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Consider a Hilbert space (or just a vector space over $\mathbb{C}$), which is a tensor product of several smaller Hilbert spaces: $$ H = H_1 \otimes \cdots \otimes H_n, $$ and let $\mathcal{H}$ be a Hermitian operator on $H$.

Can $\mathcal{H}$ always be represented as a sum of tensor products of Hermitian operators on the smaller Hilbert spaces?
i.e. can we write $$ \mathcal{H} = \sum_i \mathcal{H}^{(1)}_i \otimes \cdots \otimes \mathcal{H}^{(n)}_i, $$ where $\mathcal{H}^{(k)}_i$ is a Hermitian operator on $H_k$?

I came up with this question in the context of Quantum Mechanics. I would expect that a Hamiltonian of a composite system can be represented as a sum of Hamiltonians, where each one is a tensor product of Hamiltonians of the smaller systems.

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