Article
| Title: | On the duality of Dunford-Pettis operators on Banach lattices (English) |
| Author: | Aqzzouz, Belmesnaoui |
| Author: | Elbour, Aziz |
| Author: | Aboutafail, Othman |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 2 |
| Year: | 2025 |
| Pages: | 743-751 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We establish sufficient conditions for the duality of regular Dunford-Pettis operators on Banach lattices and necessary conditions for the duality condition ``if the adjoint of a (positive) operator is Dunford-Pettis, then the operator itself is''. In particular, we show that if each operator $T\colon E\rightarrow F$ from a Banach lattice $E$ with an order continuous norm to another Banach lattice $F$ is Dunford-Pettis whenever its adjoint $T^{\prime }\colon F^{\prime }\rightarrow E^{\prime }$ is Dunford-Pettis, then $E$ has the Schur property or $F$ is a KB-space. As consequences, we deduce a characterization of the Schur property (and KB-spaces). (English) |
| Keyword: | Dunford-Pettis operator |
| Keyword: | order continuous norm |
| Keyword: | positive Schur property |
| Keyword: | KB-space |
| MSC: | 46A40 |
| MSC: | 46B40 |
| MSC: | 46B42 |
| DOI: | 10.21136/CMJ.2024.0523-24 |
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| Date available: | 2025-05-20T11:52:34Z |
| Last updated: | 2025-05-26 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152968 |
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| Reference: | [1] Aliprantis, C. D., Burkinshaw, O.: Positive Operators.Springer, Berlin (2006). Zbl 1098.47001, MR 2262133, 10.1007/978-1-4020-5008-4 |
| Reference: | [2] Aqzzouz, B., Bourass, K., Elbour, A.: Some generalizations on positive Dunford-Pettis operators.Result. Math. 54 (2009), 207-218. Zbl 1191.47051, MR 2534444, 10.1007/s00025-009-0372-2 |
| Reference: | [3] Aqzzouz, B., Elbour, A., Wickstead, A. W.: Positive almost Dunford-Pettis operators and their duality.Positivity 15 (2011), 185-197. Zbl 1232.46019, MR 2803812, 10.1007/s11117-010-0050-3 |
| Reference: | [4] Aqzzouz, B., Nouira, R., Zraoula, L.: On the duality problem of positive Dunford-Pettis operators on Banach lattices.Rend. Circ. Mat. Palermo (2) 57 (2008), 287-294. Zbl 1166.47036, MR 2452672, 10.1007/s12215-008-0021-8 |
| Reference: | [5] Kalton, N. J., Saab, P.: Ideal properties of regular operators between Banach lattices.Ill. J. Math. 29 (1985), 382-400. Zbl 0568.47030, MR 0786728, 10.1215/ijm/1256045630 |
| Reference: | [6] Meyer-Nieberg, P.: Banach Lattices.Universitext. Springer, Berlin (1991). Zbl 0743.46015, MR 1128093, 10.1007/978-3-642-76724-1 |
| Reference: | [7] Wickstead, A. W.: Converses for the Dodds-Fremlin and Kalton-Saab theorems.Math. Proc. Camb. Philos. Soc. 120 (1996), 175-179. Zbl 0872.47018, MR 1373356, 10.1017/S0305004100074752 |
| Reference: | [8] Wnuk, W.: Banach spaces with properties of the Schur type -- A survey.Conf. Semin. Mat. Univ. Bari 249 (1993), 25 pages. Zbl 0812.46010, MR 1230964 |
| Reference: | [9] Zaanen, A. C.: Riesz Spaces II.North-Holland Mathematical Library 30. North-Holland, Amsterdam (1983). Zbl 0519.46001, MR 0704021, 10.1016/s0924-6509(08)x7015-1 |
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