Johann Boos

Gliding hump properties and some applications

In this not we consider several types of gliding bump properties for a sequence space E and we consider the various implications between these properties. By means of examples we show that most of the implications are strict and they afford a sort of structure between solid sequence spaces and those with weakly sequentially complete β-duals. Our main result is used to extend a result of Bennett and Kalton which characterizes the class of sequence spaces E with the properly that E⊂SF, whenever F is a separable FK space containing E where SF denotes the sequences in F having sectional convergence. This, in turn, is used to identify a gliding humps property as a sufficient condition for E to be in this class.

Strongly nonatomic densities defined by certain matrices

Drewnowski and Paúl proved about ten years ago that for any strongly nonatomic submeasure η on the power set P(ℕ) of the set ℕ of all natural numbers the ideal of all null sets of η has the Nikodym property (NP). They stated the problem whether the converse is true in general. By presenting an example, Alon, Drewnowski and Łuczak proved recently that the answer is negative. Nevertheless, it is of mathematical interest to identify classes of submeasures η such that η is strongly nonatomic if and only if the set of all null sets of η has the Nikodym property. In this context, the authors proved some years ago that this equivalence holds, for instance, if one restricts the attention to the case of densities defined by regular Riesz matrices or by nonnegative regular Hausdorff methods. Also sufficient and necessary conditions in terms of the matrix coefficients are given, that the defined density is strongly nonatomic. In this paper we extend these investigations to the class of generalized Riesz matrices, introduced by Drewnowski, Florencio and Paúl in 1994.

DUAL PAIRS OF SEQUENCE SPACES

The paper aims to develop for sequence spaces E a general concept for recon- ciling certain results, for example inclusion theorems, concerning generalizations of the Kothe-Toeplitz duals E × ( ×∈{ α, β}) combined with dualities (E, G), G ⊂ E × , and the SAK - property (weak sectional convergence). Taking E β : ={ (yk) ∈ ω := K N | (ykxk) ∈ cs }= : Ecs, where cs denotes the set of all summable sequences, as a starting point, then we get a general substitute of E cs by replacing cs by any locally convex sequence space S with sum s ∈ S � (in particular, a sum space) as defined by Ruckle (1970). This idea provides a dual pair (E, E S ) of sequence spaces and gives rise for a generalization of the solid topol- ogy and for the investigation of the continuity of quasi-matrix maps relative to topologies of the duality (E, E β ). That research is the basis for general versions of three types of inclusion theorems: two of them are originally due to Bennett and Kalton (1973) and gen- eralized by the authors (see Boos and Leiger (1993 and 1997)), and the third was done by Grose-Erdmann (1992). Finally, the generalizations, carried out in this paper, are justi- fied by four applications with results around different kinds of Kothe-Toeplitz duals and related section properties.

Eine Erweiterung des Satzes von Schur

As is well-known Schurs theorem gives a necessary and sufficient condition for a conservative matrix to sum all bounded sequences. The present paper deals with two further necessary and sufficient conditions and their applications to absolutely equivalent matrices.