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Sequences of 0’s and 1’s: sequence spaces with the separable Hahn property. (English) Zbl 1151.46003

Given a sequence space \(E\), denote by \(\chi(E)\) the linear span of the sequences of 0’s and 1’s contained in \(E\). We say that \(E\) has the Hahn property, the separable Hahn property, or the matrix Hahn property if the implication \(\chi(E)\subseteq F\Rightarrow E\subseteq F\) holds whenever \(F\) is an FK-space, a separable FK-space, or a matrix domain \(c_A\), respectively.
Theorem 5.2 of [G. Bennett, J. Boos and T. Leiger, Studia Math. 149, No. 1, 75–99 (2002); addendum ibid. 171, No. 3, 305–309 (2005; Zbl 0995.46010)] states that for a monotone sequence space \(E\) containing \(\varphi\), (i) the matrix Hahn property, (ii) the separable Hahn property and (iii) the equality \(\chi(E)^{\beta}=E^{\beta}\) are equivalent. However, as it was pointed out in the addendum, there is a gap in the proof of the implication (iii)\({}\Rightarrow{}\)(ii) of this result.
This ‘failed’ result was applied in the paper to three different instances, namely, the space \(|ac|_0\) of strongly almost-null sequences, the space \(\ell^\infty(|\lambda|)\) where \(\lambda\) is a lacunary sequence of indices, and the space \(\ell^\infty \cap z^\alpha\) where \(z \in c_0 \backslash \ell^1\) (see any of the papers for the precise definitions).
C. E. Stuart and P. Abraham [J. Math. Anal. Appl. 300, No. 2, 351–361 (2004; Zbl 1081.28005)] proved that for the space \(|ac|_0\) the implication (iii)\({}\Rightarrow{}\)(ii) holds true, and the purpose of the paper [J. Boos and T. Leiger, Acta Math. Hung. 115, No. 4, 341–356 (2007; Zbl 1151.46002), reviewed above] is to show, through specific gliding hump proofs that do not apply in general, that this is also the case for the remaining instances \(\ell^\infty(|\lambda|)\) and \(\ell^\infty \cap z^\alpha\).
The main result of the paper under review is a counterexample showing that the implication (iii)\({}\Rightarrow{}\)(ii) does not hold in general for monotone sequence spaces. Also, a number of new examples of spaces with the separable Hahn property are provided.

MSC:

46A35 Summability and bases in topological vector spaces
40H05 Functional analytic methods in summability
46A45 Sequence spaces (including Köthe sequence spaces)
40C05 Matrix methods for summability
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