of . We say that is a left pole of if is left Drazin invertible. Let denote the set of all left poles of . It is well know that and . Here and henceforth, for , is the set of isolated points of . An operator is called a-polaroid if or every isolated point of is a left pole of .

Following Harte and Lee [4], we say that satisfies Browder's theorem if , while, according to Djordjević and Han [5], we say that satisfies a-Browder's theorem if .

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 is said to possess property if

while is said to satisfy generalized a-Browder's theorem if

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 satisfies generalized a-Browder's theorem and is a Riesz operator commuting with , then 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 is said to have the single-valued extension property at (SVEP at for brevity) if for every open neighborhood of the only analytic function which satisfies the equation for all is the function .

Let . An operator is said to have SVEP if .

In this paper, we continue the study of property which is studied in some recent papers [6, 13–15]. We show that property is satisfied by an operator satisfying . We give a revised proof of [15, Theorem 3.10] to prove that property is preserved under commuting nilpotent perturbations. We show also that if satisfies and is a finite rank operator commuting with , then satisfies property . We show that if is an a-polaroid operator satisfying property and is a quasinilpotent operator commuting with , then satisfies property . Two counterexamples are also given to show that property 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 possesses property if and only if satisfies generalized a-Browder's theorem and .

Lemma 2.2. If the equality holds for , then possesses property .

Proof. Suppose that . If , then . This implies that . Since is always true, , that is, possesses property .

Lemma 2.3. If , then .

Proof. Let . Then is an upper semi-Weyl operator and has SVEP at . Thus, is an upper semi-B-Fredholm operator and . Hence, there exists such that is closed, is an upper semi-Fredholm operator, and . By [16, Theorem 2.11], . Thus, , by [11, Theorem 3.4(ii)], . By [11, Theorem 3.4(iv)], . Consequently, is a Browder operator. Thus, by [17, Theorem 2.9], we then conclude that is Drazin invertible, that is, . Hence, . Since the reverse inclusion obviously holds, we get .

Theorem 2.4. If satisfies , then possesses property . In particular, if has SVEP, then possesses property .

Proof. Suppose that . Then by Lemma 2.3, we get . Consequently, by Lemma 2.2, possesses property . If has SVEP, then ; the conclusion follows immediately.

The following example shows that the converse of Theorem 2.4 is not true.

Example 2.5. Let be the Hilbert space , and let be the unilateral right shift operator defined by

Then,

Hence , that is, possesses property , but .

The next theorem improves a recent result of Berkani and Zariouh [14, Theorem 2.5] by removing the extra assumption that 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 satisfies property and is a nilpotent operator that commutes with , then satisfies property .

Proof. Suppose that satisfies property and is a nilpotent operator that commutes with . By Lemma 2.1, satisfies generalized a-Browder's theorem and . Hence, satisfies generalized a-Browder's theorem. By [18], and . Hence, by [19, Theorem 2.2] and [20, Theorem 3.2], we have that By Lemma 2.1 again, satisfies property .

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 be the unilateral right shift operator defined by

Let be a nilpotent operator with rank one defined by

Then . Moreover,

It follows that and . Hence, by Lemma 2.1, does not satisfy property . But since has SVEP, satisfies a-Browder's theorem or equivalently, by [8, Theorem 2.2], satisfies generalized a-Browder's theorem. Therefore, by Lemma 2.1 again, satisfies property .

To continue the discussion of this paper, we recall some classical definitions. Using the isomorphism and following [21], a topology on is defined as follows.

Definition 2.8. Let . For every , the operator range topological on is defined by the norm such that for all ,

For a detailed discussion of operator ranges and their topologies, we refer the reader to [22, 23].

Definition 2.9. Let and let . Then has for if for all . If in addition is closed in the operator range topology of for all , then we say that has eventual topological uniform descent, and, more precisely, that has topological uniform descent for .

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 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 and is a finite rank operator commuting with , then

(1),(2).

Proof. Without loss of generality, we need only to show that if and only if . By symmetry, it suffices to prove that if .

Suppose that . Then is an upper semi-B-Fredholm operator and . Hence, it follows from [24, Theorem 3.6] and [20, Theorem 3.2] that is also an upper semi-B-Fredholm operator. Thus, by [21, Theorem 5.8], . Consequently, is an upper semi-B-Weyl operator, that is, , and this completes the proof of .

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 satisfies and is a finite rank operator commuting with , then satisfies property .

Proof. Since is a finite rank operator commuting with , by Lemma 2.10, and . Since , by Lemma 2.3, . Thus, . By Lemma 2.2, satisfies property .

The following example illustrates that property in general is not preserved under commuting finite rank perturbations.

Example 2.12. Let be the unilateral right shift operator defined by

For fixed , let be a finite rank operator defined by

We consider the operators and defined by and , respectively. Then is a finite rank operator and . Moreover,

It follows that and . Hence, by Lemma 2.1, does not satisfy property . But since has SVEP, satisfies a-Browder's theorem or equivalently, by [8, Theorem 2.2], satisfies generalized a-Browder's theorem. Therefore by Lemma 2.1 again, satisfies property .

Rashid gives in [15, Theorem 3.15] that if and is a quasinilpotent operator that commute with , then

The next example shows that this equality does not hold in general.

Example 2.13. Let denote the Volterra operator on the Banach space defined by

is injective and quasinilpotent. Hence, it is easy to see that is not closed for every . Let . It is easy to see that and , but . Hence, .

Rashid claims in [15, Theorem 3.16] that property 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 in general is not preserved under commuting quasinilpotent perturbations.

Example 2.14. Let be the unilateral right shift operator defined by

Let be a quasinilpotent operator defined by

Let be a quasinilpotent operator defined by

It is easy to verify that . We consider the operators and defined by and , respectively. Then is quasinilpotent and . Moreover,

It follows that and . Hence, by Lemma 2.1, does not satisfy property . But since has SVEP, satisfies a-Browder's theorem or equivalently, by [8, Theorem 2.2], satisfies generalized a-Browder's theorem. Therefore, by Lemma 2.1 again, satisfies property .

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 obeys property and that is a quasinilpotent operator commuting with . If is a-polaroid, then obeys .

Proof. Since satisfies property , by Lemma 2.1, satisfies generalized a-Browder's theorem and . Hence, satisfies generalized a-Browder's theorem. In order to show that satisfies property , by Lemma 2.1 again, it suffices to show that . Since is always true, one needs only to show that .

Let . Then by [18], . Since is a-polaroid, . Thus by [29, Theorem 3.12], . Therefore, , 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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