$\textbf{Context :}$ If the context helps, then the things motivated me to ask this question are action of squezing operator or displacement operator on density matrix in quantum optics and Bogolioubov transformation acting on thermal state in theory of superconductivity, Lang-Firsov transformation (Polaron transformation for spin-boson problem) in physics of dissipative quantum systems and many such canonical transformations in quantum many body physics.
Suppose we have an operator algebra ($\mathcal{O}$) consisting of operators $\{\hat{a}_{i}^{},\hat{a}_{i}^{\dagger}\}$ and all possible linear combinations (coefficients from the complex field) of arbitrary products of $\{\hat{a}_{i}^{},\hat{a}_{i}^{\dagger}\}$ acting on a Hilbert space ($\mathcal{H}$) defined over complex field.
Let $\mathbb{U}[\{\hat{a}_{i}^{},\hat{a}_{i}^{\dagger}\}]$ be an unitary operator (a function of $\{\hat{a}_{i}^{},\hat{a}_{i}^{\dagger}\}$) acting on $\mathcal{H}$ and further suppose $\mathbb{Z}[\{\hat{a}_{i}^{},\hat{a}_{i}^{\dagger}\}]$ be an arbitrary operator (a function of $\{\hat{a}_{i}^{},\hat{a}_{i}^{\dagger}\}$) also acting on $\mathcal{H}$.
Then when (conditions neccessary for $\mathbb{Z}$ and $\mathbb{U}$) is the following identity in $\mathcal{O}$ is true :
$$\mathbb{U}_{}^{}\mathbb{Z}[\{\hat{a}_{i}^{},\hat{a}_{i}^{\dagger}\}]\mathbb{U}_{}^{\dagger}\stackrel{\textbf{?}}{=}\mathbb{Z}[\{\mathbb{U}_{}^{}\hat{a}_{i}^{}\mathbb{U}_{}^{\dagger},\mathbb{U}_{}^{}\hat{a}_{i}^{\dagger}\mathbb{U}_{}^{\dagger}\}].$$
$\textbf{Note :}$ If $\mathbb{Z}[\{\hat{a}_{i}^{},\hat{a}_{i}^{\dagger}\}]$ is taylor expandable, we can clearly see the above identity.
