Journal of the Iranian Mathematical Society

Journal of the Iranian Mathematical Society

Maximal spaceability of the set of non-absolutely summing operators in quasi-Banach spaces

Document Type : Research Article

Author
Department of Mathematics, Federal Rural University of Pernambuco, 52171--900, Recife, Brazil.
Abstract
For every $0
Keywords
Subjects

[1] N. G. Albuquerque and L. Coleta, Large structures within the class of summing operators, J. Math. Anal. Appl. 526 (2023), no. 2, Paper No. 127262, 13 pp.
[2] N. G. Albuquerque, A. Raposo and G. Ribeiro, Pointwise transversal lineability, Bull. Braz. Math. Soc. (N.S.) 57 (2026), no. 2, Paper No. 24, 15 pp.
[3] G. Araújo and A. Barbosa, A general lineability criterion for complements of vector spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 118 (2024), no. 1, Paper No. 5, 10 pp.
[4] A. Arbieto and M. Saavedra, A dense lineability criterion for linear dynamics, Integral Equations Operator Theory 98 (2026), no. 3, Paper No. 26.
[5] R. M. Aron, V. I. Gurariy, and J. B. Seoane-Sepúlveda, Lineability and spaceability of sets of functions on R, Proc. Amer. Math. Soc. 133 (2005), no. 3, 795–803.
[6] R. M. Aron, L. Bernal-González, D. Pellegrino and J. B. Seoane-Sepúlveda, Lineability: The Search for Linearity in Mathematics, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, 2016.
[7] A. R. Bagheri Salec, S. M. Tabatabaie, and A. M. Juweed, Spaceability of the set of surjective bounded operators on Banach function spaces, J. Iran. Math. Soc. 6 (2025), no. 2, 107–113.
[8] L. Bernal-González, Algebraic genericity of strict-order integrability, Studia Math. 199 (2010), no. 3, 279–293.
[9] L. Bernal-González, H. J. Cabana-Méndez, G. A. Muñoz-Fernández and J. B. Seoane-Sepúlveda, On the dimension of subspaces of continuous functions attaining their maximum finitely many times, Trans. Amer. Math. Soc. 373 (2020), no. 5, 3063–3083.
[10] L. Bernal-González, M. C. Calderón-Moreno, J. Fernández-Sanchez, H. J. Cabana-Méndez, G. A. Muñoz-Fernández and J. B. Seoane-Sepúlveda, Construction of dense maximal-dimensional hypercyclic subspaces for Rolewicz operators, Chaos Solitons Fractals 162 (2022), Paper No. 112408, 6 pp.
[11] L. Bernal-González and M. O. Cabrera, Lineability criteria, with applications, J. Funct. Anal. 266 (2014), no. 6, 3997–4025.
[12] L. Bernal-González, D. Pellegrino and J. B. Seoane-Sepúlveda, Linear subsets of nonlinear sets in topological vector spaces, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 1, 71–130.
[13] L. Bernal-González, J. A. Conejero, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, [S]-linear and convex structures in function families, Linear Algebra Appl. 579 (2019) 463–483.
[14] G. Botelho, D. Diniz and D. Pellegrino, Lineability of the set of bounded linear non-absolutely summing operators, J. Math. Anal. Appl. 357 (2009), no. 1, 171–175.
[15] G. Botelho, D. Diniz, V.V. Fávaro and D. Pellegrino, Spaceability in Banach and quasi-Banach sequence spaces, Linear Algebra Appl. 434 (2011), no. 5, 1255–1260.
[16] G. Botelho, D. Cariello, V.V. Fávaro and D. Pellegrino, Maximal spaceability in sequence spaces, Linear Algebra Appl. 437 (2012), no. 12, 2978–2985.
[17] G. Botelho, V.V. Fávaro, D. Pellegrino and J. B. Seoane-Sepúlveda, $L_{p}[0,1]\setminus\bigcup_{q >p} L_{q}[0,1]$ is spaceable for every $p > 0$, Linear Algebra Appl. 436 (2012), no. 9, 2963–2965.
 [18] M. C. Calderón-Moreno, P. J. Gerlach-Mena, and J. A. Prado-Bassas, Infinite pointwise lineability: general criteria and applications, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 118 (2024), no. 1, Paper No. 25, 13 pp.
[19] J. Carmona-Tapia, J. Fernández-Sanchez, J. B. Seoane-Sepúlveda and W. Trutsching, Lineability, spaceability and latticeability of subsets of C [0, 1] and Sobolev spaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116 (2022), no. 3, Paper No. 113, 20 pp.
[20] J. A. Conejero, M. Fenoy, M. Murillo-Arcila and J. B. Seoane-Sepúlveda, Lineability within probability theory settings, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 111 (2017), no. 3, 673–684.
[21] S. Dantas, J. Falcó, M. Jung and D. L. Rodríguez-Vidanes, Linear structures in the set of non-norm-attaining operators on Banach spaces, arXiv preprint, 2023.
[22] J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math. 43, Cambridge Univ. Press, 1995.
[23] M. Fakhar and M. Soleimani-Mourchehkhorti, On the lineability and maximal lineability of the set of non-absolutely summing operators, J. Math. Anal. Appl. 551 (2025), no. 1, Paper No. 129650, 13 pp.
[24] V. V. Fávaro, D. Pellegrino and D. Tomaz, Lineability and spaceability: a new approach, Bull. Braz. Math. Soc. (N.S.) 51 (2020), no. 1, 27–46.
[25] V. V. Fávaro, D. Pellegrino, A. B. Raposo Jr., and G. Ribeiro, General criteria for a stronger notion of lineability, Proc. Amer. Math. Soc. 152 (2024), no. 3, 941–954.
[26] V. I. Gurariy, Subspaces and bases in spaces of continuous functions (Russian), Dokl. Akad. Nauk SSSR 167 (1966), 971–973.
[27] N. J. Kalton, Linear operators on Lp for 0 < p < 1, Trans. Amer. Math. Soc. 259 (1980), no. 2, 319–355.
[28] N. J. Kalton, N. T. Peck and J. W. Roberts, An F-space sampler, London Math. Soc. Lecture Note Ser. 89, Cambridge Univ. Press, Cambridge, 1984.
[29] N. J. Kalton, Quasi-Banach spaces, Handbook of the geometry of Banach spaces, Vol.2, North-Holland, Amsterdam, 2003.
[30] D. Kitson and R. M. Timoney, Operator ranges and spaceability, J. Math. Anal. Appl. 378 (2011), no. 2, 680–686.
[31] I. J. Maddox, A non-absolutely summing operator, J. Austral. Math. Soc. Ser. A 43 (1987), no. 1, 70–73.
[32] D. Pellegrino and A. Raposo Jr, Pointwise lineability in sequence spaces, Indag. Math. (N.S.) 32 (2021), no. 2, 536–546.
[33] J. O. Popoola and I. Tweddle, On the dimension of a complete metrizable topological vector space, Canad. Math. Bull. 20 (1977), no. 2, 271–272.
[34] D. Puglisi and J. B. Seoane-Sepúlveda, Bounded linear non-absolutely summing operators, J. Math. Anal. Appl. 338 (2008), no. 1, 292–298.
[35] D. Tomaz, Linear structure in certain subsets of quasi-Banach sequence spaces, Linear Multilinear Algebra 67 (2019), no. 8, 1561–1566.
Volume 7, Issue 2
This issue is in progress but all papers are fully citable
August 2026
Pages 167-182