On the hereditary character of new strong variations of weyl type theorems
Berkani and Kachad [18], [19], and Sanabria et al. [32], introduced and studied strong variations of Weyl type Theorems. In this paper, we study the behavior of these strong variations of Weyl type theorems for an operator T on a proper closed and Tinvariant subspace W ⊆ X such that T n (X) ⊆ W for...
- Autores:
-
Carpintero, C.
Malaver, A.
Rosas, E.
Sanabria, J.
- Tipo de recurso:
- Article of journal
- Fecha de publicación:
- 2019
- Institución:
- Corporación Universidad de la Costa
- Repositorio:
- REDICUC - Repositorio CUC
- Idioma:
- eng
- OAI Identifier:
- oai:repositorio.cuc.edu.co:11323/3021
- Acceso en línea:
- http://hdl.handle.net/11323/3021
https://repositorio.cuc.edu.co/
- Palabra clave:
- new Weyl-type theorems
strong variations of Weyl type theorems
restrictions of operators
spectral properties
multiplication operators
- Rights
- openAccess
- License
- Atribución – No comercial – Compartir igual
| id |
RCUC2_53c8f2df72d2d740a15b2d2e084e270e |
|---|---|
| oai_identifier_str |
oai:repositorio.cuc.edu.co:11323/3021 |
| network_acronym_str |
RCUC2 |
| network_name_str |
REDICUC - Repositorio CUC |
| repository_id_str |
|
| dc.title.spa.fl_str_mv |
On the hereditary character of new strong variations of weyl type theorems |
| title |
On the hereditary character of new strong variations of weyl type theorems |
| spellingShingle |
On the hereditary character of new strong variations of weyl type theorems new Weyl-type theorems strong variations of Weyl type theorems restrictions of operators spectral properties multiplication operators |
| title_short |
On the hereditary character of new strong variations of weyl type theorems |
| title_full |
On the hereditary character of new strong variations of weyl type theorems |
| title_fullStr |
On the hereditary character of new strong variations of weyl type theorems |
| title_full_unstemmed |
On the hereditary character of new strong variations of weyl type theorems |
| title_sort |
On the hereditary character of new strong variations of weyl type theorems |
| dc.creator.fl_str_mv |
Carpintero, C. Malaver, A. Rosas, E. Sanabria, J. |
| dc.contributor.author.spa.fl_str_mv |
Carpintero, C. Malaver, A. Rosas, E. Sanabria, J. |
| dc.subject.spa.fl_str_mv |
new Weyl-type theorems strong variations of Weyl type theorems restrictions of operators spectral properties multiplication operators |
| topic |
new Weyl-type theorems strong variations of Weyl type theorems restrictions of operators spectral properties multiplication operators |
| description |
Berkani and Kachad [18], [19], and Sanabria et al. [32], introduced and studied strong variations of Weyl type Theorems. In this paper, we study the behavior of these strong variations of Weyl type theorems for an operator T on a proper closed and Tinvariant subspace W ⊆ X such that T n (X) ⊆ W for some n ≥ 1, where T ∈ L(X) and X is an infinite-dimensional complex Banach space. The main purpose of this paper is to prove that for these subspaces (which generalize the case T n (X) closed for some n ≥ 0), these strong variations of Weyl type theorems are preserved from T to its restriction on W and vice-versa. As consequence of our results, we give sufficient conditions for which these strong variations of Weyl type Theorems are equivalent for two given operators. Also, some applications to multiplication operators acting on the boundary variation space BV [0, 1] are given. |
| publishDate |
2019 |
| dc.date.accessioned.none.fl_str_mv |
2019-04-09T19:25:15Z |
