Discrete Translates of an Operator in the Schatten $p$-Classes

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 $\mathrm{\Lambda }\subset {\mathbb{R}}^{2d}$ such that there exists an operator whose $\mathrm{\Lambda }$-translates are complete in ${\mathcal{T}}^p$ for all $p>1$.