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, 08.12.2024

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

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