  In this manuscript, we investigate the properties of systems formed by
translations of an operator in the Schatten $p$-classes $\mathcal{T}^p$. We
establish the existence of Schauder frames of integer translates in
$\mathcal{T}^p$ for $p>2$. Later, we provide an instance of a uniformly
discrete $\Lambda \subset \mathbb{R}^{2d}$ such that there exists an operator
whose $\Lambda$-translates are complete in $\mathcal{T}^p$ for all $p>1$.
