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.
📌 3 Key Takeaways
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Objective & Hypothesis: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 $p\geq 1$.
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Key Findings & Data:The topological and algebraical properties of this space are examined.
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Conclusion & Impact: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.
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.
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