| dc.date.available.none.fl_str_mv |
2019-04-09T19:25:15Z |
| dc.date.issued.none.fl_str_mv |
2019 |
| dc.type.spa.fl_str_mv |
Artículo de revista |
| dc.type.coar.fl_str_mv |
http://purl.org/coar/resource_type/c_2df8fbb1 |
| dc.type.coar.spa.fl_str_mv |
http://purl.org/coar/resource_type/c_6501 |
| dc.type.content.spa.fl_str_mv |
Text |
| dc.type.driver.spa.fl_str_mv |
info:eu-repo/semantics/article |
| dc.type.redcol.spa.fl_str_mv |
http://purl.org/redcol/resource_type/ART |
| dc.type.version.spa.fl_str_mv |
info:eu-repo/semantics/acceptedVersion |
| format |
http://purl.org/coar/resource_type/c_6501 |
| status_str |
acceptedVersion |
| dc.identifier.issn.spa.fl_str_mv |
2344-4967 1221-8421 |
| dc.identifier.uri.spa.fl_str_mv |
http://hdl.handle.net/11323/3021 |
| dc.identifier.instname.spa.fl_str_mv |
Corporación Universidad de la Costa |
| dc.identifier.reponame.spa.fl_str_mv |
REDICUC - Repositorio CUC |
| dc.identifier.repourl.spa.fl_str_mv |
https://repositorio.cuc.edu.co/ |
| identifier_str_mv |
2344-4967 1221-8421 Corporación Universidad de la Costa REDICUC - Repositorio CUC |
| url |
http://hdl.handle.net/11323/3021 https://repositorio.cuc.edu.co/ |
| dc.language.iso.none.fl_str_mv |
eng |
| language |
eng |
| dc.relation.references.spa.fl_str_mv |
[1] P. Aiena, Fredholm and Local Spectral Theory, with Application to Multipliers, Kluwer Acad. Publishers (2004). [2] P. Aiena, Classes of Operators Satisfying a-Weyl’s theorem, Studia Math. 169 (2005), 105- 122. [3] P. Aiena, Quasi-Fredholm operators and localized SVEP. Acta Sci. Math. (Szeged) 73 (2007), 251-263. [4] P. Aiena, E. Aponte and E. Balzan, Weyl type theorems for left and right polaroid operators, Int. Equa. Oper. Theory. 136 (2010), 2839-2848. [5] P. Aiena, M. T. Biondi and C. Carpintero, On Drazin invertibility, Proc. Amer. Math. Soc. 136 (2008), 2839-2848. [6] M. Amouch and M. Berkani On the property (gw), Mediterr. J. Math 5(3)(2008), 371-378. [7] F. Astudillo-Villalba and J. Ramos-Fern´andez Multiplication operators on the space of functions of bounded variation, Demonstr. Math. 50(1) (2017), 105-115. [8] B. Barnes, The spectral and Fredholm theory of extensions of bounded linear operators, Proc. Amer. Math. Soc. 105(4) (1989), 941-949. [9] B. Barnes, Restrictions of bounded linear operators: closed range, Proc. Amer. Math. Soc. 135(6) (2007), 1735-1740. [10] S. K. Berberian, An extension of Weyl’s theorem to a class of non necessarily normal operators, Michigan Math. J. 16(1969), 273-279. [11] M. Berkani, Restriction of an operator to the range of its powers, Studia Math. 140(2) (2000), 163-175. [12] M. Berkani, On a class of quasi-Fredholm operators, Int. Equa. Oper. Theory 34 (1) (1999), 244-249. [13] M. Berkani and M. Sarih, On semi B-Fredholm operators, Glasgow Math. J. 43 (2001), 457-465. [14] M. Berkani and H. Zariouh, Extended Weyl type theorems, Math. Bohemica. 134(4) (2009), 369-378. [15] M. Berkani and J. Koliha, Weyl type theorems for bounded linear operators, Acta Sci. Math. (Szeged) 69 (2003), 359-376. [16] M. Berkani and H. Zariouh, New extended Weyl type theorems, Mat. Vesnik. 62 (2010), 145-154. [17] M. Berkani, M. Sarih and H. Zariouh, Browder-type theorems and SVEP, Mediterr. J. Math. 8 (2011), 399-409. [18] M. Berkani and M. Kachad, New Weyl-type Theorems-I, Funct. Anal. Approx. Comput. 4(2) (2012), 41-47. [19] M. Berkani and M. Kachad, New Browder and Weyl-type theorems, Bull. Korean Math. Soc. 52(2) (2015), 439-452. [20] C. Carpintero, D. Mu˜noz, E. Rosas, O. Garc´ıa and J. Sanabria, Weyl type theorems and restrictions, Mediterr. J. Math. 11 (2014), 1215-1228, DOI 10.1007/s00009-013-0369-7. [21] L. A. Coburn, Weyl’s Theorem for Nonnormal Operators, Research Notes in Mathematics. 