مشبکه های باناخ گروتندیک و کلاسهایی از عملگرهای −𝒑 همگرا

نویسندگان
حلیمه اردکانی
منیژه سلیمی
چکیده
مفهوم مجموعه های −𝐿 محدود از مرتبهی 𝑝 (1≤𝑝≤∞) در دوگان فضاهای باناخ و همچنین عملگرهای −𝑝 همگرا قبلا معرفی شدهاند. در این مقاله، نتایجی در مورد مجموعه های تقریباً −𝐿 محدود از مرتبهی 𝑝 در دوگان مشبکه های باناخ و عملگرهای −𝑝 همگرای مجزا به دست می آید. به ویژه، نتیجه میگیریم که مشبکه های باناخ با این ویژگی که هر مجموعهی −𝐿 محدود ) یا −𝐿 محدود از مرتبهی 𝑝 ) در دوگان آنها یک مجموعهی فشرده ضعیف نسبی باشد، دقیقاً همان فضاهای گروتندیک هستند. به علاوه، ثابت می شود که یک مشبکه باناخ 𝑀 از فضاهای عملگری خاصیت −𝑝 شور محدود مجزا دارد اگر و تنها اگر همه عملگرهای محاسبهای روی 𝑀 مجزای −𝑝 همگرا باشند.
کلیدواژه‌ها
−𝐿 مجموعه
دنباله ی –𝑝 جمعپذیر
عملگر –𝑝 همگرا
خاصیت –𝑝 شور
مشبکه ی باناخ

عنوان مقاله English

Grothendieck Banach lattices and classes of 𝒑− convergent operators

نویسندگان English

Halimeh Ardakani
Manijeh Salimi
چکیده English

Recently, the concept of 𝐿−limited subsets of order 𝑝 (1≤𝑝<∞) in dual Banach spaces and limited p-convergent operators are introduced. The paper is devoted to some results of almost 𝐿−limited subsets of order p in dual Banach lattices and disjoint limited 𝑝−convergent operators. In particular, we derive the following result: Banach lattices with the property that 𝐿−limited subsets (or 𝐿−limited subsets of order 𝑝) in their dual are relatively weakly compact, are precisely the Grothendieck spaces. Moreover, it is established that a Banach lattice 𝑀 of some operator spaces has the disjoint limited 𝑝−Schur property if and only if all evaluation operators on 𝑀 is disjoint limited p-convergent.

کلیدواژه‌ها English

L-set
weakly p-summable sequence
p-convergent operator
p-Schur property
Banach lattice
1. C. D. Aliprantis and O. Burkishaw, Positive operators, Academic Press, New York and
London, . 1978
2. B. Aqzzouz and K. Bouras, Dunford-Pettis sets in Banach lattices, Acta Math. Univ. Comenianae,
81 (2013), 185– . 196
3. B. Aqzzouz and K. Bouras, L–sets and almost L– sets in Banach lattices, Quaestiones
Mathematicae, 36 (2013), 107–118.
4. H. Ardakani, S. M. S. Modarres Mosadegh, M. Salimi and S. M. Moshtaghioun, L-limited
and almost L-limited sets in dual Banach Lattices, J. Math. Ext., 12 (2018), 147–163.
5. H. Ardakani, S.M. Moshtaghioun, S. M. S. Modarres Mosadegh and M. Salimi, The strong
Gelfand-Phillips property in Banach lattices, Banach J. Math. Anal. 10 (2016), 15–26.
6. H. Ardakani and Kh. Taghavinejad, The strong limited p-Schur property in Banach lattices,
OaM, 16 (2022), 811-825.
7. J. Castillo and F. Sanchez, Dunford-Pettis-like properties of continuous vector function
spaces, Rev. Mat. Univ. Complut. Madrid 6 (1993), 43–59.
8. M. B. Dehghani, S. M. Moshtaghioun and M.Dehghani, On the limited p-Schur propety of
soe operator spaces, IJAA. 16 (2018), 50-61.
9. J. Diestel, Absolutely Summing Operators, Cambridge University Press, (1995).
10. J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Math. 92, Springer–
Verlag, Berlin, 1984.
11. L. Drewnowski, On Banach spaces with the Gelfand-Phillips property, Math. Z. 193 (1986), 405-411.
12. A. Elbour, Some characterizations of almost limited operators, Positivity 21 (2017), 865–
874.
13. G. Emmanuele, On Banach spaces with the Gelfand–Phillips property, III, J. Math. Pures
Appl. 72 (1993), 327–333.
14. K.E. Fahri, N. Machrafi and M. Moussa, Banach Lattices with the Positive Dunford–Pettis
Relatively Compact Property, E.M. 30 (2015), 161–179.
15. J. H. Fourie and E. D. Zeekoei, DP∗-properties of order p on Banach spaces, Quaest. Math.
37 )2014), 349-358 .
16. I. Ghenciu, The p-Gelfand-Phillips property in spaces of operators and Dunford-Pettis like
sets, Acta Math. Hungar., 155 (2018), 439–457.
17. G. Groenewegen and P. Meyer-Nieberg, An elementary and unified approach to disjoint
sequence theorems, Indag. Math. 48 (1986), 313–317.
18. P. Galindo and V. C. C. Miranda, Grothendieck-type subsets of Banach lattices, J. Math.
Anal. Appl. 506 (2022), 1–14.
19. A. El. Kaddouri, M. Moussa, About the class of ordered limited operators, Acta Universitatis
Carolinae. Mathematica et Physica, 54 (2013), 37–43.
20. P. Meyer- Nieberg, Banach lattices, Universitext, springer- Verlag, Berlin, 1991.
21. M. Salimi and S. M. Moshtaghioun, A new class of Banach spaces and its relation with
some geometric properties of Bancah spaces, Hindawi Publishing Corporation, Abstract
and Applied Analysis . 2012
22. M. Salimi and S. M. Moshtaghioun, The Gelfand-Phillips property in closed subspaces of
some operator spaces, Banach J. Math. Anal. 2 (2011), 84–92.
23. W. Wnuk , Banach Lattices with Order Continuous Norms, Polish Scientific Publishers
PWN, Warsaw, 1999.
24. W. Wnuk, Banach lattices with properties of the Schur type- a survey, Conf. Sem. Mat.
Univ. Bari 249 (1993), 1–25.
25. W. Wnuk, Banach lattices with the weak Dunford–Pettis property, Atti Semin. Mat. Fis.
Univ. Modena 42 (1994), 227–236.
26. E. Zeekoei and J. Fourie, Classes of Dunford-Pettis-type operators with applications to Banach
spaces and Banach lattices, PHD Thesis, 27 pp. (2017).