## Annals of Functional Analysis

### Some $m$th-order Difference Sequence Spaces of Generalized Means and Compact Operators

#### Abstract

In this paper, new sequence spaces $X(r, s, t ;\Delta^{(m)})$ for $X\in \{l_\infty, c,$ $c_0\}$ defined by using generalized means and difference operator of order $m$ are introduced. It is shown that these spaces are complete normed linear spaces and the spaces $c_0(r, s, t ;\Delta^{(m)})$, $c(r, s, t ;\Delta^{(m)})$ have Schauder basis. Furthermore, the $\alpha$-, $\beta$-, $\gamma$- duals of these spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $X(r, s, t ;\Delta^{(m)})$ to $X$. Finally, some classes of compact operators on the spaces $c_0(r, s, t ;\Delta^{(m)})$ and $l_{\infty}(r, s, t ;\Delta^{(m)})$ are characterized by using the Hausdorff measure of .

#### Article information

Source
Ann. Funct. Anal., Volume 6, Number 1 (2015), 170-192.

Dates
First available in Project Euclid: 19 December 2014

https://projecteuclid.org/euclid.afa/1419001458

Digital Object Identifier
doi:10.15352/afa/06-1-13

Mathematical Reviews number (MathSciNet)
MR3297795

Zentralblatt MATH identifier
1339.46005

Subjects
Primary: 46A45
Secondary: 46B15: Summability and bases [See also 46A35] 46B50: Compactness in Banach (or normed) spaces

#### Citation

Maji, Amit; Manna, Atanu; Srivastava, P. D. Some $m$th-order Difference Sequence Spaces of Generalized Means and Compact Operators. Ann. Funct. Anal. 6 (2015), no. 1, 170--192. doi:10.15352/afa/06-1-13. https://projecteuclid.org/euclid.afa/1419001458

#### References

• Z.U. Ahmad and M. Mursaleen, Köthe-Toeplitz duals of some new sequence spaces and their matrix maps, Publ. Inst. Math.(Beograd) 42 (56) (1987), 57–61.
• B. Altay and F. Başar, The fine spectrum and matrix domain of the difference operator $\Delta$ on the sequece space $\ell_p$, ($0<p< 1$) , Commun. Math. Anal. 2 (2007), no. 2, 1–11.
• B. Altay and F. Başar, Generalization of the sequence space $\ell(p)$ derived by weighted mean, J. Math. Anal. Appl. 330 (2007), 174–185.
• C. Aydin and F. Başar, Some new difference sequence spaces, Appl. Math. Comput. 157 (2004), no. 3, 677–693.
• M. Başarir and E.E. Kara, On some difference sequence spaces of weighted means and compact operators, Ann. Funct. Anal. 2 (2011), no. 2, 114–129.
• M. Başarir and E.E. Kara, On the $B$-difference sequence space derived by generalized weighted mean and compact operator, J. Math. Anal. Appl. 391 (2012), 67–81.
• R.Çolak and M. Et, On some generalized difference sequence spaces and related matrix transformations, Hokkaido Math. J. 26 (1997), no. 3, 483–492.
• I. Djolović, On the space of bounded Euler difference sequences and some classes of compact operators, Appl. Math. Comput. 182 (2006), no. 2, 1803–1811.
• I. Djolović and E. Malkowsky, Matrix transformations and compact operators on some new mth-order difference sequence spaces, Appl. Math. Comput. 198 (2008), no. 2, 700–714.
• A.M. Jarrah and E. Malkowsky, Ordinary, absolute and strong summability and matrix transformations, Filomat 17 (2003), 59–78.
• E.E. Kara and M. Başarir, On compact operators and some Euler $B^{(m)}$-difference sequence spaces, J. Math. Anal. Appl. 379 (2011), 499–511.
• H. Kizmaz, On certain sequence spaces, Canad. Math. Bull. 24 (1981), no.2, 169–176.
• E. Malkowsky and V. Rakočević, An introduction into the theory of sequence spaces and measure of , Zb. Rad. (Beogr.) 9 (17) (2000), 143–234.
• E. Malkowsky and E. Savas, Matrix transformations between sequence spaces of generalized weighted means, Appl. Math. Comput. 147 (2004), 333–345.
• E. Malkowsky and V. Rakočević, On matrix domains of triangles, Appl. Math. Comput. 189 (2007), no. 2, 1146–1163.
• M. Mursaleen and A.K. Noman, On some new difference sequence spaces of non-absolute type, Math. Comput. Modelling 52 (2010), no. 3-4, 603–617.
• M. Mursaleen and A.K. Noman, Applications of the Hausdorff measure of in some sequence spaces of weighted means, Comput. Math. Appl. 60 (2010), no. 5, 1245–1258.
• M. Mursaleen and A.K. Noman, On generalized means and some related sequence spaces, Comput. Math. Appl. 61 (2011), no. 4, 988–999.
• H. Polat and F. Başar, Some Euler spaces of difference sequences of order $m$, Acta Mathematica Scientia, 27B (2007), no. 2, 254–266.
• H. Polat, V. Karakaya and N. Simsek, Difference sequence spaces derived by using a generalized weighted mean, Appl. Math. Lett. 24 (2011), no. 5, 608–614.
• M. Stieglitz and H. Tietz, Matrix trasformationnen von Folenraumen Eine Erebisubersicht, Mathematische Zeitschrift(Math. Z.), 154 (1977), 1–16.
• A. Wilansky, Summability through Functional Analysis, North-Holland Math. Stud., vol. 85, Elsevier Science Publishers, Amsterdam, New York, Oxford, 1984.