In a separable Hilbert space (let $\mathcal{H}=\mathcal{H_1}\otimes\mathcal{H_2}$) does it have to be that every operator $\mathcal{A}$ that operates in $\mathcal{H}$ can be written as $\mathcal{A_1}\otimes \mathcal{A_2}$ so that each one acts on the subspaces?
2026-04-13 22:10:27.1776118227
Separable Operators in a Hilbert Space.
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No, a sum of operators will give you an operator. And not every sum of elementary tensors is an elementary tensor. And then you can take limits of sums of tensors, and you'll get more operators. And even then, if you are using the norm topology for instance, there are operators which are not limits of sums of tensors.