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Generalized α-Browder's and Generalized α-Weyl's Theorems for Banach Space

M. H. M. RASHID

Department of Mathematics& Statistics, Faculty of Science P.O.Box(7), Mu'tah University, Al-Karak, Jordan

Abstract: In this paper, we give necessary and sufficient conditions for a Banach space T to satisfy the generalized α-Browder’s theorem. We also prove that the spectral mapping theorem holds for the left Drazin invertible and for analytic functions on a neighborhood of σ(T). As applications, we show that if T* is algebraically ωF(p,r,q) for each p,r>0 and q≥1, or if T* is algebraically quasi-class A, then the generalized α-Weyl’s theorem hold for f(T), where f∈Hol(σ(T)), the space of functions analytic on an open neighborhoods of σ(T).

Keywords: Weyl’s theorem, α-Weyl’s theorem, generalized α-Weyl’s theorem, α-Browder’s theorem, generalized α-Browder’s theorem, reduction -isoloid, reduced subspace

Citation: M. H. M. RASHID, Generalized α-Browder's and Generalized α-Weyl's Theorems for Banach Space. Turkish Journal of Analysis and Number Theory, 4 (5). 146-154. January 2016. doi:10.12691/tjant-4-5-5

1. Introduction

Throughout this paper let denote, respectively, the algebra of bounded linear operators, the set of finite rank operators and the ideal of compact operators acting on an infinite dimensional Banach space If we shall write and (or ) for the null space and range of , respectively. Also, let , , and let denote the spectrum, approximate point spectrum and point spectrum of , respectively. An operator called Fredholm if it has closed range, finite dimensional null space, and its range has finite codimension. The index of a Fredholm operator is given by

is called Weyl if it is Fredholm of index 0, and Browder if it is Fredholm of finite ascent and descent.

Recall that the ascent, , of an operator is the smallest non-negative integer such that . If such integer does not exist we put . Analogously, the descent, , of an operator is the smallest non-negative integer such that , and if such integer does not exist we put . The essential spectrum , the Weyl spectrum and the Browder spectrum of are defined by

and

respectively. Evidently

where we write for the accumulation points of .

Following [12], we say that Weyl’s theorem holds for if where is the set of all eigenvalues of finite multiplicity isolated in . And Browder’s theorem holds for if where is the set of all poles of of finite rank.

Let be the class of all upper semi-Fredholm operators, be the class of all with , and for any let

Let be the set of all eigenvalues of of finite multiplicity which are isolated in . According to [23], we say that satisfies -Weyl’s theorem if . It follows from [[23], Corollary 2.5]] -Weyl’s theorem implies Weyl’s theorem.

In [11] Berkani define the class of B-Fredholm operators as follows. For each integer , define to be the restriction of to viewed as a map from into (in particular ). If for some the range is closed and is Fredholm (resp. Semi-Fredholm) operator, then is called a B-Fredholm (resp. Semi-B-Fredholm ) operator. In this case and from [5] is a Fredholm operator and for each . The index of a -Fredholm operator is defined as the index of the Fredholm operator , where is any integer such that the range is closed and is Fredholm operator (see [11]).

Let be the class of all B-Fredholm operators. In [5] Berkani has studied this class of operators and has proved that an operator is a B-Fredholm if and only if , where is a Fredholm operator and is a nilpotent operator.

Recall that an operator is called a B-Weyl operator(see [7]) if it is a -Fredholm operator of index . The B-Weyl spectrum of is defined by

In the case of a normal operator acting on a Hilbert space , Berkani [[11], Theorem 4.5] showed that , where is the set of all eigenvalues of which are isolated in the spectrum of . This result gives a generalization of the classical Weyl’s theorem.

Let be the class of all upper semi-B-Fredholm operators, and the class of all such that , and

Recall that an operator satisfies the generalized a-Weyl’s theorem if , where is the set of all eigenvalues of which are isolated in . Note that generalized a-Weyl’s theorem implies a-Weyl’s theorem, (see [10]).

Recall that an operator is a Drazin invertible if and only if it has a finite ascent and descent, which is also equivalent to the fact that , where is nilpotent operator and is invertible operator (see [[19], Proposition A]). The Drazin spectrum is given by

We observe that , where is the set of all poles.

