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Keywords:
Banach spaces; Schur property; hereditarily $l_p$
Banach spaces; Schur property; hereditarily $l_p$
Summary:
Hagler and the first named author introduced a class of hereditarily $l_1$ Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily $l_p$ Banach spaces for $1\leq p<\infty $. Here we use these spaces to introduce a new class of hereditarily $l_p(c_0)$ Banach spaces analogous of the space of Popov. In particular, for $p=1$ the spaces are further examples of hereditarily $l_1$ Banach spaces failing the Schur property.
References:
[1] Azimi, P.: A new class of Banach sequence spaces. Bull. of Iranian Math. Society 28 (2002), 57-68. MR 1992259 | Zbl 1035.46006
[2] Azimi, P., Hagler, J.: Examples of hereditarily $ \ell_1$ Banach spaces failing the Schur property. Pacific J. Math. 122 (1986), 287-297. DOI 10.2140/pjm.1986.122.287 | MR 0831114
[3] Bourgain, J.: $\ell_1$-subspace of Banach spaces. Lecture notes. Free University of Brussels.
[4] Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces. Vol. I sequence Spaces, Springer Verlag, Berlin. MR 0415253 | Zbl 0852.46015
[5] Popov, M. M.: A hereditarily $\ell_1$ subspace of $L_1$ without the Schur property. Proc. Amer. Math. Soc. 133 (2005), 2023-2028. DOI 10.1090/S0002-9939-05-07758-0 | MR 2137868 | Zbl 1080.46007
[6] Popov, M. M.: More examples of hereditarily $\ell _p$ Banach spaces. Ukrainian Math. Bull. 2 (2005), 95-111. MR 2172327 | Zbl 1166.46304
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