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Let $\mathscr{H}$ be a Hilbert space denoting the single-particle states and $c_k^*,c_k$ denote creation and annihilation operators of orthonormal basis $\phi_k\in \mathscr{H}$. Let $\mathscr{F}$ denote the Fock space and $H$ be a linear operator on $\mathscr{F}$. Then is $H$ a polynomial series of $c_k^*,c_k$, i.e., $H = \sum_{F,G} a_{FG} \prod_{F} c_k^* \prod_G c_l$ where the summation is over a countable collection of finite subsets $F,G$ and convergence is in norm or possibly strong operator topology?

For simplicity, we may assume that it's the fermionic Fock space so that $c_k^*,c_k$ are bounded operators and also assume that $H$ is bounded self-adjoint if necessary.

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