Some Topological and Geometric Properties of the Domain of the Double Sequential Band Matrix $B(\widetilde{r},\widetilde{s})$ in the Sequence Space $\ell(p)$

The sequence space $\ell(p)$ was introduced by Maddox [Spaces of strongly summable sequences, Quart. J. Math.Oxford (2)\textbf{18}(1967), 345--355]. In the present paper, the sequence space $\ell(\widetilde{B},p)$ of non-absolute type, the domain of the double sequential band matrix $B(\widetilde{r},\widetilde{s})$ in the sequence space $\ell(p)$, is introduced. Furthermore, the alpha-, beta- and gamma-duals of the space $\ell(\widetilde{B},p)$ are determined, and the Schauder basis is given. The classes of matrix transformations from the space $\ell(\widetilde{B},p)$ to the spaces $\ell_\infty$, $f$ and $c$ are characterized. Additionally, the characterizations of some other matrix transformations from the space $\ell(\widetilde{B},p)$ to the Euler, Riesz, difference, etc., sequence spaces are obtained by means of a given lemma. Finally, some geometric properties of the space $\ell(\widetilde{B},p)$ are examined.