We study translation surfaces in the Euclidean 3-space $\mathbb{E}^3$ and the Gauss map $N$ with respect to the so-called Cheng-Yau operator $\square$. As a result, we prove that the only translation surfaces with Gauss map $N$ satisfying $\square N=AN$ for some $3\times3$ matrix $A$ are the flat ones. We also show that the only translation surfaces with Gauss map $N$ satisfying $\square N = A N$ for some nonzero $3\times3$ matrix $A$ are the cylindrical surfaces.