# Is every operator a power series of creation and annihilation operators (in a rigorous mathematical sense)?

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.