For any operator $T$ we can define $A=\frac {T+T^{*}} 2$ and $B=\frac {T-T^{*}} {2i}$.
So $T$ can be written as $T = A + iB$ and $A,B$ are two self-adjoint operators.
Now, suppose an operator $T$ is written in the form $T = R + iM$, where $R, M$ are two self-adjoint operators. Do $R$ and $M$ have to be $A$ and $B$? In other words, is the decomposition unique?
This comes from an answer to this question.