Domain of $q$-Ces\`aro matrix in Hahn sequence space $h_d$ and the space $bv$ of the sequences of bounded variation

Abstract

Let $h_d=\big\{f=(f_k)\in \omega:\sum_k d_k|f_k-f_{k+1}|<\infty \big\}\cap c_0,$ where $d=(d_k)$ is an unbounded and monotonic increasing sequence of positive reals. We study the matrix domains $h_d(C^{q})=(h_d)_{C^q}$ and $bv(C^{q})=(bv)_{C^q},$ where $C^q$ is $q$-Ces\`aro matrix, $0<q<1$. Apart from the inclusion relations and Schauder basis, we compute $\alpha$-, $\beta$- and $\gamma$-duals of the spaces $h_d(C^{q})$ and $bv(C^{q}).$ We state and prove theorems concerning characterization of matrix classes from the spaces $h_d(C^{q})$ and $bv(C^{q})$ to any one of the space $\ell_{\infty},$ $c$, $c_0$ or $\ell_{1}.$ Finally, we obtain certain identities concerning characterization of compact operators using Hausdorff measure of non-compactness in the space $h_d(C^{q}).$