`Journal of MathematicsVolume 2013 (2013), Article ID 848176, 7 pageshttp://dx.doi.org/10.1155/2013/848176`
Research Article

Properties and for Bounded Linear Operators

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

Received 13 August 2012; Accepted 12 November 2012

Copyright © 2013 M. H. M. Rashid. 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

We shall consider properties which are related to Weyl type theorem for bounded linear operators , defined on a complex Banach space . These properties, that we call property , means that the set of all poles of the resolvent of of finite rank in the usual spectrum are exactly those points of the spectrum for which is an upper semi-Fredholm with index less than or equal to 0 and we call property , means that the set of all poles of the resolvent of in the usual spectrum are exactly those points of the spectrum for which is an upper semi--Fredholm with index less than or equal to 0. Properties and are related to a strong variants of classical Weyl’s theorem, the so-called property and property We shall characterize properties and in several ways and we shall also describe the relationships of it with the other variants of Weyl type theorems. Our main tool is localized version of the single valued extension property. Also, we consider the properties and in the frame of polaroid type operators.

1. Introduction and Preliminary

Throughout this paper, denotes an infinite-dimensional complex Banach space, the algebra of all bounded linear operators on . For an operator we shall denote by the dimension of the kernel , and by the codimension of the range . Let be the class of all upper semi-Fredholm operators, and let be the class of all lower semi-Fredholm operators. The class of all semi-Fredholm operators is defined by , while the class of all Fredholm operators is defined by . If , the index of is defined by Recall that a bounded operator is said bounded below if it injective and has closed range. Evidently, if is bounded below then and . Define The set of Weyl operators is defined by The classes of operators defined above generate the following spectra. Denote by the approximate point spectrum, and by the surjectivity spectrum of . The Weyl spectrum is defined by the Weyl essential approximate point spectrum is defined by while the Weyl essential surjectivity spectrum is defined by Obviously, and from basic Fredholm theory we have Note that is the intersection of all approximate point spectra of compact perturbations of , while is the intersection of all surjectivity spectra of compact perturbations of , see, for instance, [1, Theorem 3.65].

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 . It is well known that if and are both finite then [2, Proposition 1.49]. Moreover, precisely when is a pole of the resolvent of , see Dowson [2, Theorem 1.54].

The class of all upper semi-Browder operators is defined by while the class of all lower semi-Browder operators is defined by The class of all Browder operators is defined by We have see [1, Theorem 3.4]. The Browder spectrum of is defined by the upper Browder spectrum is defined by and analogously the lower Browder spectrum is defined by Clearly, and .

For and a nonnegative integer define to be the restriction of to viewed as a map from into (in particular, ). If for some integer the range space is closed and is an upper (a lower) semi-Fredholm operator, then is called an upper (a lower) semi--Fredholm operator. In this case the index of is defined as the index of the semi-Fredholm operator , see [3]. Moreover, if is a Fredholm operator, then is called a  -Fredholm operator. A semi--Fredholm operator is an upper or a lower semi--Fredholm operator. An operator is said to be a  -Weyl operator if it is a  -Fredholm operator of index zero. The  -Weyl spectrum of is defined by .

An operator is called Drazin invertible if it has a finite ascent and descent. The Drazin spectrum of an operator is defined by . Define also the set by and . Following [4], an operator is said to be left Drazin invertible if . We say that is a left pole of if , and that is a left pole of of finite rank if is a left pole of and . Let denotes the set of all left poles of and let denotes the set of all left poles of of finite rank. From Theorem 2.8 of [4] it follows that if is left Drazin invertible, then is an upper semi--Fredholm operator of index less than or equal to 0.

Let be the set of all poles of the resolvent of and let be the set of all poles of the resolvent of of finite rank, that is . According to [5], a complex number is a pole of the resolvent of if and only if . Moreover, if this is true then .

According also to [5], the space is closed for each . Hence we have always and . We say that Browders theorem holds for if , and that  -Browders theorem holds for if . Following [6], we say that generalized Weyls theorem holds for if , where is the set of all isolated eigenvalues of , and that generalized Browders theorem holds for if . It is proved in Theorem 2.1 of [7] that generalized Browder’s theorem is equivalent to Browder’s theorem. In [4, Theorem 3.9], it is shown that an operator satisfying generalized Weyl’s theorem satisfies also Weyl’s theorem, but the converse does not hold in general. Nonetheless and under the assumption , it is proved in Theorem 2.9 of   [8] that generalized Weyl’s theorem is equivalent to Weyl’s theorem.

