Summary: The goal of this article is to study \( \mathcal{I} \)-equivalence double nonnegative sequences. To accomplish this we present a series of theorems that are similar to the following: If \( A \) is a nonnegative four dimensional summability matrix that maps bounded double sequences to \( \ell^2 \) and let \( \mathcal{I}_2 \) and \( \mathcal{J}_2 \) be admissible ideals in \( \mathbb{N}\times\mathbb{N}\) then the following are equivalent.

1.

If \(x=(x_{k,l})\) and \(y=(y_{k,l})\) are bounded double sequences such that \(x \mathop{\sim}\limits^{\mathcal{I}_2} y \), \( (x_{k,l})\in P''_0 \) and \((y_{k,l})\in P''_\delta \), for some \( \delta >0 \), then \( R_{m,n} (Ax) \mathop{\sim}\limits^{\mathcal{J}_2} R_{m,n} (Ay) \) for each \( m \) and \( n \),

2.

\( \mathcal{J}_2$-$\lim_{m,n} \frac{\sum_{k,l\geq m,n} \sum_{p,q\in S_2}a_{k,l,p,q}}{\sum_{k,l\geq m,n} \sum^{\infty,\infty}_{p,q=1,1} a_{k,l,p,q}}=0 \) for each \( S_2\in\mathcal{I}_2 \).

In addition other variations and implications are also presented.