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\begin{document}


\begin{center}
\begin{eqnarray*}
&&\text{MATRIX TRANSFORMATIONS BETWEEN }c_{I}(\Delta )\cap l_{\infty
}(\Delta ),\text{ } \\
&&\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }c_{I}(\Delta ^{2})\cap l_{\infty
}(\Delta ^{2}),\text{ }c_{I}(\Delta ^{m})\cap l_{\infty }(\Delta ^{m})\text{
AND }c_{I}
\end{eqnarray*}%
\\[0pt]

\bigskip

Jeff Connor$^{\ast }$ and Hafize G\"{o}k$^{\ast \ast }$\\[0pt]

\bigskip

$\ast $ Ohio University, Department of Mathematics, Athens, Ohio, USA

connorj@ohio.edu

$\ast \ast $ Afyonkarahisar Kocatepe University, Department of Mathematics,
Afyonkarahisar/TURKEY

hafize\_1409@hotmail.com

\bigskip

\bigskip

ABSTRACT
\end{center}

Let $l_{\infty },$ $c$ and $c_{0}$ be the linear spaces of bounded,
convergent and null sequences, respectively. In 1981, H. K\i zmaz defined $%
l_{\infty }(\Delta ),$ $c(\Delta ),$ and $c_{0}(\Delta )$ spaces with $%
\Delta x=(\Delta x_{n})=(x_{n}-x_{n+1})$ where $n\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
=\left\{ 1,2,...\right\} .$ He then characteri-\newline
zed some matrix transformations such that $A\in (E^{^{\prime }},F)$ and $%
A\in (E,F^{^{\prime }})$ where $E$ and $F$ denote one of the sequence spaces 
$l_{\infty }$ and $c;$ $E^{^{\prime }}$ and $F^{^{\prime }}$ denote one of
the sequence spaces $l_{\infty }(\Delta )$ and $c(\Delta ).$ In this study
we establish some results for matrix transformations such that $A\in
(c_{I}(\Delta )\cap l_{\infty }(\Delta ),c_{I})$ where $A=(a_{nk})$ is a
nonnegative matrix, $I\subseteq 2^{%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
}$ is an admissible ideal, $c_{I}$ is the space of all $I-$convergent
sequences and,%
\begin{eqnarray*}
c_{I}(\Delta ) &=&\left\{ x=(x_{n}):(\Delta x_{n})\in c_{I}\right\}  \\
l_{\infty }(\Delta ) &=&\left\{ x=(x_{n}):(\Delta x_{n})\in l_{\infty
}\right\} 
\end{eqnarray*}

Then we establish some results for matrix transformations such that $A\in
(c_{I}(\Delta ^{2})\cap l_{\infty }(\Delta ^{2}),c_{I})$ where,%
\begin{eqnarray*}
c_{I}(\Delta ^{2}) &=&\left\{ x=(x_{n}):(\Delta ^{2}x_{n})\in c_{I}\right\}
\\
l_{\infty }(\Delta ^{2}) &=&\left\{ x=(x_{n}):(\Delta ^{2}x_{n})\in
l_{\infty }\right\}
\end{eqnarray*}

In conclusion we generalize these results for%
\begin{eqnarray*}
c_{I}(\Delta ^{m}) &=&\left\{ x=(x_{n}):(\Delta ^{m}x_{n})\in c_{I}\right\} 
\\
l_{\infty }(\Delta ^{m}) &=&\left\{ x=(x_{n}):(\Delta ^{m}x_{n})\in
l_{\infty }\right\} 
\end{eqnarray*}%
where $m\in 
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
$ and we establish some results for $A\in (c_{I}(\Delta ^{m})\cap l_{\infty
}(\Delta ^{m}),c_{I})$ matrix transformations.

\newpage

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