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

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.