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\begin{document}
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\begin{center}
Asymptotically Equivalent Matrices in an Ideal Setting.
\end{center}

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\begin{center}
Jeff Connor* and Hafize G\"{o}k**

\bigskip

*Ohio University, Department of Mathematics, Athens, Ohio, USA

connorj@ohio.edu\bigskip

**Afyonkarahisar Kocatepe University, Department of Mathematics, Afyonkarahisar/TURKEY

hafize\_1409@hotmail.com\bigskip
\end{center}

Two sequences $x$ and $y$ are defined to be asymptotically equivalent if
\linebreak$\lim_{k}\left(  x_{k}/y_{k}\right)  =1$ and a summability matrix is
defined to be \textit{asymptotically regular} if it maps nonnegative
asymptotically equivalent sequences to asymptotically equivalent sequences.
Pobyvanets(1980) introduced asymptotically equivalent matrices and provided a
characterization of nonnegative asymptotically equivalent matrices. More
recently Marouf(1993), Li(1997), Patterson(2003), and Patterson and Savas
(2009) have explored variations of the definition of asymptotically equivalent
sequences and studied nonnegative matrices which preserve these variations of
asymptotic equivalence. Given an ideal $I$ of subsets of $\mathbb{N}$ that
contains the finite subsets of $\mathbb{N}$, a sequence is said to be
$I$-convergent to $l$ provided $\left\{  k:\left\vert x_{k}-l\right\vert
\geq\varepsilon\right\}  \in I$ for all $\varepsilon>0,$ and two nonnegative
sequences $x$ and $y$ are said to be $I-$asymptotically equivalent provided
$x_{n}/y_{n}$ is $I$-convergent to $1.$ In this note, given ideals $I$ and
$J,$ we characterize nonnegative matrices which map $I$-asymptotically
equivalent sequences to $J$-asymptotically equivalent sequences and establish
similar results analogous to those of Marouf and Li.

To date, characterizations of matrices that preserve asymptotic equivalence
(or its variants) are typically restricted to nonnegative matrices and
nonnegative sequences, where in some cases the sequences are bounded below by
a positive number. We also discuss the necessity of these hypotheses.


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