Assume we are given a densely defined and closed operator $T$ on a Hilbert space such that $D(T)\cap D(T^*)$ is dense. I am looking for sufficient conditions for $T+T^*$ to be selfadjoint as well as nice counter examples. Especially: Is $T+T^*$ always selfadjoint, when $T$ is normal?
Edit after some discussion in the comments: Selfadjointness seems to much to ask for, so let me rephrase: I would like to know whether for all normal operators $T$ there exist two (commuting) selfadjoint operators $A$ and $B$ such that $T=A+ i B$? [If $A=0$, then $T$ is normal but $T+T^* = 0|_{D(T)}$ is not selfadjoint but only essentially selfadjoint.]