Eberhard Malkowsky

Characterizations of compact operators on some Euler spaces of difference sequences of order m

Abstract In the past, several authors studied spaces of m -th order difference sequences, among them, H.Polat and F.Basar ([17]) defined the Euler spaces of m -th order difference sequencesn e 0 r ( Δ ( m ) ) , e c r ( Δ ( m ) ) andn e ∞ r ( Δ ( m ) ) and characterized some classes of matrix transformations on them. In our paper, we add a new supplementary aspect to their research by characterizing classes of compact operators on those spaces. For that purpose, the spaces are treated as the matrix domains of a triangle in the classical sequence spaces c 0 , c and l ∞ . The main tool for our characterizations is the Hausdorff measure of noncompactness.

On matrix mappings into some strong Cesàro sequence spaces

We determine the classes (X, YT) of matrix transformations from X into YT where X is one of the classical sequence spaces c0, c, l∞ and l1 of all null, convergent and bounded complex sequences and all absolutely convergent complex series, T is a triangle, YT is the matrix domain of T in Y and Y is any of the sets of all sequences that are summable, summable to zero or bounded by the strong Cesaro method of order 1, with index 1 ⩽ p < ∞. Furthermore, we determine the representations of the general bounded linear operators from c into Y. We also establish estimates for the norms of the operators in each case.

Banach algebras of matrix transformations between some sequence spaces related to Λ-strong convergence and boundedness

We study the spaces c0(@L),c(@L) and c~(@L) of sequences that are @L-strongly convergent to zero, @L-strongly convergent and @L-strongly bounded, and the related spaces v0(@L) and v~(@L). In particular, we give the characterizations of several classes of matrix transformations between those spaces. Furthermore, we use our results to prove that the classes of matrix transformations from v~(@L),c~(@L) and c(@L) into themselves are Banach algebras. As an application of our results, we establish an estimate for the Hausdorff measure of noncompactness of matrix operators that map c(@L) into itself, give a characterization of the subclass of compact operators, and a sufficient condition for those operators to be Fredholm.

Visualization of Wulff’s crystals

In this extended abstract, we deal with Wulff’s construction and the graphical representation of Wulff’s crystals and their surface energy functions as potential surfaces.

Visualization of the spaces W (u, v; ℓp) and their duals

We consider the weighted means spaces W (u, v; lp) and their α–, β– and γ–duals. The duality of the spaces is visualized in three dimensional real space by representing the norm as a potential surface and the dual norm as the corresponding Wulff’s crystal.

Visualization of neighbourhoods in some FK spaces

In this extended abstract, we present the graphical representations of some neighbourhoods in certain FK spaces that have recently been studied. These visualizations strongly support the understanding of the topological and geometric structures of the spaces. We emphasize that the graphics in this paper were created by our own software package and its extensions [1–4].