Let be the class of all upper semi--Fredholm operators, . The upper  -Weyl spectrum of is defined by . We say that generalized  -Weyls theorem holds for if , where is the set of all eigenvalues of which are isolated in and that obeys generalized  -Browders theorem if . It is proved in [7, Theorem 2.2] that generalized  -Browder’s theorem is equivalent to  -Browder’s theorem, and it is known from [4, Theorem 3.11] that an operator satisfying generalized  -Weyl’s theorem satisfies  -Weyl’s theorem, but the converse does not hold in general and under the assumption it is proved in [8, Theorem 2.10] that generalized  -Weyl’s theorem is equivalent to  -Weyl’s theorem.

Following [9], we say that possesses property if . The property has been studied in [1]. In Theorem 2.8 of [10], it is shown that property implies Weyl’s theorem, but the converse is not true in general. We say that possesses property if . Property has been introduced and studied in [11, 12]. Property extends property to the context of  -Fredholm theory, and it is proved in [11] that an operator possessing property possesses property but the converse is not true in general. According to [13], an operator is said to possess property if , and is said to possess property if . It is shown in Theorem 2.3 of [13] that an operator possessing property possesses property but the converse is not true in general, see also [14]. Following [15], we say an operator is said to be satisfies property if . In Theorem 2.4 of [15], it is shown that satisfies property if and only if satisfies -Browder’s theorem and satisfies property .

The single valued extension property plays an important role in local spectral theory, see the recent monograph of Laursen and Neumann [16] and Aiena [1]. In this article we shall consider the following local version of this property, which has been studied in recent papers, [10, 17] and previously by Finch [18].

Let be the space of all functions that analytic in an open neighborhoods of . Following [18] 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 [17, Proposition 1.8], Laursen proved that if is of finite ascent, then has SVEP.

Theorem 1 (see [19, Theorem 1.3]). If the following statements are equivalent: (i)has SVEP at ; (ii); (iii) does not cluster at ; (iv) is finite dimensional.

By duality we have.

Theorem 2. If the following statements are equivalent: (i) has SVEP at ; (ii); (iii) does not cluster at .

In this paper we shall consider properties which are related to Weyl type theorem for bounded linear operators , defined on a complex Banach space . These properties, that we call property , means that the set of all poles of the resolvent of of finite rank in the usual spectrum are exactly those points of the spectrum for which is an upper semi-Fredholm with index less than or equal to 0 (see Definition 3) and we call property , means that the set of all poles of the resolvent of in the usual spectrum are exactly those points of the spectrum for which is an upper semi--Fredholm with index less than or equal to 0 (see Definition 3). Properties and are related to a strong variants of classical Weyl’s theorem, the so-called property and property introduced by Berkani and Zariouh [13] and more extensively studied in recent papers [12, 14, 20, 21]. We shall characterize properties and in several ways and we shall also describe the relationships of it with the other variants of Weyl type theorems. Our main tool is localized version of the single valued extension property. Also, we consider the properties and in the frame of polaroid type operators.

2. Properties () and ()

Definition 3. Let . We say that satisfies (i)property if ;(ii)property if .

Theorem 4. Let . If satisfies property , then satisfies property .

Proof. Suppose that satisfies property , then . If , then . Since is an upper semi-Fredholm, then . As , then . So, it follows from [1, Theorem 3.4] that . Hence and this implies that . To show the other inclusion, let be arbitrary. Then and hence . Therefore, , that is, satisfies property .

Theorem 5. Let . Then the following assertions hold. (i)If satisfies property , then satisfies property .(ii)If satisfies property , then satisfies property .

Proof. (i) Assume that satisfies property , then . If , then and so . To show the other inclusion, let be an arbitrary. Then and hence . Therefore, , that is, satisfies property .
(ii) Assume that satisfies property , then . If , then and so . To show the other inclusion, let be an arbitrary. Then and hence . Therefore, , that is, satisfies property .

The following example shows the converse of the previous theorem does not hold in general.

