Matrix transformations and compact operators on the binomial sequence spaces
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Abstract
In this work, we characterize some matrix classes concerning the Binomial sequence spaces $b_{\infty}^{r,s}$ and $b_{p}^{r,s}$, where $1\leq p<\infty$. Moreover, by using the notion of Hausdorff measure of noncompactness, we characterize the class of compact matrix operators from $b_{0}^{r,s}$, $b_{c}^{r,s}$ and $b_{\infty}^{r,s}$ into $c_{0}$, $c$ and $\ell_{\infty}$, respectively.
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[1] Altay, B, Ba ̧sar, F., Some Euler sequence spaces of non-absolute type, Ukrainian Math. J. 57 (1) (2005), 1–17. Google Scholar
[2] Altay, B, Ba ̧sar, F, Mursaleen, M., On the Euler sequence spaces which include the spaces lp and l∞ I, Inform. Sci. 176 (10) (2006), 1450–1462. Google Scholar
[3] Altay, B, Polat, H., On some new Euler difference sequence spaces, Southeast Asian Bull. Math. 30 (2) (2006), 209–220. Google Scholar
[4] Ba ̧sar, F, Altay, B., On the space of sequences of p-bounded variation and related matrix mappings, Ukrainian Math. J. 55(1)(2003), 136–147. Google Scholar
[5] Bi ̧sgin, M.C., The Binomial sequence spaces of nonabsolute type, J. Inequal. Appl., 2016:309, (2016), doi: 10.1186/s13660-016-1256-0. Google Scholar
[6] Bi ̧sgin, M.C., The Binomial sequence spaces which include the spaces lp and l∞ and Geometric Properties, J. Inequal. Appl., 2016:304, (2016), doi:10.1186/s13660-016-1252-4. Google Scholar
[7] Choudhary, B, Nanda, S, Functional Analysis with Applications, John Wiley & sons Inc., New Delhi (1989). Google Scholar
[8] Djolovi c, I, Malkowsky, E., Matrix transformations and compact operators on some new m-th order difference sequences, Appl. Math. Comput. 198 (2) (2008), 700-714. Google Scholar
[9] Djolovi c, I, Malkowsky, E., A note on compact operators on matrix domains, j. Math. Anal. Appl. 340 (1) (2008), 291-303. Google Scholar
[10] Jarrah, A, M, Malkowsky, E., Ordinary, absolute and strong summability and matrix transformation, Filomat. 17 (2003), 59–78. Google Scholar
[11] Kızmaz, H., On certain sequence spaces, Canad. Math. Bull. 24 (2) (1981), 169–176. Google Scholar
[12] Lorentz, G.G., A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167–190. Google Scholar
[13] Maddox, I. J., Elements of Functional Analysis, Cambridge University Press (2nd edition), (1988). Google Scholar
[14] Malkowsky, E, Rako cevi c, V., On matrix domains and triangels, Appl. Math. Comput. 189 (2) (2007), 1146-1163. Google Scholar
[15] Malkowsky, E, Rako cevi c, V., An introduction into the theory of sequence spaces measure of noncompactness, Zbornik Radova, Matemati cki Institut Sanu, Bel- grade. 9 (17) (2000), 143-234. Google Scholar
[16] Mursaleen, M, Ba ̧sar, F, Altay, B., On the Euler sequence spaces which include the spaces lp and l∞ II, Nonlinear Anal. 65 (3) (2006), 707–717. Google Scholar
[17] Mursaleen, M, Noman, A.K., Compactness by the Hausdorff measure of non- compactness, Nonlinear Anal. 73 (2010) (2010), 2541–2557. Google Scholar
[18] Mursaleen, M, Noman, A.K., The Hausdorff measure of noncompactness of matrix operators on some BK spaces, Oper. Matrices 5 (3) (2011), 473–486. Google Scholar
[19] Mursaleen, M, Noman, A.K., Compactness of matrix operators on some new difference sequence spaces, Lineer Algebra Appl. 436 (1) (2012), 41–52. Google Scholar
[20] Mursaleen, M, Noman, A.K., Applications of Hausdorff measure of noncompactness in the spaces of generalized means, Math. Ineq. Appl. 16 (2013) (2013), 207–220. Google Scholar
[21] Mursaleen, M, Noman, A.K., Hausdorff measure of noncompactness of certain matrix operators on the sequence spaces of generalized means, Jour. Math. Anal. Appl. 16 (2013) (2013), 207–220. Google Scholar
[22] Ng, P. -N, Lee, P. -Y., Ces`aro sequence spaces of non-absolute type, Comment. Math. (Prace Mat.) 20 (2) (1978), 429–433. Google Scholar
[23] Polat, H, Ba ̧sar, F., Some Euler spaces of difference sequences of order m, Acta Math. Sci. Ser. B, Engl. Ed. 27B (2) (2007), 254–266. Google Scholar
[24] Rako cevi c, V., Measures of non compactness and some applications, Filomat. 12 (1998), 87-120. Google Scholar
[25] Stieglitz, M, Tietz, H., Matrix transformationen von folgenr ̈aumen eine ergeb- nisu ̈bersicht, Math. Z. 154 (1977), 1–16. Google Scholar
[26] S ̧eng ̈onu ̈l, M, Ba ̧sar, F., Some new Ces`aro sequence spaces of non–absolute type which include the spaces c0 and c, Soochow J. Math. 31 (1) (2005), 107–119. Google Scholar