Research Article | Open Access
Birsen Sağır, Oğuz Oğur, "On Vector-Valued Generalized Lorentz Difference Sequence Space", Journal of Operators, vol. 2014, Article ID 545346, 4 pages, 2014. https://doi.org/10.1155/2014/545346
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.
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  in 1981 as follows: for , , where for all . Et and Çolak in  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 , Malkowsky and Parashar , Et and Basarir , 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  and examined comprehensively by Kato in .
A weight sequence is a positive decreasing sequence such that , and , where for every . Popa  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 ). 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 ). 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 .
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.
- H. Kızmaz, “On certain sequence spaces,” Canadian Mathematical Bulletin, vol. 24, no. 2, pp. 169–176, 1981.
- M. Et and R. Çolak, “On some generalized difference sequence spaces,” Soochow Journal of Mathematics, vol. 21, no. 4, pp. 377–386, 1995.
- Z. U. Ahmad and Mursaleen, “Köthe-toeplitz duals of some new sequence spaces and their matrix maps,” Publications de l'Institut Mathématique, vol. 42(56), pp. 57–61, 1987.
- E. Malkowsky and S. D. Parashar, “Matrix transformations in spaces of bounded and convergent difference sequences of order ,” Analysis, vol. 17, no. 1, pp. 87–97, 1997.
- M. Et and M. Basarir, “On some new generalized difference sequence spaces,” Periodica Mathematica Hungarica, vol. 35, no. 3, pp. 169–175, 1997.
- K. Miyazaki, “(p,q)-nuclear and (p,q)-integral operators,” Hiroshima Mathematical Journal, vol. 4, pp. 99–132, 1974.
- M. Kato, “On Lorentz spaces ,” Hiroshima Mathematical Journal, vol. 6, no. 1, pp. 73–93, 1976.
- N. Popa, “Basic sequences and subspaces in Lorentz sequence spaces without local convexity,” Transactions of the American Mathematical Society, vol. 263, no. 2, pp. 431–456, 1981.
- M. Nawrocki and A. Ortynski, “The Mackey topology and complemented subspaces of Lorentz sequence spaces for ,” Transactions of the American Mathematical Society, vol. 287, no. 2, pp. 713–722, 1985.
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University Press, Cambridge, UK, 1967.
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.