Example 6. Consider the operator that defined on , where is the right unilateral shift operator and . Then , where is the unit disc of . Hence, and so, . Moreover, , where is the unit circle of . Since and , then satisfies both property and property . On the other hand, since and , then does not satisfy property nor the property .

As a consequence of Theorem 5, we have.

Corollary 7. Let . Then the following assertions hold. (i)If satisfies property , then -Browder’s theorem holds for . (ii)If satisfies property , then generalized -Browder’s theorem holds for .

Theorem 8. Let . Then the following assertions hold. (i) satisfies property if and only if   satisfies property and .(ii)If satisfies property if and only if satisfies property and .

Proof. (i) If satisfies property , then satisfies property and consequently, and . Hence and . This implies that . Conversely, assume that satisfies property and . Then That is, satisfies property .
(ii) If satisfies property , then satisfies property and consequently, and . Hence and . This implies that . Conversely, assume that satisfies property and . Then That is, satisfies property .

Theorem 9. Let . Then the following assertions are equivalent. (i) satisfies property . (ii) satisfies property and .

Proof. (i)(ii) Assume that satisfies property , then satisfies property . As satisfies property , then satisfies property , and hence it follows from [13, Corollary 2.9] that satisfies generalized -Browder’s theorem and .
(ii)(i) Assume that satisfies property and then by Corollary 7, satisfies -Browder’s theorem. As we know from [7, Theorem 2.2] that -Browder’s theorem is equivalent to generalized -Browder’s theorem, then satisfies generalized -Browder’s theorem. Hence we have . Since satisfies property , then Theorem 8 implies that satisfies property and . By hypothesis , it then follows that and satisfies property .

Theorem 10. Suppose that has SVEP at every . Then the following assertions are equivalent: (i); (ii); (iii).
Consequently, property , property , property , generalized -Weyl’s theorem and generalized Weyl’s theorem are equivalent for .

Proof. Suppose that has SVEP at every . We prove first the equality . If then is an upper semi--Fredholm operator and . As has SVEP, then it follows from Corollary 2.8 of [22] that is a  -Weyl operator and so . Therefore, . Since the other inclusion is always verified, we have the equality. Now we prove that . Since is always verified. Then has SVEP at every . This implies that satisfies Browder’s theorem. As we know from Theorem 2.1 of [7] that Browder’s theorem is equivalent to generalized Browder’s theorem, we have . On the other hand, as has SVEP at every , then . From this we deduce that and from which the equivalence of (i), (ii) and (iii) easily follows. To show the last statement observed that the SVEP of at the points entails that generalized -Browder’s theorem (and hence generalized Browder’s theorem) holds for , see [20, Corollary 2.7]. Therefor, That is, property , property , property , generalized -Weyl’s theorem and generalized Weyl’s theorem are equivalent for .

Dually, we have.

Theorem 11. Suppose that has SVEP at every . Then the following assertions are equivalent: (i); (ii); (iii).
Consequently, property , property , property , generalized -Weyl’s theorem and generalized Weyl’s theorem are equivalent for .

Proof. Suppose that has SVEP at every . We prove first the equality . If then is a lower semi--Fredholm operator and . As has SVEP, then it follows from Theorem 2.5 of [22] that is a  -Weyl operator and so . As . Then . So . As , then . Since the other inclusion is always verified, it then follows that . Now we show that . Since we have always , then has SVEP at every . Hence satisfies generalized Browder’s theorem. So . Finally, we have and , from which we obtain and . The SVEP at every ensure by Corollary 2.7 of [20] that generalized -Browder’s theorem (and hence generalized Browder’s theorem) holds for . Hence That is, property , property , property , generalized -Weyl’s theorem and generalized Weyl’s theorem are equivalent for .

Corollary 12. Suppose that has SVEP at every . Then the following assertions are equivalent: (i); (ii); (iii).
Consequently, property , property , property , property ,  -Weyl’s theorem and Weyl’s theorem are equivalent for .

Proof. Assume that has SVEP at every . Then it follows from [1, Corollary 2.5] that and by Corollary 3.53 of [1], we then have . Then the SVEP of at every entails that -Browder’s theorem (and hence Browder’s theorem) holds for , see [23, Theorem 2.3]. Then it follows by Theorem 2.19 of [15] that are equivalent. Hence Therefore, property , property , property , property ,  -Weyl’s theorem and Weyl’s theorem are equivalent for .

