Let $A$ be self-adjoint and $B$ be symmetric with $A$-bound less than $1$. By Kato-Rellich, I know that $(A+B)^*=A+B$. Could I also get something like $(A+iB)^*=A-iB$ or is there a counterexample to this?
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$\begingroup$ Can't you prove that [A+i(B-B*)] is self adjoint directly from Kato-Rellich? Note that if B is A bounded, then so is i(B-B*)--use polar decomposition to show B*=UBU for some unitary U. Or does this not work in the unbounded setting? $\endgroup$– DerekCommented Mar 24, 2022 at 12:17
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