On The Difference Sequence Space $l_p(\hat{T}^q)$

Year 2019, Volume: 7 Issue: 2, 161 - 173, 15.10.2019

Merve İlkhan , Pınar Zengin Alp

https://doi.org/10.36753/mathenot.597703

Cited By: 15

Abstract

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 $\lambda\in\{\ell_1,c_0,c,\ell_\infty\}$.

Keywords

sequence spaces, matrix transformations, Schauder basis, $\alpha- \beta- \gamma-duals

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