The characterization of compact operators on spaces of strongly summable and bounded sequences (Q628857)

Let \(A=(a_{nk})_{n,k=0}^\infty\) be an infinite matrix of complex numbers, \(X\) and \(Y\) be subsets of \(\omega\). Let \((X,Y)\) denote the class of all matrices \(A\) such that \(A_n=(a_{nk})_{k=0}^\infty\in X^\beta\) for all \(n\in\mathbb N\) and \(Ax=(A_nx)_{n=0}^\infty\in Y\) for all \(x\in X\), where \(X^\beta\) is the \(\beta\) dual of \(X\) and \(A_nx=\sum_{k=0}^\infty a_{nk}x_k\). \textit{I. J. Maddox} in [J. Lond. Math. Soc. 43, 285--290 (1968; Zbl 0155.38802)] introduced and studied the following sets of sequences that are strongly summable and bounded with index \(p\) (\(1\leq p<\infty\)) by the Cesàro method of order 1: \[ w_0^p= \bigg\{x\in\omega:\;\lim_{n\to \infty} \frac{1}{n}\;\sum_{k=1}^n\,|x_k|^p=0\bigg\},\quad w_\infty^p= \bigg\{x\in\omega:\;\sup_{n\in {\mathbb N}} \frac{1}{n}\;\sum_{k=1}^n\,|x_k|^p<\infty\bigg\} \] and \[ w^p= \bigg\{x\in\omega:\;\lim_{n\to \infty}\;\frac{1}{n}\,\sum_{k=1}^n|x_k-\xi|^p=0 \text{\;for some\;} \xi\in {\mathbb C}\bigg\}. \] In the paper under review, the authors use the characterizations given in [\textit{F. Başar, E. Malkowsky} and \textit{B. Altay}, Publ. Math. 73, No.~1--2, 193--213 (2008; Zbl 1164.46003)] of the classes \((w^p_0,c_0)\), \((w^p,c_0)\), \((w^p_\infty,c_0)\), \((w^p_0,c)\), \((w^p,c)\) and \((w^p_\infty,c)\) and the Hausdorff measure of noncompactness to characterize the classes of compact operators from \(w^p_0\), \(w^p\) and \(w^p_\infty\) into \(c_0\) and \(c\).