An operator is called left Drazin invertible if and is closed (see [[8], Definition 2.4]). The left Drazin spectrum is given by

Recall [[8], Definition 2.5] that is a left pole of if is left Drazin invertible operator and is a left pole of finite rank if is a left pole of and . We will denote the set of all left pole of , and by the set of all left pole of of finite rank. We have . Note that if , then it is easily seen that is an operator of topological uniform descent. Therefore it follows from ([[10], Theorem 2.5]) that is isolated in . Following  [8] if and is an isolated in , then if and only if and if and only if . The quasinilpotent part and the analytic core of are defined by

and

We note that and are generally non-closed hyper-invariant subspaces of such that for all and . Recall that if , then , where is the glocal spectral subspace consisting of all for which there exists an analytic function that satisfies for all (see [14]).

Let be the space of all functions that analytic in an open neighborhoods of . Following [16], we say that has the single-valued extension property (SVEP) at point if for every open neighborhood of , the only analytic function which satisfies the equation is the constant function . It is well-known that has SVEP at every point of the resolvent . Moreover, from the identity Theorem for analytic function it easily follows that has SVEP at every point of the boundary of the spectrum. In particular, has SVEP at every isolated point of . In [[21], Proposition 1.8], Laursen proved that if is of finite ascent, then has SVEP.

Proposition 1.1.  [20] Let .

(i) If has the SVEP, then for every

(ii) If has the SVEP, then for every

(iii) If has the SVEP, then

In [27] H. Weyl examined the spectra of all compact perturbations of a Hermitian operator on a Hilbert space and proved that their intersection coincides with the isolated point of the spectrum which are the eigenvalues of finite multiplicity. Weyl’s theorem has been extended to several classes of Hilbert space operators including seminormal operators [3,4]. In [6] M. Berkani introduced the concepts of the generalized Weyl’s theorem and generalized Browder’s theorem, and they showed that satisfies the generalized Weyl’s theorem whenever is a normal operator on Hilbert space. More recently, [9] extended this result to hyponormal operators. In  [26] extended this result to -hyponormal operators. Recently, Rashid et al. [25] showed that if is quasi-class then generalized Weyl’s theorem holds for every More recently, I. J. An and Y. M. Han [2] showed that Weyl’s theorem holds for algebraically quasi-class operators.

In this paper we extend this result to several classes much larger than that of normal operators. we first find necessary and sufficient conditions for a Banach space operator to satisfy the generalized -Browder’s theorem. We then characterize the smaller class of operators satisfying the generalized -Weyl’s theorem. A long the way we prove that the spectral mapping theorem always holds for the left Drazin spectrum and for analytic functions on an open neighborhood of we have three main applications of our results: if is an algebraically operators with and , or if is an algebraically quasi-class , then the generalized -Weyl’s theorem holds for , for each and if is reduced by each of its eigenspaces, then generalized -Browder’s theorem holds for for each Throughout of this paper, we will use to signify that an operator obeys generalized a-Weyl’s theorem, analogous meaning is attached to the abbreviations with respect to Browder’s theorem.

2. Structural Properties of Operators in gaB and gaW

Theorem 2.1. Let Then the following statements are equivalent:

(i)

(ii)

(iii)

(iv)

(v)

Proof. (i) (ii). Suppose that . Then Let Then , and so is left Drazin invertible. Therefore and hence, On the other hand, since is always verified for any operator [[10], Lemma 2.12.].

(ii) (i). We assume that and we will establish that Suppose first that Then and so is left Drazin invertible. Therefore and is closed. Since we have Thus

Conversely, suppose that . Then is left Drazin invertible but not bounded below. Since is an isolated point of then Therefore Thus

(ii) (iii). Let Then and so is left Drazin invertible but not bounded below. Therefore Thus Since the other inclusion is always true, we must have

(iii) (ii). Suppose To show that it suffices to show that . Suppose that Then but not invertible. Since we see that In particular, is an isolated point of Hence, is left Drazin invertible, and so

(i) (iv). Suppose . Then Let Then and so is an isolated point of Therefore and hence, .

Conversely, let Since it follows that and . It follows from [[10], Theorem 2.8.] that Therefore For the converse, suppose Then is a left pole of the resolvent of , and so is an isolated point of Therefore It follows from [[10], Theorem 2.11.] that Thus and so .

