The Norm of Certain Matrix Operators On the New Block Sequence Space

Abstract

The purpose of the this study is to introduce the sequence space $$ \ell_{p}(E,B(r,s))=\bigg\{x=(x_{n})\in \omega: \sum_{n=1}^{\infty} \bigg|\sum_{j\in E_n}rx_{j}+\sum_{j\in E_{n+1}}sx_{j}\bigg|^{p}<\infty\bigg\}, $$ where $E=(E_n)$ is a partition of finite subsets of the positive integers, $r,s\in \mathbb{R}\backslash \{0\}$ and $p\geq 1$. The topological and algebraical properties of this space are examined. Furthermore, we establish some inclusion relations. Finally, the problem of finding the norm of certain matrix operators such as Copson and Hilbert from $\ell_p$ into $\ell_{p}(E,B(r,s)) $ is investigated.