Infinite matrices and Cesàro sequence spaces
Given positive sequences \(q\) and \(p\), with \(1<p_n=O(1)\) assumed throughout in the paper, a complete linear metric space \(\operatorname{ces}(p,q)\) is defined here to comprise all \(x=(x_n)\) such that \(\sum u_n<\infty\) for \(u_n=(\sum_{(n)}q_k|x_k|/\sum_{(n)}q_k)^{p_n}\), where \(\sum_{(n)}\) indicates summation over \(k=2^n,\dots,2^{n+1}-1\). (Reference is made to the space ``\(\operatorname{ces}(p,q)\)'' considered by \textit{P.D. Johnson jun.} and \textit{R.N. Mohapatra} [Math. Japonica 24, 253-262 (1979; Zbl 0418.46005)], which differs from the present one not only by its norm as the authors presume.) The Köthe-Toeplitz dual of \(\operatorname{ces}(p,q)\) is made explicit and thus proved isomorphic to the one of continuous linear functionals. Furthermore, those matrices are characterized for which the transform \(\sum_{k=1}^\infty a_{nk}x_k\) is bounded or convergent, respectively, whenever \(x\in\operatorname{ces}(p,q)\). In the case of \(q_n\equiv 1\), the results specialize to work of \textit{K.P. Lim} [Tamkang J. Math. 8, no. 2, 213-220 (1977; Zbl 0405.46005)].
scientific article; zbMATH DE number 1099669