I attempted to separate the real and imaginary parts of the function $Tr(R_T(i\lambda)[A] B)$, where $R_T$ represents the resolvent and $T$ is a super-operator. The parameter $\lambda$ is a real number, and both $A$ and $B$ are Hermitian operators. However, I encountered difficulties in achieving this separation using the methods I employed. One approach I tried was utilizing the Neumann series expansion, but it turned out to be applicable only when the norm of $T$ is less than or equal to the absolute value of $\lambda$. This limitation posed a challenge for me in successfully separating the real and imaginary components of the function in all cases.
Is there any idea that I can separate the parts without any limitation?