Journal of Operators

Volume 2014 (2014), Article ID 545346, 4 pages

http://dx.doi.org/10.1155/2014/545346

Research Article

On Vector-Valued Generalized Lorentz Difference Sequence Space

Birsen Sağır and Oğuz Oğur

Department of Mathematics, Art and Science Faculty, Ondokuz Mayis University, Kurupelit Campus, 55139 Samsun, Turkey

Received 19 June 2014; Accepted 15 September 2014; Published 21 September 2014

Academic Editor: Lingju Kong

Copyright © 2014 Birsen Sağır and Oğuz Oğur. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 Kızmaz in [1] in 1981 as follows:

for , , where for all . Et and Çolak 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 1 we have

where and denote the nonincreasing rearrangements of and , respectively.

Let be an element in and let be a vector in . Hence we have

Theorem 5. The space for is complete with respect to its norm.

Proof. Let be an arbitrary Cauchy sequence in with for all . Then we have

Hence we obtain for each and so , for a fixed , is a Cauchy sequence in .

Then, there exists such that as . Let . Since for each   , by Lemma 2 we have . Therefore we can choose the nonincreasing rearrangement of . Also, for an arbitrary there exists such that

for . Hence we get

for . Let be an arbitrary positive integer with and fixed. If we put

then we have

Thus by Lemma 2 we get

for each ; that is,

for each . Hence, by (17), (21) we get

Also, since is a linear space we have . Hence the space is complete with respect to its norm.

Theorem 6. Let . Then, the inclusion holds.

Proof. Let . Then we have

where denotes the nonincreasing rearrangements of . Since is decreasing, by Lemma 1 we get

where denotes the nonincreasing rearrangements of and . This completes the proof.

Theorem 7. If , then .

Proof. Let . Since is decreasing we have

for every . Hence we get

for every . Thus

This implies that .

3. Conclusion

If we put instead of , where and , , for all in the definition of , we obtain generalized Lorentz difference sequence space of order . It can be shown that the sequence space is a Banach space with norm

where denotes the nonincreasing rearrangements of and properties in this work.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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