Ask for the name of a condition on commutator of two operators

Let $T, S$ be two bounded linear operators on a Hilbert space. I wonder whether there is a standard way referring the following condition:

$$\text{The commutator [T, S] is in the Hilbert-Schmidt class.}$$

Note that, if the Hilbert-Schmidt class is replaced by compact operator, then we call that $T$ and $S$ essentially commute.

• I think that there is no special name for the mentioned condition. Usualy, for an ideal ${\mathcal I}$ (or even for an arbitrary set) of operators, one says that $S, T$ commute modulo ${\mathcal I}$ if $[T,S]\in {\mathcal I}$. – Janko Bracic Jan 13 '15 at 4:54
• That is OK, then I will say that T, S commute modulo HS class. – Yanqi QIU Jan 13 '15 at 7:52