On Vector-Valued Generalized Lorentz Difference Sequence Space

We introduce generalized Lorentz difference sequence spaces . Also we study some topologic properties of this space and obtain some inclusion relations. 1. Introduction Throughout this work, , , and denote the set of positive integers, real numbers, and complex numbers, respectively. The notion of difference sequence space was introduced by K？zmaz in [1] in 1981 as follows: for , , where for all . Et and ？olak in [2] defined the sequence space for , , where , , for all , and showed that this space is a Banach space with norm Subsequently difference sequence spaces has been discussed in Ahmad and Mursaleen [3], Malkowsky and Parashar [4], Et and Basarir [5], and others. Let be a Banach space. The Lorentz sequence space for , is the collection of all sequences such that is finite, where is nonincreasing rearrangement of (we can interpret that the decreasing rearrangement is obtained by rearranging in decreasing order). This space was introduced by Miyazaki in [6] and examined comprehensively by Kato in [7]. A weight sequence is a positive decreasing sequence such that , and , where for every . Popa [8] defined the generalized Lorentz sequence space for as follows: where ranges over all permutations of the positive integers and is a weight sequence. It is known that and hence for each there exists a nonincreasing rearrangement of and (see [8, 9]). Let be a Banach space and let be a weight sequence. We introduce the vector-valued generalized Lorentz difference sequence space for . The space is the collection of all -valued -sequences ？？ such that is finite, where is nonincreasing rearrangement of and for all . We will need the following lemmas. Lemma 1 (see [10]). Let and be the nonincreasing and nondecreasing rearrangements of a finite sequence of positive numbers, respectively. Then for two sequences and of positive numbers we have Lemma 2 (see [7]). Let be an -valued double sequence such that for each and let be an -valued sequence such that (uniformly in ). Then and for each where and are the nonincreasing rearrangements of and , respectively. 2. Main Results Theorem 3. The space for is a linear space over the field or . Proof. Let and let and be the nonincreasing rearrangements of the sequences and , respectively. Since is nonincreasing, by Lemma 1 we have where . Let . Hence we get This shows that and so is a linear space. Theorem 4. The space for is normed space with the norm where denotes the nonincreasing rearrangements of . Proof. It is clear that . Let . Then we have and for all . Hence we get . Let . Since weight sequence is decreasing, by Lemma