Limited type subsets of locally convex spaces

Saak Gabriyelyan; Saak Gabriyelyan

doi:10.3934/math.20241513

AIMS Mathematics

2024, Volume 9, Issue 11: 31414-31443. doi: 10.3934/math.20241513

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Research article

Limited type subsets of locally convex spaces

Saak Gabriyelyan ^,

Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, P.O. 653, Israel

Received: 19 July 2024 Revised: 12 October 2024 Accepted: 25 October 2024 Published: 05 November 2024
MSC : 46A03, 46E10

Let $ 1\leq p\leq q\leq\infty. $ Being motivated by the classical notions of limited, $ p $-limited, and coarse $ p $-limited subsets of a Banach space, we introduce and study $ (p, q) $-limited subsets and their equicontinuous versions and coarse $ p $-limited subsets of an arbitrary locally convex space $ E $. Operator characterizations of these classes are given. We compare these classes with the classes of bounded, (pre)compact, weakly (pre)compact, and relatively weakly sequentially (pre)compact sets. If $ E $ is a Banach space, we show that the class of coarse $ 1 $-limited subsets of $ E $ coincides with the class of $ (1, \infty) $-limited sets, and if $ 1 < p < \infty $, then the class of coarse $ p $-limited sets in $ E $ coincides with the class of $ p $-$ (V^\ast) $ sets of Pełczyński. We also generalize a known theorem of Grothendieck.
- $ (p, q) $-limited set,
- coarse $ p $-limited set,
- $ p $-$ (V^\ast) $ set,
- $ p $-convergent operator,
- $ p $-barrelled space
Citation: Saak Gabriyelyan. Limited type subsets of locally convex spaces[J]. AIMS Mathematics, 2024, 9(11): 31414-31443. doi: 10.3934/math.20241513

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Abstract

Let $ 1\leq p\leq q\leq\infty. $ Being motivated by the classical notions of limited, $ p $-limited, and coarse $ p $-limited subsets of a Banach space, we introduce and study $ (p, q) $-limited subsets and their equicontinuous versions and coarse $ p $-limited subsets of an arbitrary locally convex space $ E $. Operator characterizations of these classes are given. We compare these classes with the classes of bounded, (pre)compact, weakly (pre)compact, and relatively weakly sequentially (pre)compact sets. If $ E $ is a Banach space, we show that the class of coarse $ 1 $-limited subsets of $ E $ coincides with the class of $ (1, \infty) $-limited sets, and if $ 1 < p < \infty $, then the class of coarse $ p $-limited sets in $ E $ coincides with the class of $ p $-$ (V^\ast) $ sets of Pełczyński. We also generalize a known theorem of Grothendieck.