Dually, we have.

Corollary 13. Suppose that has SVEP at every . Then the following assertions are equivalent: (i); (ii); (iii).
Consequently, property , property , property , property , Weyl’s theorem and -Weyl’s theorem are equivalent for .

Proof. Assume that has SVEP at every . Then it follows from [1, Corollary 2.5] that and by Corollary 3.53 of [1], we then have . Then the SVEP of at every entails that -Browder’s theorem (and hence Browder’s theorem) holds for , see [23, Theorem 2.3]. Then it follows by Theorem 2.20 of [15] that are equivalent. Hence Therefore, property , property , property , property , Weyl’s theorem and -Weyl’s theorem are equivalent for .

Definition 14 (see [21]). A bounded linear operator is said to satisfy property if and is said to satisfy property if .

Definition 15 (see [24]). A bounded linear operator is said to satisfy property if and is said to satisfy property if .

Theorem 16. Let . Then satisfies property if and only if satisfies property and .

Proof. Assume that satisfies property , then . As satisfies property , then we conclude from [24, Theorem 2.7] that satisfies property ,   and from [21, Theorem 3.5] that . Therefore, . That is, satisfies property . Conversely, if satisfies property and . Then . That is, satisfies property .

Similarly we have the following result in the case of property , which we give without proof.

Theorem 17. Let . Then satisfies property if and only if satisfies property and .

3. Properties () and () for Polaroid Type Operators

In this section we consider classes of operators for which the isolated points of the spectrum are poles of the resolvent.

An operator is said to be polaroid if isolated point of is a pole of the resolvent of . is said to be -polaroid if every isolated of is a pole of the resolvent of .

It is easily seen that if is -polaroid, then is polaroid, while in general the converse is not true. It is well known that is a pole of the resolvent of if and only if is a pole of the resolvent of . Since we then have

From the proof of Theorem 10 we know that if has SVEP, then . Therefore, if has SVEP then If has SVEP, we know that . Therefore, if   has SVEP, then

Theorem 18. Suppose that is -polaroid. Then the following assertions hold. (i) satisfies property if and only if  satisfies property .(ii) satisfies property if and only if   satisfies property .

Proof. (i) Note first that if   is -polaroid then . In fact, if then is isolated in and hence . Moreover, , so by Theorem 3.4 of [1] it follows that is also finite, thus . This shows . Since the other inclusion is always verified, we have . Therefore, it follows from Theorem 17 that satisfies property if and only if satisfies property .
(ii) Note first that if   is -polaroid then . In fact, if then is isolated in and hence . Therefore, . This shows . Since we have always , and so . Therefore, it follows from Theorem 16 that satisfies property if and only if satisfies property .

Theorem 19. Suppose that is polaroid. (i)If   has SVEP, then property holds for if and only if   satisfies property .(ii)If   has SVEP, then property holds for if and only if   satisfies property .

Proof. (i) Since has SVEP, then it follows from [1, Corollary 2.45] that , , see also [25, Theorem 1.5]. Since is polaroid and has SVEP, then by equivalence (29), we have is -polaroid and so the result follows now from Theorem 18.
(ii) The SVEP for implies by Corollary 2.45 of [1] that and . Since is polaroid by equivalence (28) and has SVEP, then is -polaroid and so the result follows now from Theorem 18.

Similarly we have the following result in the case of property , which we give without proof.

Theorem 20. Suppose that is polaroid. (i)If   has SVEP, then property holds for if and only if   satisfies property .(ii)If   has SVEP, then property holds for if and only if   satisfies property .

Theorem 21. Suppose that is polaroid and .(i)If   has SVEP, then property holds for if and only if property holds for . (ii)If   has SVEP, then property holds for if and only if property holds for .

Proof. (i) It follows from Lemma 3.11 of [26] that is polaroid. By Theorem 2.40 of [1], we have has SVEP, hence from equivalence (29) we conclude that is -polaroid and hence it then follows by Theorem 19 that property holds for if and only if property holds for .
(ii) From the equivalence (28) we know that is polaroid and hence by Lemma 3.11 of [26] that is polaroid. By Theorem 2.40 of [1], we have has SVEP, hence from equivalence (29) we conclude that is -polaroid and hence it then follows by Theorem 19 that property holds for if and only if property holds for .

