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.

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    $\begingroup$ 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 $[T,S]\in {\mathcal I}$. $\endgroup$ – Janko Bracic Jan 13 '15 at 4:54
  • $\begingroup$ That is OK, then I will say that T, S commute modulo HS class. $\endgroup$ – Yanqi QIU Jan 13 '15 at 7:52

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