Some Topological and Geometric Properties of the Domain of the Generalized Difference Matrix B(r, s) in the Sequence Space ℓ(p)∗

Cafer Aydın, Feyzi Basar

Abstract

The sequence space  ℓ(p) was introduced by Maddox [Spaces ofstrongly summable sequences, Quart. J. Math. Oxford 18 (2) (1967) 345–355]. Quite recently, the domain of the generalized difference matrix  B(r, s) in the sequence space  ℓp has been investigated by Kiri¸s¸ci and Ba¸sar [Some new sequence spaces derived by the domain of generalized difference matrix, Comput. Math.Appl. 60 (5) (2010) 1299–1309]. In the present paper, the sequence space  bℓ(p) of non-absolute type is studied which is the domain of the generalized difference matrix  B(r, s) in the sequence space ℓ(p). Furthermore, the alpha-, beta- and gamma-duals of the space  bℓ(p) are determined, and the Schauder basis is constructed.The classes of matrix transformations from the space  bℓ(p) to the spaces ℓ ∞, c and c0 are characterized. Additionally, the characterizations of some other matrix transformations from the space  bℓ(p) to the Euler, Riesz, difference, etc., sequence spaces are obtained by means of a given lemma. The last two sections ofthe paper are devoted to some results about the rotundity of the space  bℓ(p) and conclusion.