  The initial value problem of the Zakharov system on two-dimensional torus
with general period is considered in this paper. We apply the $I$-method with
some 'resonant decomposition' to show global well-posedness results for
small-in-$L^2$ initial data belonging to some spaces weaker than the energy
class. We also consider an application of our ideas to the initial value
problem on $\mathbb{R}^2$ and give an improvement of the best known result by
Pecher (2012).
