MULTIDIMENSIONAL CHARACTERIZATION OF ASYMPTOTICALLY I s 2− EQUIVALENT SEQUENCES

Abstract

The goal of this paper includes the presentation of matrix characterization of
asymptotically $\mathcal{I}_{2}-$equivalent and asymptotically $\mathcal{I}%
_{2}-$statistical equivalent double sequences. To accomplish these goals we presented the following natural
theorem. Let us consider $\mathcal{I}_{2}^{s}$ and $\mathcal{J}_{2}^{s}$ the
following ideals in $%
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\mathbb{N}
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\times
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\mathbb{N}
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$. Let four dimensional matrix $A$ be a nonnegative $c_{0}^{\prime \prime
}-c_{0}^{\prime \prime }$\ summability matrix, also let $x$ and $y$ be
element of $\ell ^{2}$. Suppose that $x\overset{\mathcal{I}_{2}^{s}}{\sim }y$
with $x,$ $y\in P_{\delta }^{\prime \prime }$ for some $\delta >0,$ then $%
\mu _{m,n}\left( Ax\right) \overset{\mathcal{J}_{2}^{s}}{\sim }\mu
_{m,n}\left( Ay\right) $ if and only if for each $i,j=1,2,3,...$ and for
some $\varepsilon >0$ and $\gamma >0$ such that
\begin{equation*}
\mathcal{J}_{2}^{s}-\lim_{m,n}\frac{1}{mn}\left\{ \left\vert \left\{ k\leq m%
\text{, }l\leq n:\left\vert \frac{a_{m,n,i,j}}{\underset{r,s=1,1}{\overset{%
\infty ,\infty }{\sum }}a_{m,n,r,s}}\right\vert \geq \varepsilon \right\}
\right\vert \geq \gamma \right\} =0
\end{equation*}%
for $\left\{ \left( k,l\right) \in
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\mathbb{N}
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\times
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\mathbb{N}
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:\frac{1}{mn}\left\vert \left\{ k\leq m,l\leq n:\left\vert \frac{x_{k,l}}{%
y_{k,l}}-L\right\vert \geq \varepsilon \right\} \right\vert \geq \gamma
\right\} \in \mathcal{I}_{2}^{s}$. Additionally, we also presented conditions on
four-dimensional matrix $A=\left( a_{m,n,k,l}\right) $ that assures
regularity in the asymptotically $\mathcal{I}_{2}-$ sense.