(iv) (v). Suppose that and let Then but not bounded below. Since is an isolated point of It follows from [[10], Theorem 2.8.] that is a left pole of of the resolvent of . Therefore and hence, Conversely, suppose that and let Then and so is an isolated point of Therefore which implies that

Recall that , see [10]. However, the reverse inclusion does not hold, as the following example shows.

Example 2.2. Let , let be given by and and let on

Then and . Therefore

The next result gives simple necessary and sufficient conditions for an operator to belong to the smaller class .

Theorem 2.3. Let The following statements are equivalent:

(i)

(ii)

(iii)

Proof. (i)(ii). Assume that is, It then easily that as required for (ii).

(ii)(iii). Let . The condition in (ii) implies that , and since we must have It follows that and since the reverse inclusion always holds, we obtain (iii).

(iii)(i). Since we know that and since we are assuming it follows that that is,

Theorem 2.4. Let . Then

Proof. Suppose that Then is left Drazin invertible but not bounded below. Then . Therefore, and is closed. Hence it follows from [[10], Remark 2.7] that is an isolated point of Hence

Conversely, suppose that Then is a semi-B-Fredholm and is an isolated point of . Since is semi--Fredholm, it follows from [[10], Corollary 2.10] that can be decompose as where is an upper semi-Fredholm operator with and is nilpotent. We consider two cases.

Case I. Suppose that is bounded below. Then is left Drazin invertible, and so

Case II. Suppose that is not bounded below. Then 0 is an isolated point of But is an upper semi-Fredholm operator, hence it follows from the punctured neighborhood theorem that is -Browder. Therefore there exists a finite rank operator such that is bounded below and Put Then is a finite rank operator, and is left Drazin invertible. Hence

As shown in [11] that the spectral mapping theorem holds for the Drazin spectrum. We prove here the spectral mapping theorem holds for left Drazin spectrum.

Theorem 2.5. Let and let If , then

Proof. Suppose that and set Then has no zeros in Since by Theorem 2.4, we conclude that has finitely many zeros in Now we consider two cases.

Case I. Suppose that has no zeros in . Then is bounded below, and so

Case II. Suppose that has at least one zero in Then

where and is a nonvanishing analytic function on an open neighborhood. Therefore

where is invertible. Since Therefore is left Drazin invertible, and hence each But each is an isolated point of it follows from [10, Theorem 2.8] that each is a left pole of the resolvent of . Therefore and is closed so

and is closed. Since isinvertible, and is closed. Therefore is left Drazin invertible, and so Hence, It follows from case I and II that

Conversely, suppose that Then is left Drazin invertible. We again consider two cases.

Case I. Suppose that is bounded below. Then and hence Case II. Suppose that Write

where and is invertible. Since is left Drazin invertible

has finite ascent say and is closed. Hence, has finite ascent say and is closed for every Therefore each is left Drazin invertible, and so

We now wish to prove that Assume not; then there exists such that Since , we must have for some , which implies a contradiction. Hence, and so . This completes the proof.

Corollary 2.6. Let and let If , then

Proof. Since it follows from Theorem 2.1 that and . By Theorem 2.5 we have

Hence

Theorem 2.7. Let If has SVEP and then for every In particular, if has SVEP, then

Proof. Suppose that has SVEP. Since it follows from the proof of [[13], Theorem 3.2] that has SVEP. We now show that Let then but not bounded below. Since , it follows from from [[10], Corollary 2.10] that where is an upper semi-Fredholm operator with and is nilpotent. Since has SVEP, and also have SVEP. Therefore -Browder’s theorem holds for and hence, Since is semi-Fredholm with is -Browder’s. Hence, is an isolated point of It follows from Theorem 2.1 that

Now let since the SVEP is stable under the functional calculus, then has the SVEP. Therefore by the first part of the proof.

We now recall that the generalized -Weyl’s theorem may not hold for quasinilpotent operators, and that it does not necessarily transfer to or from adjoints.

Example 2.8. Let defined on by

Then is quasinilpotent operator and

Thus does not obey generalized -Weyl’s theorem.

Now and Therefore .