51 (1966). [22] L. Chen and W. Su, A note on Weyl-type theorems and restrictions, Ann. Funct. Anal. 8(2) (2017), 190-198. [23] J. K. Finch, The single valued extension property on a Banach space, Pacific J. Math. 58 (1975), 61-69. [24] A. Gupta and K. Mamtani, Weyl-type theorems for restrictions of closed linear unbounded operators, Acta Univ. M. Belli Ser. Math. 2015, 72-79. [25] R. E. Harte and W. Y. Lee, Another note on Weyl’s theorem, Trans. Amer. Math. Soc. 349 (1997), 2115-2124. [26] H. Heuser, Functional Analysis, Marcel Dekker, New York 1982. [27] M. Mbekhta and V. M¨uller, On the axiomatic theory of the spectrum II, Studia Math. 119 (1996), 129-147. [28] J. P. Labrousse, Les op´erateurs quasi Fredholm: une g´en´eralization des op´erateurs semi Fredholm, Rend. Circ. Mat. Palermo. 29(2) (1980), 161-258. [29] V. Rakoˇcevi´c, Operators obeying a-Weyl’s theorem, Rev. Roumaine Math. Pures Appl. 34(10) (1989), 915-919. [30] V. Rakoˇcevi´c, On the essential approximate point spectrum II, Math. Vesnik. 36 (1984), 89-97. [31] J. Sanabria, C. Carpintero, E. Rosas and O. Garc´ıa, On generalized property (v) for bounded linear operators, Studia Math. 212 (2012), 141-154. [32] J. Sanabria, L. V´asquez, C. Carpintero, E. Rosas and O. Garc´ıa, On Strong Variations of Weyl Type Theorems, Acta Math. Univ. Comenianae. 86(2) (2017), 129-147. [33] H. Zariouh, Property (gz) for bounded linear operators, Mat. Vesnik. 65(1) (2013), 94-103 [34] H. Zariouh, New version of property (az), Mat. Vesnik. 66(3) (2014), 317-322 [35] H. Weyl, Uber beschrankte quadratiche Formen, deren Differenz vollsteigist, Rend. Circ. Mat. Palermo, 27 (1909), 373-392. |
| dc.rights.spa.fl_str_mv |
Atribución – No comercial – Compartir igual |
| dc.rights.accessrights.spa.fl_str_mv |
info:eu-repo/semantics/openAccess |
| dc.rights.coar.spa.fl_str_mv |
http://purl.org/coar/access_right/c_abf2 |
| rights_invalid_str_mv |
Atribución – No comercial – Compartir igual http://purl.org/coar/access_right/c_abf2 |
| eu_rights_str_mv |
openAccess |
| dc.publisher.spa.fl_str_mv |
Analele Stiintifice ale Universitatii Al I Cuza din Iasi - Matematica |
| institution |
Corporación Universidad de la Costa |
| bitstream.url.fl_str_mv |
https://repositorio.cuc.edu.co/bitstream/11323/3021/1/On%20the%20hereditary%20character%20of%20new%20strong%20variations%20of%20weyl%20type%20theorems.pdf https://repositorio.cuc.edu.co/bitstream/11323/3021/2/license.txt https://repositorio.cuc.edu.co/bitstream/11323/3021/4/On%20the%20hereditary%20character%20of%20new%20strong%20variations%20of%20weyl%20type%20theorems.pdf.jpg https://repositorio.cuc.edu.co/bitstream/11323/3021/5/On%20the%20hereditary%20character%20of%20new%20strong%20variations%20of%20weyl%20type%20theorems.pdf.txt |
| bitstream.checksum.fl_str_mv |
db68e8a3b670056bcdc0d018797b0108 8a4605be74aa9ea9d79846c1fba20a33 84e3d6ebc1b1dc331a45e938bc149d49 4023f46486e1b2c022f2b45ba72d70a9 |
| bitstream.checksumAlgorithm.fl_str_mv |
MD5 MD5 MD5 MD5 |
| repository.name.fl_str_mv |
Repositorio Universidad de La Costa |
| repository.mail.fl_str_mv |
bdigital@metabiblioteca.com |
| _version_ |
1808400134590955520 |
| spelling |
