Abstract
Let \(\mathbb {Y}\) be either an Orlicz sequence space or a Marcinkiewicz sequence space. We take advantage of the recent advances in the theory of factorization of the identity carried on by Lechner (Stud Math 248(3):295–319, 2019) to provide conditions on \(\mathbb {Y}\) that ensure that, for any \(1\le p\le \infty \), the infinite direct sum of \(\mathbb {Y}\) in the sense of \(\ell _p\) is a primary Banach space. This way, we enlarge the list of Banach spaces that are known to be primary.
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References
Albiac, F., Ansorena, J.L.: Lorentz spaces and embeddings induced by almost greedy bases in Banach spaces. Constr. Approx. 43(2), 197–215 (2016)
Albiac, F., Ansorena, J.L., Dilworth, S.J., Kutzarova, D.: Banach spaces with a unique greedy basis. J. Approx. Theory 210, 80–102 (2016)
Albiac, F., Ansorena, J.L., Wallis, B.: Garling sequence spaces. J. Lond. Math. Soc. (2) 98(1), 204–222 (2018)
Albiac, F., Kalton, N.J.: Topics in Banach Space Theory, 2nd edn, Graduate Texts in Mathematics, vol. 233. Springer, Cham (2016). With a foreword by Gilles Godefory
Ansorena, J.L.: A note on subsymmetric renormings of Banach spaces. Quaest. Math. 41(5), 615–628 (2018)
Capon, M.: Primarité de certains espaces de Banach. Proc. Lond. Math. Soc. (3) 45(1), 113–130 (1982)
Carro, M.J., Raposo, J.A., Soria, J.: Recent developments in the theory of Lorentz spaces and weighted inequalities, Mem. Amer. Math. Soc., vol. 187, no. 877, xii+128 (2007)
Casazza, P.G., Kottman, C.A., Lin, B.L.: On some classes of primary Banach spaces. Can. J. Math. 29(4), 856–873 (1977)
Casazza, P.G., Lin, B.L.: Projections on Banach spaces with symmetric bases. Stud. Math. 52, 189–193 (1974)
James, R.C.: Bases and reflexivity of Banach spaces. Ann. Math. 2(52), 518–527 (1950)
Kadec, M.I., Pełczyński, A.: Bases, Lacunary sequences and complemented subspaces in the spaces \(L_{p}\). Stud. Math. 21, 161–176 (1961/1962)
Lechner, R.: Subsymmetric weak* Schauder bases and factorization of the identity. Stud. Math. 248(3), 295–319 (2019)
Lindenstrauss, J., Pełczyński, A.: Absolutely summing operators in \(L_{p}\)-spaces and their applications. Stud. Math. 29, 275–326 (1968)
Lindenstrauss, J., Rosenthal, H.P.: The \(\cal{L}_{p}\) spaces. Isr. J. Math. 7, 325–349 (1969)
Lindenstrauss, J., Tzafriri, L.: Classical Banach spaces. I. Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92. Springer, Berlin (1977)
Musielak, J.: Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics. Springer, Berlin (1983)
Samuel, C.: Primarité des produits d’espaces de suites. Colloq. Math. 39(1), 123–132 (1978)
Singer, I.: Basic sequences and reflexivity of Banach spaces. Stud. Math. 21, 351–369 (1961/1962)
Singer, I.: Some characterizations of symmetric bases in Banach spaces. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys. 10, 185–192 (1962)
Acknowledgements
José L. Ansorena is partially supported by the Spanish Research Grant Análisis Vectorial, Multilineal y Aproximación, reference number PGC2018-095366-B-I00.
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Communicated by Dirk Werner.
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Ansorena, J.L. Primarity of direct sums of Orlicz spaces and Marcinkiewicz spaces. Banach J. Math. Anal. 14, 950–969 (2020). https://doi.org/10.1007/s43037-019-00047-9
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DOI: https://doi.org/10.1007/s43037-019-00047-9
Keywords
- Subsymmetric basis
- Primary Banach space
- Factorization of the identity
- Marcinkiewicz space
- Lorentz space
- Orlicz space
- Sequence space