Integrated rate space $\int {\ell_\pi}$

Abstract

This paper deals with the BK-AK property of the integrated rate space $\int{\ell _\pi}$. Importance of $\delta ^{\left( k \right)}$ in this content is pointed out. We investigate a determining set for the integrated rate space $\int{\ell_\pi}$. The set of all infinite matrices transforming $\int {\ell _\pi }$ into BK-AK space $Y$ is denoted $\left( {\int{\ell _\pi :Y}}\right)$. We characterize the classes $\left( {\int{\ell_\pi :Y}}\right)$. When $Y=\ell _\infty, c_0, c, \ell^p, bv, bv_0, bs, cs, \ell_\rho, \ell _\pi$. In summary we have the following table: \medskip \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline &\hspace{0.2em} $\ell_\infty$ \hspace{0.2em}&\hspace{0.2em} $c_0$ \hspace{0.2em}&\hspace{0.2em} $c$ \hspace{0.2em}&\hspace{0.2em} $\ell^p$ \hspace{0.2em}& \hspace{0.2em} $bv$ \hspace{0.2em}&\hspace{0.2em} $bv_0$ \hspace{0.2em}&\hspace{0.2em} $bs$ \hspace{0.2em}& \hspace{0.2em}$cs$ \hspace{0.2em}& \hspace{0.2em}$\ell_\rho$\hspace{0.2em} & \hspace{0.2em}$\ell_\pi$ \\ \hline$\int{\ell_\pi}$&\multicolumn{10}{|c|}{Necessary and sufficient conditions on the matrix are obtained} \\ \hline \end{tabular} \label{tab1} \end{center}