Bounded linear and compact operators between the Hahn space and spaces of strongly summable and bounded sequences

Eberhard Malkowsky ; Vladimir Rakočević ; Vesna Veličković

Bulletin de l'Académie serbe des sciences. Classe des sciences mathématiques et naturelles, Tome 45 (2020) no. 1

Résumé

We establish the characterisations of the classes of bounded linear operators from the generalised Hahn sequence space $h_{d}$, where $d$ is an unbounded monotone increasing sequence of positive real numbers, into the spaces \wop, \wcp\hspace*{0.5pt} and \wip\hspace*{0.5pt} of sequences that are strongly summable to zero, strongly summable and strongly bounded by the Cesàro method of order one and index $p$ for $1\le p\infty$. Furthermore, we prove estimates for the Hausdorff measure of noncompactness of bounded linear operators from $h_{d}$ into \wcp, and identities for the Hausdorff measure of noncompactness of bounded linear operators from $h_{d}$ to \wop. We use these results to characterise the classes of compact operators from $h_{d}$ to \wcp\hspace*{0.5pt} and \wop. Finally, we provide an example for some applications of our results and visualisations in crystallography.