We introduce the notion of decoupling for operators, and prove an equivalence between classical $\ell^qL^p$ decoupling for functions and $\ell^q\mathcal{ S}^p$ decoupling for operators on bounded sets in $\mathbb{ R}^{ 2d}$. We also show that the equivalence only depends on the bounded set $\Omega$, and not the values of $p,q$ nor the partition of $\Omega$. The proof relies on a quantum version of Wiener's division lemma.
Categories: math.FA math.CA