On \(b\)-Weakly Demicompact Operators on Banach Lattices

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/b8543-3760-0663-r On \(b\)-Weakly Demicompact Operators on Banach Lattices Benkhaled, H. , Jeribi, A. Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 4. Abstract: Aqzzouz and Elbour proved that an operator \(T\) on a Banach lattice \(E\) is \(b\)-weakly compact if and only if \(\\|Tx_{n}\\|\rightarrow 0\) as \(n\rightarrow \infty\) for each \(b\)-order bounded weakly null sequence \(\{x_{n}\}\) in \(E_{+}\). In this present paper, we introduce and study new concept of operators that we call \(b\)-weakly demicompact, use it to generalize known classes of operators which defined by \(b\)-weakly compact operators. An operator \(T\) on a Banach lattice \(E\) is said to be \(b\)-weakly demicompact if for every \(b\)-order bounded sequence \(\{x_{n}\}\) in \(E_{+}\) such that \(x_{n}\rightarrow 0\) in \(\sigma(E,E')\) and \(\\|x_{n}-Tx_{n}\\|\rightarrow 0\) as \(n\rightarrow \infty\), we have \(\\|x_{n}\\|\rightarrow 0\) as \(n\rightarrow \infty\). As consequence, we obtain a characterization of \(KB\)-spaces in terms of \(b\)-weakly demicompact operators. After that, we investigate the relationships between \(b\)-weakly demicompact operators and some other classes of operators on Banach lattices espaciallly their relationships with demi Dunford-Pettis operators and order weakly demicompact operators. Keywords: Banach lattice, \(KB\)-space, \(b\)-weakly demicompact operator, order weakly demicompact operator, demi Dunford-Pettis operator Language: English Download the full text For citation: Benkhaled, H. and Jeribi, A. On \(b\)-Weakly Demicompact Operators on Banach Lattices, Vladikavkaz Math. J., 2023, vol. 25, no. 4, pp. 20-28. DOI 10.46698/b8543-3760-0663-r + References 1. Alpay, S., Altn, B. and Tonyal, C. On Property (b) of Vector Lattices, Positivity, 2003, vol. 7, pp. 135-139. DOI: 10.1023/A:1025840528211. 2. Alpay, S. and Altin, B. A Note on \(b\)-Weakly Compact Operators, Positivity, 2007, vol. 11, pp. 575-582. DOI: 10.1007/s11117-007-2110-x. 3. Altin, B. On \(b\)-Weakly Compact Operators on Banach Lattices, Taiwanese Journal of Mathematics, 2007, vol. 11, no. 1, pp. 143-150. 4. Aqzzouz, B. and Elbour, A. Some Properties of the Class of \(b\)-Weakly Compact Operators, Complex Analysis and Operator Theory, 2012, vol. 6, pp. 1139-1145. DOI: 10.1007/s11785-010-0108-z. 5. Petryshyn, W. V. Construction of Fixed Points of Demicompact Mappings in Hilbert Space, Journal of Mathematical Analysis and Applications, 1996, vol. 14, no. 2, pp. 276-284. DOI: 10.1016/0022-247x(66)90027-8. 6. Jeribi, A. Spectral Theory and Applications of Linear Operators and Block Operator Matrices, Springer, 2015. 7. Krichen, B. and O'Regan, D. Weakly Demicompact Linear Operators and Axiomatic Measures of Weak Noncompactness, Mathematica Slovaca, 2019, vol. 69, no. 6, pp. 1403-1412. DOI: 10.1515/ms-2017-0317. 8. Benkhaled, H., Hajji., M. and Jeribi, A. On the Class of Demi Dunford-Pettis Operators, Rendiconti del Circolo Matematico di Palermo Series 2, vol. 72, no. 2, 2022, pp. 901-911. DOI: 10.1007/s12215-021-00702-x 9. Benkhaled, H., Elleuch, A. and Jeribi, A. The Class of Order Weakly Demicompact Operators, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 2020, vol. 114, no. 2. DOI: 10.1007/s13398-020-00808-4. 10. Aliprantis, C. D. and Burkinshaw, O. Positive Operators, Berlin, Springer, 2006. 11. Meyer-Nieberg, P. Banach Lattices, Berlin, Heidelberg, New York, Springer-Verlag, 1991. 12. Altin, B. Some Properties of \(b\)-Weakly Compact Operators, Gazi University Journal of Science, 2005, vol. 18, no. 3, pp. 391-395. 13. Abramovich, Y. A. and Aliprantis, C. D. An Invitation to Operator Theory, Graduate Studies in Mathematics, vol. 50, Providence, American Mathematical Society, 2002. 14. Aqzzouz, B., Elbour, A. and Hmichane, J. The Duality Problem for the Class of \(b\)-Weakly Compact Operators, Positivity, 2009, vol. 13, no. 4, pp. 683-692. DOI: 10.1007/s11117-008-2288-6. 15. Aqzzouz, B., Moussa, M. and Hmichane, J. Some Characterizations of \(b\)-Weakly Compact Operators on Banach Lattices, Mathematical Reports, 2010, vol. 62, pp. 315-324. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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