Genelleştirilmiş Mutlak Cesàro Seri Uzaylarında Nonkompaktlık Ölçüsünün Uygulamaları

Year 2020, Volume: 10 Issue: 1, 60 - 73, 15.06.2020

G. Canan H. Güleç

https://doi.org/10.31466/kfbd.723446

Cited By: 2

Abstract

Bu çalışmada, |C_(λ,μ) |_p (p≥1) genelleştirilmiş mutlak Cesàro seri uzaylarından l_∞,c ve c_0 klasik dizi uzaylarına bazı matris dönüşümleri karakterize edilmiştir. Bunun yanı sıra, bu matris dönüşümlerine karşılık gelen sınırlı lineer operatörlerin normları için bazı özdeşlikler veya tahminler verilmiştir. Ayrıca, nonkompaktlık Hausdorff ölçüsünün uygulaması ile bu operatörlerin kompakt olması için gerek ve yeter şartlar elde edilmiştir.

Keywords

Dizi Uzayları, Matris Operatörler, BK Uzayları, Kompakt Operatörler, Hausdorff Nonkompaktlık Ölçüsü

Applications of Measure of Noncompactness in the Series Spaces of Generalized Absolute Cesàro Means

Abstract

In this study, we characterize some matrix transformations from the generalized absolute Cesàro series spaces |C_(λ,μ) |_p (p≥1) to the classical sequence spaces l_∞,c and c_0. Besides this, we obtain some identities or estimates for the norms of the bounded linear operators corresponding these matrix transformations. Further, by applying the Hausdorff measure of noncompactness, we give the necessary and sufficient conditions for such operators to be compact.

Keywords

Sequence Spaces, Matrix Operators, BK Spaces, Compact Operators, Hausdorff Measure of Noncompactness

References

• 1 : Altay, B., Başar, F. and Malkowsky E., Matrix transformations on some sequence spaces related to strong Cesàro summability and boundedness, Appl. Math. Comput, 211 (2) (2009) 255-264. 2 : Altay, B. and Başar, F., Generalization of the sequence space l(p) derived by weighted mean, J. Math. Anal. Appl. 330 (2007) 174-185. 3 : Başarır M. and Kara E. E., On the mth order difference sequence space of generalized weighted mean and compact operators, Acta Math. Sci., 33 (2013), 797--813. 4 : Borwein, D., Theorems on some methods of summability, Quart. J. Math. Oxford Ser. 9 (1958), 310-314. 5 : Çanak, İ., A Tauberian Theorem for a Weighted Mean Method of Summability in Ordered Spaces, National Academy Science Letters- India, (2020). 6 : Das, G., A Tauberian theorem for absolute summability, Proc. Cambridge Philos. 67 (1970), 321-326. 7 : Djolović, I. , On compact operators on some spaces related to matrix B(r,s), Filomat 24 (2) (2010) 41--51 8 : Et, M. and Işık, M., On pα-dual spaces of generalized difference sequence spaces, Applied Math. Letters 25 (2012) 1486--1489. 9 : Flett, T.M., On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc. 7 (1957), 113-141. 10 : Goldenstein L.S., Gohberg I.C. and Markus A.S., Investigation of some properties of bounded linear operators in connection with their q-norms, Ucen. Zap. Kishinevsk. Univ. 29 (1957) 29--36. 11 : Hazar, G. C. and Sarıgöl M. A., "Absolute Cesàro series spaces and matrix operators", Acta App. Math., 154, 153--165 (2018) 12 : Hazar Güleç, G.C. and Sarıgöl M. A., Compact and Matrix Operators on the Space |C,-1|_{k}, J. Comput. Anal. Appl., 25(6), (2018), 1014-1024. 13 : Hazar Güleç, G.C. and Sarıgöl, M.A., Hausdorff measure of noncompactness of matrix mappings on Cesàro spaces, Bol. Soc. Paran. Mat. (in press). 14 : Hazar Güleç, G. C., Compact Matrix Operators on Absolute Cesàro Spaces, Numer. Funct. Anal. Optim., DOI: 10.1080/01630563.2019.1633665 15 : Kara, E. E. and İlkhan, M., Some properties of generalized Fibonacci sequence spaces, Linear and Multilinear Algebra, (2016), Vol. 64, No. 11, 2208--2223. 16 : Karakaya, V., Noman, A., K. and Polat, H., On paranormed λ- sequence spaces of non-absolute type, Mathematical and Comp. Modelling 54 (2011), 1473--1480. 17 : Maddox, I.J., Elements of functinal analysis, Cambridge University Press, London,New York, (1970).1 18 : Malkowsky E., Rakočević, V., S. Zivković, Matrix transformations between the sequence spaces bv^{p }and certain BK spaces, Bull. Cl. Sci. Math. Nat. Sci. Math. 123 (27) (2002) 33-46. 19 : Malkowsky, E. and Rakočević, V., An introduction into the theory of sequence space and measures of noncompactness, Zb. Rad. (Beogr) 9, (17),(2000), 143-234. 20 : Mazhar, S.M., On the absolute summability factors of infinite series, Tohoku Math. J. 23 (1971), 433-451. 21 : Mehdi, M.R., Summability factors for generalized absolute summability I, Proc. London Math. Soc. 10 (1960), 180-199. 22 : Mohapatra, R.N. and Sarıgöl, M.A.,On Matrix Operators on the Series Space |N_{p}^{θ}|_{k} , Ukr Math J (2018) 69 (11), 1772-1783. 23 : Mursaleen, M. and Noman, A. K., Hausdorff measure of noncompactness of certain matrix operators on the sequence spaces of generalized means, Journal of Math. Anal. and Appl., 417 (2014) 96-111. 24 : Mursaleen, M. and Noman, A. K., The Hausdorff measure of noncompactness of matrix operators on some BK spaces, Operator and Matrices, 5(3) (2011), 473-486. 25 : Mursaleen, M. and Noman, A. K., Compactness by the Hausdorff measure of noncompactness, Nonlinear Analysis: TMA, 73, 8 (2010), 2541-2557. 26 : Nur, M. Gunawan, H., Three Equivalent n-Norms on the Space of p-Summable Sequences, Fundamental Journal of Mathematics and Applications, 2 (2), (2019), 123-129. 27 : Rakočević, V., Measures of noncompactness and some applications, Filomat, 12 (2), (1998), 87-120. 28 : Sarıgöl, M.A., Spaces of Series Summable by Absolute Cesàro and Matrix Operators, Comm. Math Appl. 7 (1) (2016) 11-22. 29 : Sarıgöl, M.A., Extension of Mazhar's theorem on summability factors, Kuwait J. Sci. 42 (3) (2015), 28-35. 30 : Sezer, S.A. and Çanak, İ., On a Tauberian theorem for the weighted mean method of summability, Kuwait Journal of Science, 42, (2015) 1-9. 31 : Stieglitz, M. and Tietz, H., Matrixtransformationen von folgenraumen eine ergebnisüberischt, Math Z., 154 (1977), 1-16. 32 : Wilansky, A., Summability Through Functional Analysis, North-Holland Mathematical Studies, vol. 85, Elsevier Science Publisher, 1984.