References

[1]	C. Alonso, Limited sets in Fréchet spaces, Rev. Real Acad. Cienc. Exact. Fs. Natur. Madrid, 89 (1995), 53–60.
[2]	T. Banakh, S. Gabriyelyan, On free locally convex spaces, Filomat, 36 (2022), 6393–6401. http://dx.doi.org/10.2298/FIL2218393B doi: 10.2298/FIL2218393B
[3]	T. Banakh, S. Gabriyelyan, The $b$-Gelfand-Phillips property for locally convex spaces, Collect. Math., 75 (2024), 715–734. http://dx.doi.org/10.1007/s13348-023-00409-5 doi: 10.1007/s13348-023-00409-5
[4]	F. Bombal, On $(V^\ast)$ sets and Pelczynski's property $(V^\ast)$, Glasgow Math. J., 32 (1990), 109–120. http://dx.doi.org/10.1017/S0017089500009113 doi: 10.1017/S0017089500009113
[5]	J. Bourgain, J. Diestel, Limited operators and strict cosingularity, Math. Nachr., 119 (1984), 55–58. http://dx.doi.org/10.1002/mana.19841190105 doi: 10.1002/mana.19841190105
[6]	J. Castillo, F. Sanchez, Dunford-Pettis-like properties of continuous vector function spaces, Rev. Mat. Univ. Complut. Madrid, 6 (1993), 43–59.
[7]	D. Chen, J. Chávez-Domínguez, L. Li, $p$-Converging operators and Dunford-Pettis property of order $p$, J. Math. Anal. Appl., 461 (2018), 1053–1066. http://dx.doi.org/10.1016/j.jmaa.2018.01.051 doi: 10.1016/j.jmaa.2018.01.051
[8]	J. Diestel, Sequences and series in Banach spaces, New York: Springer, 1984. http://dx.doi.org/10.1007/978-1-4612-5200-9
[9]	J. Diestel, H. Jarchow, A. Tonge, Absolutely summing operators, Cambridge: Cambridge University Press, 1995. http://dx.doi.org/10.1017/CBO9780511526138
[10]	M. Fabian, P. Habala, P. Hájek, V. Montesinos, V. Zizler, Banach space theory: the basis for linear and nonlinear analysis, New York: Springer, 2010. http://dx.doi.org/10.1007/978-1-4419-7515-7
[11]	J. Fourie, E. Zeekoei, On weak-star $p$-convergent operators, Quaest. Math., 40 (2017), 563–579. http://dx.doi.org/10.2989/16073606.2017.1301591 doi: 10.2989/16073606.2017.1301591
[12]	S. Gabriyelyan, The Mackey problem for free locally convex spaces, Forum Math., 30 (2018), 1339–1344. http://dx.doi.org/10.1515/forum-2018-0067 doi: 10.1515/forum-2018-0067
[13]	S. Gabriyelyan, Locally convex properties of free locally convex spaces, J. Math. Anal. Appl., 480 (2019), 123453. http://dx.doi.org/10.1016/j.jmaa.2019.123453 doi: 10.1016/j.jmaa.2019.123453
[14]	S. Gabriyelyan, Locally convex spaces and Schur type properties, Ann. Fenn. Math., 44 (2019), 363–378. http://dx.doi.org/10.5186/aasfm.2019.4417 doi: 10.5186/aasfm.2019.4417
[15]	S. Gabriyelyan, Maximally almost periodic groups and respecting properties, In: Descriptive topology and functional analysis II, Cham: Springer, 2019,103–136. http://dx.doi.org/10.1007/978-3-030-17376-0_7
[16]	S. Gabriyelyan, Dunford-Pettis type properties of locally convex spaces, Ann. Funct. Anal., 15 (2024), 55. http://dx.doi.org/10.1007/s43034-024-00359-4 doi: 10.1007/s43034-024-00359-4
[17]	S. Gabriyelyan, Pełczyński's type sets and Pełczyński's geometrical properties of locally convex spaces, arXiv: 2402.08860. http://dx.doi.org/10.48550/arXiv.2402.08860
[18]	S. Gabriyelyan, Gelfand-Phillips type properties of locally convex spaces, arXiv: 2406.07178. http://dx.doi.org/10.48550/arXiv.2406.07178
[19]	S. Gabriyelyan, J. K$ \text{a}̧ $kol, A. Kubzdela, M. Lopez Pellicer, On topological properties of Fréchet locally convex spaces with the weak topology, Topol. Appl., 192 (2015), 123–137. http://dx.doi.org/10.1016/j.topol.2015.05.075 doi: 10.1016/j.topol.2015.05.075
[20]	P. Galindo, V. Miranda, A class of sets in a Banach space coarser than limited sets, Bull. Braz. Math. Soc., New series, 53 (2022), 941–955. http://dx.doi.org/10.1007/s00574-022-00290-z doi: 10.1007/s00574-022-00290-z
[21]	I. Ghenciu, Limited sets and bibasic sequences, Can. Math. Bull., 58 (2015), 71–79. http://dx.doi.org/10.4153/CMB-2014-014-6 doi: 10.4153/CMB-2014-014-6
[22]	I. Ghenciu, The $p$-Gelfand-Phillips property in spaces of operators and Dunford-Pettis like sets, Acta Math. Hungar., 155 (2018), 439–457. http://dx.doi.org/10.1007/s10474-018-0836-5 doi: 10.1007/s10474-018-0836-5
[23]	A. Grothendieck, Sur les applications linéaires faiblement compactes d'espaces du type $C(K)$, Can. J. Math., 5 (1953), 129–173. http://dx.doi.org/10.4153/CJM-1953-017-4 doi: 10.4153/CJM-1953-017-4
[24]	A. Grothendieck, Topological vector spaces, New York: Gordon and Breach, 1973.
[25]	P. Hájek, V. Montesinos, J. Vanderwerff, V. Zizler, Biorthogonal systems in Banach spaces, New York: Springer, 2008. http://dx.doi.org/10.1007/978-0-387-68915-9
[26]	J. Horváth, Topological vector spaces and distributions, I, Reading: Addison-Wesley, 1966.
[27]	H. Jarchow, Locally Convex Spaces, Stuttgart: B. G. Teubner, 1981.
[28]	A. Karn, D. Sinha, An operator summability in Banach spaces, Glasgow Math. J., 56 (2014), 427–437. http://dx.doi.org/10.1017/S0017089513000360 doi: 10.1017/S0017089513000360
[29]	L. Li, D. Chen, J. Chávez-Domínguez, Pełczyński's property $(V^\ast)$ of order $p$ and its quantification, Math. Nachr., 291 (2018), 420–442. http://dx.doi.org/10.1002/mana.201600335 doi: 10.1002/mana.201600335
[30]	M. Lindström, Th. Schlumprecht, On limitedness in locally convex spaces, Arch. Math., 53 (1989), 65–74. http://dx.doi.org/10.1007/BF01194874 doi: 10.1007/BF01194874
[31]	A. Markov, On free topological groups (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 9 (1945), 3–64.
[32]	L. Narici, E. Beckenstein, Topological vector spaces, 2 Eds., New York: Chapman and Hall/CRC, 2010. http://dx.doi.org/10.1201/9781584888673
[33]	C. Nicolescu, Weak compactness in Banach lattices, J. Operator Theory, 6 (1981), 217–213.
[34]	P. Pérez Carreras, J. Bonet, Barrelled locally convex spaces, Amsterdam: North-Holland, 1987.
[35]	J. Pryce, A device of R. J. Whitley's applied to pointwise compactness in spaces of continuous functions, Proc. Lond. Math. Soc., 23 (1971), 532–546. http://dx.doi.org/10.1112/plms/s3-23.3.532 doi: 10.1112/plms/s3-23.3.532
[36]	W. Rudin, Real and complex analysis, 3 Eds., New York: McGraw-Hill, 1987.
[37]	W. Ruess, Locally convex spaces not containing $\ell_{1}$, Funct. Approx. Comment. Math., 50 (2014), 351–358. http://dx.doi.org/10.7169/facm/2014.50.2.9 doi: 10.7169/facm/2014.50.2.9
[38]	T. Schlumprecht, Limited sets in Banach spaces, Ph.D Thesis, Ludwig Maximilian University of Munich, 1987.

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