Carpintero, C.9cf96a93ba15567673667cbe23d33f41Malaver, A.24ea70024ffda0ffa11033bbed84e162300Rosas, E.0c0095c9dc3ff6cdd2a5220fc988ed43Sanabria, J.d7aaf0486681396f394a6f5b3a6d65ba2019-04-09T19:25:15Z2019-04-09T19:25:15Z20192344-49671221-8421http://hdl.handle.net/11323/3021Corporación Universidad de la CostaREDICUC - Repositorio CUChttps://repositorio.cuc.edu.co/Berkani and Kachad [18], [19], and Sanabria et al. [32], introduced and studied strong variations of Weyl type Theorems. In this paper, we study the behavior of these strong variations of Weyl type theorems for an operator T on a proper closed and Tinvariant subspace W ⊆ X such that T n (X) ⊆ W for some n ≥ 1, where T ∈ L(X) and X is an infinite-dimensional complex Banach space. The main purpose of this paper is to prove that for these subspaces (which generalize the case T n (X) closed for some n ≥ 0), these strong variations of Weyl type theorems are preserved from T to its restriction on W and vice-versa. As consequence of our results, we give sufficient conditions for which these strong variations of Weyl type Theorems are equivalent for two given operators. Also, some applications to multiplication operators acting on the boundary variation space BV [0, 1] are given.engAnalele Stiintifice ale Universitatii Al I Cuza din Iasi - MatematicaAtribución – No comercial – Compartir igualinfo:eu-repo/semantics/openAccesshttp://purl.org/coar/access_right/c_abf2new Weyl-type theoremsstrong variations of Weyl type theoremsrestrictions of operatorsspectral propertiesmultiplication operatorsOn the hereditary character of new strong variations of weyl type theoremsArtículo de revistahttp://purl.org/coar/resource_type/c_6501http://purl.org/coar/resource_type/c_2df8fbb1Textinfo:eu-repo/semantics/articlehttp://purl.org/redcol/resource_type/ARTinfo:eu-repo/semantics/acceptedVersion[1] P. Aiena, Fredholm and Local Spectral Theory, with Application to Multipliers, Kluwer Acad. Publishers (2004). [2] P. Aiena, Classes of Operators Satisfying a-Weyl’s theorem, Studia Math. 169 (2005), 105- 122. [3] P. Aiena, Quasi-Fredholm operators and localized SVEP. Acta Sci. Math. (Szeged) 73 (2007), 251-263. [4] P. Aiena, E. Aponte and E. Balzan, Weyl type theorems for left and right polaroid operators, Int. Equa. Oper. Theory. 136 (2010), 2839-2848. [5] P. Aiena, M. T. Biondi and C. Carpintero, On Drazin invertibility, Proc. Amer. Math. Soc. 136 (2008), 2839-2848. [6] M. Amouch and M. Berkani On the property (gw), Mediterr. J. Math 5(3)(2008), 371-378. [7] F. Astudillo-Villalba and J. Ramos-Fern´andez Multiplication operators on the space of functions of bounded variation, Demonstr. Math. 50(1) (2017), 105-115. [8] B. Barnes, The spectral and Fredholm theory of extensions of bounded linear operators, Proc. Amer. Math. Soc. 105(4) (1989), 941-949. [9] B. Barnes, Restrictions of bounded linear operators: closed range, Proc. Amer. Math. Soc. 135(6) (2007), 1735-1740. [10] S. K. Berberian, An extension of Weyl’s theorem to a class of non necessarily normal operators, Michigan Math. J. 16(1969), 273-279. [11] M. Berkani, Restriction of an operator to the range of its powers, Studia Math. 140(2) (2000), 163-175. [12] M. Berkani, On a class of quasi-Fredholm operators, Int. Equa. Oper. Theory 34 (1) (1999), 244-249. [13] M. Berkani and M. Sarih, On semi B-Fredholm operators, Glasgow Math. J. 43 (2001), 457-465. [14] M. Berkani and H. Zariouh, Extended Weyl type theorems, Math. Bohemica. 134(4) (2009), 369-378. [15] M. Berkani and J. Koliha, Weyl type theorems for bounded linear operators, Acta Sci. Math. (Szeged) 69 (2003), 359-376. [16] M. Berkani and H. Zariouh, New extended Weyl type theorems, Mat. Vesnik. 62 (2010), 145-154. [17] M. Berkani, M. Sarih and H. Zariouh, Browder-type theorems and SVEP, Mediterr. J. Math. 8 (2011), 399-409. [18] M. Berkani and M. Kachad, New Weyl-type Theorems-I, Funct. Anal. Approx. Comput. 