This is perhaps a particular case of the question discussed here.

Given a fermionic Fock space $H$ of dimension $2^n$, that is, with $n$ fermionic modes, let $H_n$, $n \in \mathbb{N}$ be the subspace of states with $n$ occupied fermions. Let $\hat{c}^\dagger_i$ be the creation operator for the fermionic mode $i$.

Question: Can all operators $\hat{a}^\dagger$ that map $H_n \rightarrow H_{n+1}$ be written in the form \begin{align} \hat{a}^\dagger &= \sum_{i_1} A_{i_1} \hat{c}^\dagger_{i_1} + \sum_{i_1 i_2 j_1} A_{i_1 i_2 j_1} \hat{c}^\dagger_{i_1} \hat{c}^\dagger_{i_2} \hat{c}_{j_1} + ~ ... ~ + \sum_{i_1...i_{n} j_1 ... j_{n-1}} A_{i_1 i_2 j_1} \hat{c}^\dagger_{i_1} ... \hat{c}^\dagger_{i_n} \hat{c}_{j_1} ... \hat{c}_{j_{n-1}} \text{?} \end{align} You may consider some normal ordering assumption on the coefficients $A_{i_1...i_m j_1...j_{m-1}} \in \mathbb{C}$.

As a bonus question, can we therefore associate such an operator to the quasiparticles of any number conserving fermionic Hamiltonian?

I am looking for a formal proof or a discussion with pointing to a proof in the literature.

This is perhaps a particular case of the question discussed here.

Given a fermionic Fock space $H$ of dimension $2^N$, that is, with $N$ fermionic modes, let $H_n$ be the subspace of states with $n$ occupied fermions, where $n<N$. Let $\hat{c}^\dagger_i$ be the creation operator for the fermionic mode $i$.

Question: Can all operators $\hat{a}^\dagger$ that map $H_n \rightarrow H_{n+1}$ be written in the form \begin{align} \hat{a}^\dagger &= \sum_{i_1} A_{i_1} \hat{c}^\dagger_{i_1} + \sum_{i_1 i_2 j_1} A_{i_1 i_2 j_1} \hat{c}^\dagger_{i_1} \hat{c}^\dagger_{i_2} \hat{c}_{j_1} + ~ ... ~ + \sum_{i_1...i_{N} j_1 ... j_{N-1}} A_{i_1 i_2 j_1} \hat{c}^\dagger_{i_1} ... \hat{c}^\dagger_{i_N} \hat{c}_{j_1} ... \hat{c}_{j_{N-1}} \text{?} \end{align} You may consider some normal ordering assumption on the coefficients $A_{i_1...i_m j_1...j_{m-1}} \in \mathbb{C}$.

As a bonus question, can we therefore associate such an operator to the quasiparticles of any number conserving fermionic Hamiltonian?

I am looking for a formal proof or a discussion with pointing to a proof in the literature.