Characterizations of Some New Classes of Four-Dimensional Matrices on the Double Series Spaces of First Order Cesaro Means

Year 2024, Volume: 12 Issue: 4, 196 - 206

Okan Bodur , Canan Hazar Gulec

https://doi.org/10.36753/mathenot.1471156

Cited By: 1

Abstract

The main purpose in this study is to investigate some topological and algebraic properties of the absolutely double series spaces $\left\vert C_{1,1}\right\vert _{k}$ defined by combining the first order Cesaro means with the concept of absolute summability for $k\geq 1$. Beside this, we determine the $\alpha -$dual of the space $\left\vert C_{1,1}\right\vert _{1}$ and the $\beta \left( bp\right) -$ and $\gamma -$duals of the spaces $% \left\vert C_{1,1}\right\vert _{k}$ for $k\geq 1.$ Finally, we characterize some new four-dimensional matrix classes $\left( \left\vert C_{1,1}\right\vert _{k},\upsilon \right) ,$ $\left( \left\vert C_{1,1}\right\vert _{1},\upsilon \right) $, $\left( \left\vert C_{1,1}\right\vert _{1},\mathcal{L}_{k}\right) ,$ $\left( \left\vert C_{1,1}\right\vert _{k},\mathcal{L}_{u}\right) ,$ $\left( \mathcal{L} _{u},\left\vert C_{1,1}\right\vert _{k}\right) $ and $\left( \mathcal{L} _{k},\left\vert C_{1,1}\right\vert _{1}\right) $, where $\upsilon \in \left\{ \mathcal{M}_{u},\mathcal{C}_{bp}\right\} $ for $1\leq k<\infty $. Hence, some important results concerned on Ces\`{a}ro matrix summation methods have been extended to double sequences.

Keywords

Double sequences, Dual spaces, Four dimensional Cesaro matrix, Four dimensional matrix transformations, Pringsheim convergence

References

• [1] Bromwich, T. J.:An Introduction to the Theory of Infinite Series. Macmillan, New York, NY, USA, (1965).
• [2] Demiriz, S., Duyar, O.:Domain of difference matrix of order one in some spaces of double sequences. Gulf Journal of Mathematics. 3(3), 85–100 (2015).
• [3] Demiriz, S., Duyar, O.:Domain of the generalized double Cesàro matrix in some paranormed spaces of double sequences. Tbilisi Mathematical Journal. 10(2), 43-56 (2017).
• [4] Demiriz, S., Erdem, S.:Domain of Euler-totient matrix operator in the space Lp. Korean Journal of Mathematics. 28(2), 361-378 (2020).
• [5] Demiriz, S., Erdem, S.:Domain of binomial matrix in some spaces of double sequences. Punjab University Journal of Mathematics. 52(11), 65-79 (2020).
• [6] Erdem, S., Demiriz, S.:A study on strongly almost convergent and strongly almost null binomial double sequence spaces. Fundamental Journal of Mathematics and Applications. 4(4), 271-279 (2021).
• [7] Erdem, S., Demiriz, S.:A new RH-regular matrix derived by Jordan’s function and its domains on some double sequence spaces. Journal of Function Spaces. 2021, Article ID 5594751, 9 pages, (2021).
• [8] Erdem, S., Demiriz, S.:q−Cesàro double sequence space ˜ Lq derived by q−analog.Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica. 22, 111-126 (2023).
• [9] Móricz, F.:Extensions of the spaces c and c0 from single to double sequences. Acta Mathematica Hungarica. 57(1-2), 129–136 (1991).
• [10] Móricz, F., Rhoades, B. E.:Almost convergence of double sequences and strong regularity of summability matrices. Mathematical Proceedings of the Cambridge Philosophical Society. 104(2), 283–294 (1988).
• [11] Mursaleen, M.:Almost strongly regular matrices and a core theorem for double sequences. Journal of Mathematical Analysis and Applications. 293(2), 523–531 (2004).
• [12] Mursaleen, M., Ba¸sar, F.:Domain of Cesàro mean of order one in some spaces of double sequences. Studia Scientiarum Mathematicarum Hungarica. 51, 335-356 (2014).
• [13] Hazar Güleç, G. C.:Compact matrix operators on absolute Cesàro spaces. Numerical Functional Analysis and Optimization. 41(1), 1-15 (2020).
• [14] Hazar Güleç, G. C.:Characterization of some classes of compact and matrix operators on the sequence spaces of Cesàro means. Operators and Matrices. 13(3), 809-822 (2019).
• [15] Hazar Güleç, G. C., Sarigöl, M. A.:Matrix mappings and norms on the absolute Cesàro and weighted spaces. Quaestiones Mathematicae. 43(1), 117-130 (2020).
• [16] Sarıgöl, M. A.:On equivalence of absolute double weighted mean methods. Quaestiones Mathematicae. 44(6), 755-764 (2021).
• [17] Başar, F., Sever, Y.:The space Lk of double sequences. Mathematical Journal of Okayama University. 51, 149–157 (2009).
• [18] Zeltser, M.:On conservative matrix methods for double sequence spaces. Acta Mathematica Hungarica. 95(3), 225-242 (2002).
• [19] Hardy, G. H.:On the convergence of certain multiple series. Proceedings of the Cambridge Philosophical Society. 19, 86–95 (1917).
• [20] Rhoades, B. E.:Absolute comparison theorems for double weighted mean and double Cesàro means. Mathematica Slovaca. 48, 285-291 (1998).
• [21] Boos, J.:Classical and Modern Methods in Summability. Oxford University Press, New York, (2000).
• [22] Yeşilkayagil, M., Ba¸sar, F.:On the domain of Riesz mean in the space L∗ k. Filomat. 31(4), 925-940 (2017).
• [23] Sarıgöl, M. A.:Four dimensional matrix mappings on double summable spaces. Filomat. 37(4), 1277-1290 (2023).