Spaceability of the sets of surjective and injective operators between sequence spaces
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Spaceability of the sets of surjective and injective operators between sequence spaces Diogo Diniz1 · Vinícius V. Fávaro2 · Daniel Pellegrino3
· Anselmo Raposo Jr.4
Received: 14 July 2020 / Accepted: 12 August 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract We investigate algebraic structures within sets of surjective and injective linear operators between sequence spaces, completing results of Aron et al. Keywords Spaceability · Lineability · Sequence spaces Mathematics Subject Classification 15A03 · 47B37 · 47L05 · 46B87
1 Introduction If V is a vector space and α is a cardinal number, a subset A of V is called α-lineable in V if A ∪ {0} contains an α-dimensional linear subspace W of V . When V has a topology and the subspace W can be chosen to be closed, we say that A is spaceable. This line of research has its starting point with the seminal paper [2] by Aron, Gurariy, and Seoane-Sepúlveda. Nowadays, this theme has profound inroads in Set Theory and Real Analysis, with investi-
D. Diniz was partially supported by CNPq 301704/2019-8 and Grant 2019/0014 Paraíba State Research Foundation (FAPESQ). V. V. Fávaro was supported by CNPq 310500/2017-6 and FAPEMIG Grant PPM-00217-18. D. Pellegrino was partially supported by CNPq 307327/2017-5 and Grant 2019/0014 Paraíba State Research Foundation (FAPESQ).
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Daniel Pellegrino [email protected] Diogo Diniz [email protected] Vinícius V. Fávaro [email protected] Anselmo Raposo Jr. [email protected]
1
Unidade Acadêmica de Matemática e Estatística, Universidade Federal de Campina Grande, Campina Grande 58109-970, Brazil
2
Faculdade de Matemática, Universidade Federal de Uberlândia, Uberlândia 38400-902, Brazil
3
Departamento de Matemática, Universidade Federal da Paraíba, João Pessoa 58051-900, Brazil
4
Departamento de Matemática, Universidade Federal do Maranhão, São Luís 65085-580, Brazil 0123456789().: V,-vol
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gations involving themes such as undecidability and consistency (see, for instance [11] and the references therein). It has been also successfully explored in several research branches, with increasingly relevant applications in areas such as norm-attaining operators, multilinear forms, homogeneous polynomials, sequence spaces, holomorphic mappings, absolutely summing operators, Peano curves, fractals, topological dynamical systems and many others (see, for instance, [1,4–10,12,14] and the references therein). From now on all vector spaces are considered over a fixed scalar field K which can be either R or C. For any set X we shall denote by card (X ) the cardinality of X ; in particular, we denote c = card (R) and ℵ0 = card (N). In this paper we are interested in lineability and spaceability properties of sets of injective and surjective continuous linear operators between sequence spaces. The following results were recently proved in [3]: Theorem 1.1 [3, Theorem 4.1 and Corollary 3.4] The set S = T : p → p : T is linear, continuous and surjective is spaceable in L p ; p
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