3. Operators Reduced by Their Eigenspaces

Let denote, the algebra of bounded linear operator acting on an infinite dimensional separable Hilbert space . Let . Suppose that is reduced by each of its eigenspaces. If we let

it follows that reduces . Let and By [[4], Proposition 4.1] we have:

(i) is normal operator with pure point spectrum;

(ii)

(iii)

(iv)

Corollary 3.1. Suppose that is reduced by each of its eigenspaces. Then for every In particular, .

Proof. Since is reduced by each of its eigenspaces, has finite ascent for each Therefore has SVEP, and since the SVEP is stable under functional calculus, then has SVEP for every It follows that

Corollary 3.2. Suppose that is reduced by each of its eigenspaces. Assume that has no isolated points. Then . Moreover, if .

Proof. We first prove that . Since is reduced by each of its eigenspaces, it follows from Corollary 3.1 that By Theorem 2.1, But hence, which implies Therefore On the other hand, observe that and Hence, Let Since is reduced by each of its eigenspaces, has SVEP, and so has SVEP. Therefore

Thus But has no isolated points, hence, is -isoloid. It follows from Theorem 2.3 that generalized -Weyl’s theorem holds for

Lemma 3.3. Suppose that is -isoloid. Then for any we have

Proof. Let then . We distinguish two cases:

Case I. , then there is an infinite sequence such that and . But , therefore and .

Case II. , since then is not an eigenvalue of . Then

where are scalars and is invertible. Since is not an eigenvalue of , then for each , is not an eigenvalue of . Hence, and .

Conversely, Let then . Assume that . Then

where are scalars and is invertible. if , then . Since is a-isoloid , is an eigenvalue of . Hence, . So this leads a contraction to the fact that .

Theorem 3.4. Suppose that is -isoloid and Then for any

Proof. Suppose Then

Since is -isoloid, it follows from Lemma 3.3 that But hence, which implies . Therefore

Suppose that Since is -isoloid, it follows from Lemma 3.3 that

Since , we have Therefore

and hence,

Definition 3.5. Let . Then is said to be reduction--isoloid if the restriction of to every reducing subspace is -isoloid.

Theorem 3.6. Suppose that is both reduction--isoloid and reduced by each of its eigenspaces. Then for every

Proof. We first show that . In view of Theorem 3.4, it suffices to show that Suppose that Then

If then since is -isoloid we have But hence, we must have Since is normal, Hence, and so is which implies Therefore and hence, Now, let Since is reduced by each of its eigenspaces, it follows from the proof of Corollary 3.2 that Therefore by Theorem 3.4.

4. Applications

In this section we show that the generalized -Weyl’s theorem holds to algebraically operators with and and to algebraically quasi-class operators, using the results in section 2 and 3. We begin with the following definition.

4.1. Algebraically Class wF(p,r,q) operators with p, r>0 and q>1

Definition 4.1. For and , an operator is called of class if

We say that is an algebraically class for each and if there exists a nonconstant complex polynomial such that is of class with and .

An operator is called isoloid if every isolated point of is an eigenvalue of . An operator is called normaloid if , where is the spectral radius of . An operator is said to be convexoid if

where means the convex hull of the spectrum of . is called a quasiaffinity if it has trivial kernel and dense range. is said to be a quasiaffine transform of (notation: ) if there is a quasiaffinity such that If both and then we say that and are quasisimilar.

In general, the following implications hold: class algebraically class for each and .

The following facts follow from the above definition and some well known facts about class for each and .

Lemma 4.2. For each and .

(i) If be an algebraically class and is invariant under , then is an algebraically class .

(ii) If is algebraically class then so is for each .

Remark. In what follows, we use the notation to denote the class operators with and

Lemma 4.3. Let belong to class with and . Let . Assume that . Then

Proof. We consider two cases:

Case (I). : Since belongs class for each and , is normaloid. Therefore .

Case (II). : Here is invertible, and since belongs class for each and , we see that is also belongs class for each and . Therefore is normaloid. On the other hand, , so . It follows that is convexoid, so . Therefore .

Proposition 4.4. Let be a quasinilpotent algebraically operator. Then is nilpotent.

