Sequences of 0’s and 1’s: Special sequence spaces with the separable Hahn property

Abstract

As pointed out in [4] the paper [2], authored by G. Bennett, J. Boos and T. Leiger, contains a nontrivial gap in the argumentation of the proof of Theorem 5.2 which is one of main results of that paper and has been applied three times. Till now neither the gap is closed nor a counterexample has been stated. That is why the authors have examined in [4] the situation around thè gap' aiming to a better understanding for the gap. The aim of this paper is to prove the mentioned applications of the theorem in doubt by using gliding hump arguments (quite similar to the classical proofs of the Theorems of Schur and Hahn in the rst case (cf. [3]) and a very technical and artful construction, being of independent mathematical interest, in the second case). Concerning notations and preliminary results we refer to the papers [2]and [4], and to the monographs [10], [11] and [3]. Let χ be the set of all sequences of 0's and 1's, and, if E is any sequence space, let χ(E) denote the linear hull of χ ∩ E. Let A = (a nk ) be an innite matrix with real entries and E be any sequence space. The matrix domain E A is dened byof T. Leiger supported by Estonian Science Foundation Grant 5376.where the denition of Ax includes the convergence of the series. We will apply this denition to E ∈ { c 0 , c, ∞ } . If E is one-dimensional, say E = Sp {z}, we shall write z α in place of Sp {z} α .