Domain of the Double Sequential Band Matrix B ( r ~ , s ~ ) in the Sequence Space ℓ ( p ) ∗

Abstract

The sequence space $\ell (p)$ was introduced by Maddox (1967). Quite recently, the domain of the generalized difference matrix $B(r,s)$ in the sequence space ${\ell }_p$ has been investigated by Kirişçi and Başar (2010). In the present paper, the sequence space $\ell (\tilde{B},p)$ of nonabsolute type has been studied which is the domain of the generalized difference matrix $B(\tilde{r},\tilde{s})$ in the sequence space $\ell (p)$. Furthermore, the alpha-, beta-, and gamma-duals of the space $\ell (\tilde{B},p)$ have been determined, and the Schauder basis has been given. The classes of matrix transformations from the space $\ell (\tilde{B},p)$ to the spaces ${\ell }_{\infty }$, c and c₀ have been characterized. Additionally, the characterizations of some other matrix transformations from the space $\ell (\tilde{B},p)$ to the Euler, Riesz, difference, and so forth sequence spaces have been obtained by means of a given lemma. The last section of the paper has been devoted to conclusion.