Proof. Assume that is operator for some nonconstant polynomial . Since the operator is quasinilpotent. Thus Lemma 4.3 would imply that

where Since is invertible for every we must have

In [28] they showed that every operator is isoloid. We can prove more:

Proposition 4.5. Let be an algebraically operator. Then is polaroid.

Proof. Suppose is an algebraically operator. Then is for some nonconstant polynomial . Let Using the spectral projection where is a closed disk of center which contains no other points of , we can represent as the direct sum

Since is algebraically class and . But it follows from Proposition 4.4 that is nilpotent. Therefore has finite ascent and descent. On the other hand, since is invertible, clearly it has finite ascent and descent. Therefore has finite ascent and descent. Therefore is a pole of the resolvent of . Thus if implies , and so . Hence, is polaroid.

Corollary 4.6. Let be an algebraically operator. Then is isoloid.

Following [17] we say that belongs to class if Recall [15,18] that is called a quasi-class operator if

Definition 4.7. [2] An operator is called an algebraically quasi-class operator if there exists a nonconstant complex polynomial such that is a quasi-class operator.

In general, the following implications hold:

class quasi-class algebraically quasi-class .

The following facts follow from the above definition and some well known facts about quasi-class operators.

(i) If is algebraically quasi-class then so is for each .

(ii) If is algebraically quasi-class and is a closed -invariant subspace of then is algebraically quasi-class .

Lemma 4.8. [18] Let be quasi-class and not have dense range. Then

where , the restriction of to , belongs to class . Moreover,

Lemma 4.9. [[2], Lemma 2.2.] Suppose is a quasinilpotent algebraically quasi-class operator. Then is nilpotent.

Proposition 4.10. Suppose is an algebraically quasi-class operator. If , then is nilpotent.

Proof. Assume is quasi-class for some non-constant complex polynomial . Since

the operator is nilpotent by Lemma 4.9. Let

where for , is an integer, Then

and hence,

Let be the set of all such that is an algebraically quasi-class or algebraically class operator with and .

Theorem 4.11. Let or . Then for all .

Proof. If or . Then or has SVEP. Hence . Now the result follows from Corollary 3.4.

Proposition 4. 12. If . Then is -isoloid.

Proof. Let be an isolated point of Since has SVEP, is an isolated point of . But is polaroid, hence is polaroid. Therefore it is isoloid, and hence Thus is -isoloid.

Corollary 4.13. Let . If , then for every . Consequently,

Theorem 2.4 of [29] affirms that if or has the SVEP and if is -isoloid and generalized -Weyl’s holds for then generalized -Weyl’s theorem holds for , for every . If , then we have:

Theorem 4.14. Let . Then generalized -Weyl’s holds for , for every .

Proof. If . Then has SVEP then it follows from Corollary 2.45 of  [1] that and consequently . Let be given, then is semi--Fredholm and . Then Proposition 1.1 implies that and consequently is -Weyl. Hence, . Hence, it follows from [[29], Theorem 3.1] that .

For the converse, let . Then . Since has SVEP then it follows from Corollary 2.45 of [1] . Hence, . Now we represent as the direct sum , where and . Since then so does , and so we have two cases:

Case I. : then is quasinilpotent. Hence, it follows that is nilpotent. Since is invertible, Then is a -Weyl’s.

Case II. : Since , then is nilpotent and is invertible, it follows from  [29] that is -Weyl’s. Thus in any case

Let Since is -isoloid, then it follows from Theorem 4.11 that

Thus generalized -Weyl’s theorem holds for

An operator is called -polaroid if . In general, if is -polaroid then it is polaroid. However, the converse is not true. Consider the following example.

Example 4.15. Let be the unilateral right shift on and define

Clearly, is a quasi-nilpotent operator. Let We have is the unit disc of , so and hence, is polaroid. Moreover, Since does not cluster at 0, then has the SVEP at 0, as well as at the points Since has SVEP at all points it then follows that T has SVEP. Finally, so Hence, is not -polaroid.

In [2] they showed that every algebraically quasi-class operator is polaroid. We can prove more:

Theorem 16. Let Then is -polaroid.

Proof. Since T satis_es generalized a-Weyl's theorem by Theorem 4.14 and a-isoloid by Proposition 4.12. Then it follows from [[8], Lemma 3.2] that T is a-polaroid.

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