On absolute Nörlund spaces and matrix operators (Q1637063)

An absolute Nörlund space \(|N_p^{\theta}|_k\) is introduced as an extension of absolute Cesàro space \(|C_\alpha|_k\). For \(k\geq 1\) the space \(|N_p^{\theta}|_k\) is defined as: \[ |N_p^{\theta}|_k = \Big\{a=(a_\upsilon): \displaystyle\sum_{n=1}^{\infty}\theta_n^{k-1}\Big|\sum_{\upsilon=1}^{n}\Big(\frac{P_{n-\upsilon}}{P_n}-\frac{P_{n-1-\upsilon}}{P_{n-1}}\Big)a_{\upsilon}\Big|^k<\infty\Big\},\tag{1} \] where \((a_\upsilon)\) and \((p_n)\) are sequences of real numbers such that \(P_n=p_0+p_1+\cdots+p_n\rightarrow\infty\) and \(\theta_n=\frac{P_n}{p_n}\). Certain topological structures are studied for the space \(|N_p^{\theta}|_k\). The \(\alpha\)-, \(\beta\)- and \(\gamma\)-duals are obtained for \(|N_p^{\theta}|_k\) and it is shown that it has a Schauder basis. A characterization of certain matrix operators on this space is also obtained and hence the exact or approximate values of the operator norms are estimated. The obtained results extend many earlier results.