4(2) (2012), 41-47. [19] M. Berkani and M. Kachad, New Browder and Weyl-type theorems, Bull. Korean Math. Soc. 52(2) (2015), 439-452. [20] C. Carpintero, D. Mu˜noz, E. Rosas, O. Garc´ıa and J. Sanabria, Weyl type theorems and restrictions, Mediterr. J. Math. 11 (2014), 1215-1228, DOI 10.1007/s00009-013-0369-7. [21] L. A. Coburn, Weyl’s Theorem for Nonnormal Operators, Research Notes in Mathematics. 51 (1966). [22] L. Chen and W. Su, A note on Weyl-type theorems and restrictions, Ann. Funct. Anal. 8(2) (2017), 190-198. [23] J. K. Finch, The single valued extension property on a Banach space, Pacific J. Math. 58 (1975), 61-69. [24] A. Gupta and K. Mamtani, Weyl-type theorems for restrictions of closed linear unbounded operators, Acta Univ. M. Belli Ser. Math. 2015, 72-79. [25] R. E. Harte and W. Y. Lee, Another note on Weyl’s theorem, Trans. Amer. Math. Soc. 349 (1997), 2115-2124. [26] H. Heuser, Functional Analysis, Marcel Dekker, New York 1982. [27] M. Mbekhta and V. M¨uller, On the axiomatic theory of the spectrum II, Studia Math. 119 (1996), 129-147. [28] J. P. Labrousse, Les op´erateurs quasi Fredholm: une g´en´eralization des op´erateurs semi Fredholm, Rend. Circ. Mat. Palermo. 29(2) (1980), 161-258. [29] V. Rakoˇcevi´c, Operators obeying a-Weyl’s theorem, Rev. Roumaine Math. Pures Appl. 34(10) (1989), 915-919. [30] V. Rakoˇcevi´c, On the essential approximate point spectrum II, Math. Vesnik. 36 (1984), 89-97. [31] J. Sanabria, C. Carpintero, E. Rosas and O. Garc´ıa, On generalized property (v) for bounded linear operators, Studia Math. 212 (2012), 141-154. [32] J. Sanabria, L. V´asquez, C. Carpintero, E. Rosas and O. Garc´ıa, On Strong Variations of Weyl Type Theorems, Acta Math. Univ. Comenianae. 86(2) (2017), 129-147. [33] H. Zariouh, Property (gz) for bounded linear operators, Mat. Vesnik. 65(1) (2013), 94-103 [34] H. Zariouh, New version of property (az), Mat. Vesnik. 66(3) (2014), 317-322 [35] H. Weyl, Uber beschrankte quadratiche Formen, deren Differenz vollsteigist, Rend. Circ. Mat. Palermo, 27 (1909), 373-392.ORIGINALOn the hereditary character of new strong variations of weyl type theorems.pdfOn the hereditary character of new strong variations of weyl type theorems.pdfapplication/pdf923812https://repositorio.cuc.edu.co/bitstream/11323/3021/1/On%20the%20hereditary%20character%20of%20new%20strong%20variations%20of%20weyl%20type%20theorems.pdfdb68e8a3b670056bcdc0d018797b0108MD51open accessLICENSElicense.txtlicense.txttext/plain; charset=utf-81748https://repositorio.cuc.edu.co/bitstream/11323/3021/2/license.txt8a4605be74aa9ea9d79846c1fba20a33MD52open accessTHUMBNAILOn the hereditary character of new strong variations of weyl type theorems.pdf.jpgOn the hereditary character of new strong variations of weyl type theorems.pdf.jpgimage/jpeg50009https://repositorio.cuc.edu.co/bitstream/11323/3021/4/On%20the%20hereditary%20character%20of%20new%20strong%20variations%20of%20weyl%20type%20theorems.pdf.jpg84e3d6ebc1b1dc331a45e938bc149d49MD54open accessTEXTOn the hereditary character of new strong variations of weyl type theorems.pdf.txtOn the hereditary character of new strong variations of weyl type theorems.pdf.txttext/plain38960https://repositorio.cuc.edu.co/bitstream/11323/3021/5/On%20the%20hereditary%20character%20of%20new%20strong%20variations%20of%20weyl%20type%20theorems.pdf.txt4023f46486e1b2c022f2b45ba72d70a9MD55open access11323/3021oai:repositorio.cuc.edu.co:11323/30212023-12-14 14:57:03.446open accessRepositorio Universidad de La Costabdigital@metabiblioteca.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 |