{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,2]],"date-time":"2025-11-02T02:55:11Z","timestamp":1762052111056,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,12]],"date-time":"2022-05-12T00:00:00Z","timestamp":1652313600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Escuela Superior Polit\u00e9cnica del Litoral"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>For bounded linear operators defined on complex infinite-dimensional Banach space, H. Zariouh, in an article [1] introduced and studied the property (gaz). In this study, through techniques using the local spectral theory of operators, we discover the sufficient conditions that allow the transfer of the property (gaz) from two tensor factors T and S to their tensor product T\u2297S. The stability of the property (gaz) in the tensor product under perturbations is also investigated. The theory is exemplified by considering suitable classes of operators such as shift operators, convolution operators, and m-invertible contractions.<\/jats:p>","DOI":"10.3390\/axioms11050225","type":"journal-article","created":{"date-parts":[[2022,5,12]],"date-time":"2022-05-12T21:46:53Z","timestamp":1652392013000},"page":"225","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Tensor Product of Operators Satisfying Zariouh\u2019s Property (gaz), and Stability under Perturbations"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5745-3233","authenticated-orcid":false,"given":"Elvis","family":"Aponte","sequence":"first","affiliation":[{"name":"Escuela Superior Polit\u00e9cnica del Litoral, F.C.N.M., Campus Gustavo Galindo Km. 30.5 V\u00eda Perimetral, Guayaquil EC090112, Ecuador"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8056-1710","authenticated-orcid":false,"given":"Narayanapillai","family":"Jayanthi","sequence":"additional","affiliation":[{"name":"Government Arts College (Autonomous), Coimbatore 641018, TamilNadu, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9516-8716","authenticated-orcid":false,"given":"Domingo","family":"Quiroz","sequence":"additional","affiliation":[{"name":"Escuela Superior Polit\u00e9cnica del Litoral, F.C.N.M., Campus Gustavo Galindo Km. 30.5 V\u00eda Perimetral, Guayaquil EC090112, Ecuador"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7139-6319","authenticated-orcid":false,"given":"Ponraj","family":"Vasanthakumar","sequence":"additional","affiliation":[{"name":"Government Arts College (Autonomous), Coimbatore 641018, TamilNadu, India"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,12]]},"reference":[{"key":"ref_1","first-page":"94","article-title":"Property (gz) for bounded linear operators","volume":"65","author":"Zariouh","year":"2013","journal-title":"Mat. Vesn."},{"key":"ref_2","first-page":"314","article-title":"The Zariouh\u2019s property (gaz) through localized SVEP","volume":"72","author":"Aiena","year":"2020","journal-title":"Mat. Vesn."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Aponte, E., Mac\u00edas, J., Sanabria, J., and Soto, J. (2021). B-Fredholm spectra of Drazin invertible operators and applications. Axioms, 10.","DOI":"10.3390\/axioms10020111"},{"key":"ref_4","unstructured":"Aponte, E. (2022). Property (az) through Topological Notions and Some Applications, Transactions of A. Razmadze Mathematical Institute. for appearing."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1007\/s12215-010-0035-x","article-title":"On a-Browder and a-Weyl spectra of tensor products","volume":"59","author":"Duggal","year":"2010","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1017\/S0017089512000407","article-title":"On Weyl\u2019s theorem for tensor products","volume":"55","author":"Kubrusly","year":"2013","journal-title":"Glasgow Math. J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/s12215-011-0023-9","article-title":"Tensor products and property (w)","volume":"60","author":"Duggal","year":"2011","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1007\/s11253-013-0729-2","article-title":"Generalized Weyl\u2019s theorem and tensor product","volume":"64","author":"Rashid","year":"2013","journal-title":"Ukr. Math. J."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1007\/s11253-016-1245-y","article-title":"Stability of versions of the Weyl-type theorems under the tensor product","volume":"68","author":"Rashid","year":"2016","journal-title":"Ukrainian Math. J."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Aiena, P. (2018). Fredholm and Local Spectral Theory II with Applications to Weyl-Type Theorems, Springer.","DOI":"10.1007\/978-3-030-02266-2"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"61","DOI":"10.2140\/pjm.1975.58.61","article-title":"The single valued extension property on a Banach space","volume":"58","author":"Finch","year":"1975","journal-title":"Pac. J. Math."},{"key":"ref_12","unstructured":"Aiena, P. (2004). Fredholm and Local Spectral Theory, with Applications to Multipliers, Kluwer Academic Publishers."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1016\/j.jmaa.2010.12.051","article-title":"Weyl\u2019s theorem and tensor products: A counterexample","volume":"378","author":"Kitson","year":"2011","journal-title":"J. Math. Anal. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1007\/s00020-003-1331-z","article-title":"Weyl\u2019s theorem for some classes of operators","volume":"53","author":"Aiena","year":"2005","journal-title":"Int. Equ. Oper. Theory"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1515\/dema-2020-0020","article-title":"Structure of n-quasi left m-invertible and related classes of operators","volume":"53","author":"Duggal","year":"2020","journal-title":"Demonstr. Math."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/5\/225\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:09:53Z","timestamp":1760137793000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/5\/225"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,12]]},"references-count":15,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["axioms11050225"],"URL":"https:\/\/doi.org\/10.3390\/axioms11050225","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,5,12]]}}}