## Abstract and Applied Analysis

### On the Riesz Almost Convergent Sequences Space

#### Abstract

The purpose of this paper is to introduce new spaces $\stackrel{̂}{f}$ and ${\stackrel{̂}{f}}_{0}$ that consist of all sequences whose Riesz transforms of order one are in the spaces $f$ and ${f}_{0}$, respectively. We also show that $\stackrel{̂}{f}$ and ${\stackrel{̂}{f}}_{0}$ are linearly isomorphic to the spaces $f$ and ${f}_{0}$, respectively. The $\beta \text{-}$ and $\gamma \text{-}$duals of the spaces $\stackrel{̂}{f}$ and ${\stackrel{̂}{f}}_{0}$ are computed. Furthermore, the classes $(\stackrel{̂}{f}:\mu )$ and $(\mu :\stackrel{̂}{f})$ of infinite matrices are characterized for any given sequence space $\mu$ and determine the necessary and sufficient conditions on a matrix $A$ to satisfy ${B}_{R}-\text{c}\text{o}\text{r}\text{e}(Ax)\subseteq K-\text{c}\text{o}\text{r}\text{e}(x)$, ${B}_{R}-\text{c}\text{o}\text{r}\text{e}(Ax)\subseteq st-\text{c}\text{o}\text{r}\text{e}(x)$ for all $x\in {\mathcal{l}}_{\infty }$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 691694, 18 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.aaa/1355495698

Digital Object Identifier
doi:10.1155/2012/691694

Mathematical Reviews number (MathSciNet)
MR2922916

Zentralblatt MATH identifier
1250.46005

#### Citation

Şengönül, Mehmet; Kayaduman, Kuddusi. On the Riesz Almost Convergent Sequences Space. Abstr. Appl. Anal. 2012 (2012), Article ID 691694, 18 pages. doi:10.1155/2012/691694. https://projecteuclid.org/euclid.aaa/1355495698

#### References

• J. Boos, Classical and Modern Methods in Summability, Oxford University Press, Oxford, UK, 2000.
• B. Altay and F. Başar, “The fine spectrum and the matrix domain of the difference operator $\Delta$ on the sequence space ${l}_{p}$, $(0<p<1)$,” Communications in Mathematical Analysis, vol. 2, no. 2, pp. 1–11, 2007.
• M. Başarir, “On some new sequence spaces and related matrix transformations,” Indian Journal of Pure and Applied Mathematics, vol. 26, no. 10, pp. 1003–1010, 1995.
• C. Ayd\in and F. Başar, “Some generalizations of the sequencespace ${a}_{r}^{p}$,” Iranian Journal of Science and Technology, vol. 20, no. 2, pp. 175–190, 2006.
• M. Kirişçi and F. Başar, “Some new sequence spaces derived by the domain of generalized difference matrix,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1299–1309, 2010.
• M. Şengönül and F. Başar, “Some new Cesàro sequence spaces of non-absolute type which include the spaces ${c}_{0}$ and $c$,” Soochow Journal of Mathematics, vol. 31, no. 1, pp. 107–119, 2005.
• H. Polat and F. Başar, “Some Euler spaces of difference sequences of order m,” Acta Mathematica Scientia. Series B, vol. 27, no. 2, pp. 254–266, 2007.
• E. Malkowsky, M. Mursaleen, and S. Suantai, “The dual spaces of sets of difference sequences of order m and matrix transformations,” Acta Mathematica Sinica, vol. 23, no. 3, pp. 521–532, 2007.
• B. Altay and F. Başar, “Certain topological properties and duals of the domain of a triangle matrix in a sequence space,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 632–645, 2007.
• M. Kirişçi and F. Başar, “Some new sequence spaces derived by the domain of generalized difference matrix,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1299–1309, 2010.
• G. G. Lorentz, “A contribution to the theory of divergent sequences,” Acta Mathematica, vol. 80, pp. 167–190, 1948.
• D. Butković, H. Kraljević, and N. Sarapa, “On the almost convergence,” in Functional Analysis, II, vol. 1242 of Lecture Notes in Mathematics, pp. 396–417, Springer, Berlin, Germany, 1987.
• J. A. Fridy and C. Orhan, “Statistical limit superior and limit inferior,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3625–3631, 1997.
• J. S. Connor, “The statistical and strong p-Cesàro convergence of sequences,” Analysis, vol. 8, no. 1-2, pp. 47–63, 1988.
• S. L. Mishra, B. Satapathy, and N. Rath, “Invariant means and $\sigma$-core,” The Journal of the Indian Mathematical Society. New Series, vol. 60, no. 1-4, pp. 151–158, 1994.
• K. Kayaduman and H. Çoşkun, “On the ${\sigma }^{(A)}$-summability and ${\sigma }^{(A)}$-core,” Demonstratio Mathematica, vol. 40, no. 4, pp. 859–867, 2007.
• M. Mursaleen, “On some new invariant matrix methods of summability,” Quarterly Journal of Math-ematics, vol. 24, pp. 77–86, 1983.
• K. Kayaduman and M. Şengönül, “On the Cesàro almost čommentComment on ref. [13?]: Please provide more information for this reference, if possible. convergent sequences spaces,” under comminication.
• F. Móricz and B. E. Rhoades, “Some characterizations of almost convergence for single and double sequences,” Publications De L'institut Mathématique, Nouvelle série tome, vol. 48, no. 62, pp. 61–68, 1990.
• J. A. S\idd\iq\i, “Infinite matrices summing every almost periodic sequence,” Pacific Journal of Math-ematics, vol. 39, pp. 235–251, 1971.
• F. Başar, “Matrix transformations between certain sequence spaces of ${X}_{p}$ and ${l}_{p}$,” Soochow Journal of Mathematics, vol. 26, no. 2, pp. 191–204, 2000.
• F. Başar and R. Çolak, “Almost-conservative matrix transformations,” Turkish Journal of Mathematics, vol. 13, no. 3, pp. 91–100, 1989.
• B. Kuttner, “On dual summability methods,” Proceedings of the Cambridge Philosophical Society, vol. 71, pp. 67–73, 1972.
• G. G. Lorentz and K. Zeller, “Summation of sequences and summation of series,” Proceedings of the Cambridge Philosophical Society, vol. 60, pp. 67–73, 1972.
• G. Das, “Sublinear functionals and a class of conservative matrices,” Bulletin of the Institute of Math-ematics. Academia Sinica, vol. 15, no. 1, pp. 89–106, 1987.