Let $\{a_n\},\{b_m\}\in\ell^2(\mathbb{Z}^2)$ such that $\sum_{(n,m)\in\mathbb{Z}^4}a_nb_m<\infty$. Does it mean that for every permutation $\sigma$ of $\mathbb{Z}^2$, $\sum a_{\sigma(n)}b_m<\infty$?
I thought maybe using the convolution theorem with Poisson summation formula but I can't see how summation on functions evaluated on a lattice can be useful here.
EDIT/Motivation: I had a function $f,g_n\in L^2(\mathbb T^2)$ with $\sum_{n\in\mathbb{Z}^2}\hat f(n)g_n(x)=0$ and then I wanted to show that $\sum \hat f(\sigma(n))g_n(x)=0$ as well by expanding $g_n$ into a Fourier series.