Let the triangle matrix $A^{ru}$ be a generalization of the Cesàro matrix and $U\in\{c_{0},c,\ell_{\infty}\}$. In this study, we essentially deal with the space $U(A^{ru})$ defined by the domain of $A^{ru}$ in the space $U$ and give the bases, and determine the Köthe-Toeplitz, generalized Köthe-Toeplitz and bounded-duals of the space $U(A^{ru})$. We characterize the classes $(\ell_{\infty}(A^{ru}):\ell_{\infty})$, $(\ell_{\infty}(A^{ru}):c)$, $(c(A^{ru}):c)$, and $(U:V(A^{ru}))$ of infinite matrices, where $V$ denotes any given sequence space. Additionally, we also present a Steinhaus type theorem. As an another result of this study, we investigate the $\ell_p$-norm of the matrix $A^{ru}$ and as a result obtaining a generalized version of Hardy's inequality, and some inclusion relations. Moreover, we compute the norm of well-known operators on the matrix domain $\ell_p(A^{ru})$.
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