Similarly we have the following result in the case of property , which we give without proof.

Theorem 22. Suppose that is polaroid and .(i)If has SVEP, then property holds for if and only if property holds for . (ii)If has SVEP, then property holds for if and only if property holds for .

Acknowledgments

The author would like to express their appreciations to the referees for their careful and kind comments.

References

1. P. Aiena, Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2004.
2. H. R. Dowson, Spectral Theory of Linear Operators, vol. 12, Academic Press, London, UK, 1978.
3. M. Berkani and M. Sarih, “On semi B-Fredholm operators,” Glasgow Mathematical Journal, vol. 43, no. 3, pp. 457–465, 2001.
4. 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.
5. H. Heuser, Functional Analysis, Marcel Dekker, New York, NY, USA, 1982.
6. M. Berkani, “$B$-Weyl spectrum and poles of the resolvent,” Journal of Mathematical Analysis and Applications, vol. 272, no. 2, pp. 596–603, 2002.
7. 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.
8. M. Berkani, “On the equivalence of Weyl theorem and generalized Weyl theorem,” Acta Mathematica Sinica, vol. 23, no. 1, pp. 103–110, 2007.
9. V. Rakočević, “On a class of operators,” Matematički Vesnik, vol. 37, no. 4, pp. 423–426, 1985.
10. P. Aiena and P. Peña, “Variations on Weyl's theorem,” Journal of Mathematical Analysis and Applications, vol. 324, no. 1, pp. 566–579, 2006.
11. M. Amouch and M. Berkani, “On the property $\left(gw\right)$,” Mediterranean Journal of Mathematics, vol. 5, no. 3, pp. 371–378, 2008.
12. M. H. M. Rashid, “Property (gw) and perturbations,” Bulletin of the Belgian Mathematical Society, vol. 18, no. 4, pp. 635–654, 2011.
13. M. Berkani and H. Zariouh, “Extended Weyl type theorems,” Mathematica Bohemica, vol. 134, no. 4, pp. 369–378, 2009.
14. M. H. M. Rashid, “Property (gb) and perturbations,” Journal of Mathematical Analysis and Applications, vol. 383, no. 1, pp. 82–94, 2011.
15. P. Aiena, J. R. Guillén, and P. Peña, “Property $\left(R\right)$ for bounded linear operators,” Mediterranean Journal of Mathematics, vol. 8, no. 4, pp. 491–508, 2011.
16. K. B. Laursen and M. M. Neumann, An Introduction to Local Spectral Theory, vol. 20, The Clarendon Press Oxford University, New York, NY, USA, 2000.
17. K. B. Laursen, “Operators with finite ascent,” Pacific Journal of Mathematics, vol. 152, no. 2, pp. 323–336, 1992.
18. J. K. Finch, “The single valued extension property on a Banach space,” Pacific Journal of Mathematics, vol. 58, no. 1, pp. 61–69, 1975.
19. P. Aiena and C. Carpintero, “Weyl's theorem, $\alpha$-Weyl's theorem and single-valued extension property,” Extracta Mathematicae, vol. 20, no. 1, pp. 25–41, 2005.
20. M. Berkani, M. Sarih, and H. Zariouh, “Browder-type theorems and SVEP,” Mediterranean Journal of Mathematics, vol. 8, no. 3, pp. 399–409, 2011.
21. M. Berkani and H. Zariouh, “New extended Weyl type theorems,” Matematichki Vesnik, vol. 62, no. 2, pp. 145–154, 2010.
22. 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.
23. P. Aiena, C. Carpintero, and E. Rosas, “Some characterizations of operators satisfying $a$-Browder's theorem,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 530–544, 2005.
24. H. Zariouh, “Property (gz) for bounded linear operator,” Mathematica Bohemica, vol. 65, no. 1, pp. 94–103, 2013.
25. P. Aiena, “Property $\left(w\right)$ and perturbations. II,” Journal of Mathematical Analysis and Applications, vol. 342, no. 2, pp. 830–837, 2008.
26. P. Aiena, E. Aponte, and E. Balzan, “Weyl type theorems for left and right polaroid operators,” Integral Equations and Operator Theory, vol. 66, no. 1, pp. 1–20, 2010.