Approximation of lower bound for matrix operators on the weighted sequence space $C_{p}^r(w)(p \in (0,1))$

Abstract

Let $A=(a_{n,k})_{n,k\geq 0}$ be a non-negative matrix. We denote by $L_{\ell_p(w),C_{q}^r(w)}(A)$ the supremum of those $\ell,$ satisfying the following inequality: \[{\left( {\sum\limits_{n = 0}^\infty {{w_n}{{\left( {\frac{1}{{{{\left( {1 + r} \right)}^n}}}\sum\limits_{k = n}^\infty {\frac{{{{\left( {1 + r} \right)}^k}}}{{1 + k}}\sum\limits_{j = 0}^\infty {{a_{k,j}}{x_j}} } } \right)}^q}} } \right)^{1/q}} \ge \ell {\left( {\sum\limits_{n = 0}^\infty {{w_n}x_n^p} } \right)^{1/p}},\] where $x\geq 0$, $x\in \ell_p(w)$, $r\in (0,1)$, $q\le p$ are numbers in $(0,1)$ and $\left(w_n \right)_{n=0}^\infty$ is a non-negative and non-increasing sequence of real numbers. In this paper, first we introduce the weighted sequence space $C_{p}^r(w)~(p \in (0,1))$ of non-absolute type which is a $p$-normed space and is isometrically isomorphic to the space $\ell_p(w)$. Then we focus on the evaluation of $L_{\ell_p(w),C_{q}^r(w)}(A^t)$ for a lower triangular matrix $A$, where $q\le p$ are in $(0,1)$. A lower estimate is obtained. Moreover, in this paper a Hardy type formula is obtained for $L_{\ell_p,C_{q}^r}(H_\mu ^\alpha )$ where $H^\alpha_\mu$ is the generalized Hausdorff matrix, $q\le p\le1$ are positive numbers and $\alpha\geq 0.$ A similar result is also established for the case in which $H^\alpha_\mu$ is replaced by ${(H_\mu ^\alpha )^t}$.