# Method to factorize/diagonalize differential operators with field redefinitions.

Assuming that $$h_{ab}$$ is some metric perturbation and $$h_{}$$ means the traceless part, I have the LHS of the next equation and I want to find a field redefinition $$t_{ab}$$ as a function of $$h$$ and $$h_{ab}$$ such that I can rewrite the expression as the last line : $$E_{}= ( \Box+m_1^2) ( \Box+m_2^2) \Box h_{}+ ( \Box+m_3^2) ( \Box+m_4^2) \partial_{}h = ^{?} ( \Box+M_1^2) ( \Box+M_2^2) ( \Box+M_3^2) t_{ab}$$ Exist some particular method to do it? ( $$h=h_{ab}g^{ab}$$ is simply the trace of $$h_{ab}$$ and $$g_{ab}$$ is the background / zero order metric. ) Thanks.