Lastly, we analyze the necessary and sufficient conditions for an infinite matrix to be in the classes $(\ell_p(\hat{T}^q),\lambda)$ or $(\lambda,\ell_p(\hat{T}^q))$, where $\lambda\in\{\ell_1,c_0,c,\ell_\infty\}$.
📌 3 Key Takeaways
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Objective & Hypothesis:In this study, we introduce a new matrix $\hat{T}^q=(\hat{t}^q_{nk})$ by\[\hat{t}^q_{nk}=\left \{\begin{array}[c]{ccl}%\frac{q_n}{Q_n} t_n & , & k=n\\\frac{q_k}{Q_n}t_k-\frac{q_{k+1}}{Q_n} \frac{1}{t_{k+1}} & , & k<n\\0 & , & k>n .\end{array}\right.\] where $t_k>0$ for all $n\in\mathbb{N}$ and $(t_n)\in c\backslash c_0$.
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Key Findings & Data:By using the matrix $\hat{T}^q$, we introduce the sequence space $\ell_p(\hat{T}^q)$ for $1\leq p\leq\infty$.
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Conclusion & Impact:Lastly, we analyze the necessary and sufficient conditions for an infinite matrix to be in the classes $(\ell_p(\hat{T}^q),\lambda)$ or $(\lambda,\ell_p(\hat{T}^q))$, where $\lambda\in\{\ell_1,c_0,c,\ell_\infty\}$.
In this study, we introduce a new matrix $\hat{T}^q=(\hat{t}^q_{nk})$ by\[\hat{t}^q_{nk}=\left \{\begin{array}[c]{ccl}%\frac{q_n}{Q_n} t_n & , & k=n\\\frac{q_k}{Q_n}t_k-\frac{q_{k+1}}{Q_n} \frac{1}{t_{k+1}} & , & k<n\\0 & , & k>n .\end{array}\right.\] where $t_k>0$ for all $n\in\mathbb{N}$ and $(t_n)\in c\backslash c_0$. By using the matrix $\hat{T}^q$, we introduce the sequence space $\ell_p(\hat{T}^q)$ for $1\leq p\leq\infty$. In addition, we give some theorems on inclusion relations associated with $\ell_p(\hat{T}^q)$ and find the $\alpha$-, $\beta$-, $\gamma$- duals of this space. Lastly, we analyze the necessary and sufficient conditions for an infinite matrix to be in the classes $(\ell_p(\hat{T}^q),\lambda)$ or $(\lambda,\ell_p(\hat{T}^q))$, where $\lambda\in\{\ell_1,c_0,c,\ell_\infty\}$.
İLKHAN, M., ALP, P. Z. (2019). On The Difference Sequence Space $l_p(\hat{T}^q)$. Mathematical Sciences and Applications E-Notes, Cilt: 7 Sayı: 2, 161–173. https://ajindex.com/articles/931133
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