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Abstract: Drewnowski and Paúl proved in [L. Drewnowski, P.J. Paúl, The Nikodým property for ideals of sets defined by matrix summability methods, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 94 (2000) 485–503] that for any strongly nonatomic submeasure η on the power set of the ideal has the Nikodym property (NP); in particular, this result applies to densities defined by strongly regular matrices A. Grahame Bennett and the authors stated in [G. Bennett, J. Boos, T. Leiger, Sequences of 0''s and 1''s, Studia Math. 149 (2002) 75–99] that the strong null domain of any strongly regular matrix A has the Hahn property (HP). Moreover, Stuart and Abraham [C.E. Stuart, P. Abraham, Generalizations of the Nikodym boundedness and Vitali–Hahn–Saks theorems, J. Math. Anal. Appl. 300 (2) (2004) 351–361] pointed out that the said results are in some sense dual and that the last one follows from the first one by considering the density (defined by A) as submeasure on and the ideal as well by identifying with the set χ of sequences of 0''s and 1''s. In this paper we aim at a better understanding of the intimated duality and at a characterization of those members of special classes of matrices A such that has NP (equivalently, has HP). [Copyright &y& Elsevier]

The authors proceed their investigation of dual pairs (E, E[sup s]), where E is a sequence space, S is a N-space on which a sum s is defined in the sense of Ruckle, and E[sup s] is the space of all corresponding factor sequences. Here, the particular case is considered that the sum s has the representation s(z): limγ ∑[sub k] v[sub γk] Z[sub k] (z ∈ S), where F is a directed set of indices γ and (v[sub γk])[sub k] is a finite sequence for each γ ∈.... On the basis of this representation the S-sections of any sequence x = (x[sub k]) and both, their convergence (AN(S)) and boundedness (AB(S)) in N-spaces E are studied. Further, inclusion theorems due to Bennett and Kalton are proved in this more general situation. Following an idea of Schaefer to consider "section convergence barrels", the notion of AN(S)-barrelled N-spaces is introduced which leads to the result that a Mackey N-space E containing all finite sequences is AN(S)-barrelled if and only if E[sup s] ⊂ E'. The paper covers some results concerning the Köthe-Toeplitz duals and related section properties, for example, the β(T)-dual and the STN-property (considered by Buntinas and Meyers). [ABSTRACT FROM AUTHOR]

This paper continues the joint investigation by Bennett et al. (2001) of the extent to which sequence spaces are determined by the sequences of 0''s and 1''s that they contain. The first main result gives a negative answer to Question 6 in their paper: There exists a sequence space E such that each matrix domain containing all of the sequences of zeros and ones in E contains all of E, but such that this statement fails, if we replace matrix domains by separable FK-spaces. The second main result goes on from Hahn''s theorem that tells us that each matrix domain including χ, the set of all sequences of 0''s and 1''s, contains all of the bounded sequences: It is shown that there exists a really ‘small’ subset χ of χ such that Hahn''s theorem remains true when χ is replaced with it. The proofs of both results have in common that, by identifying sequence spaces and double sequence spaces, the constructions and the required investigations are done in double sequence spaces that allow the description of finer structures. [Copyright &y& Elsevier]