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.