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Prove that the decomposition of a linear operator into a sum of a self-adjoint and anti-self-adjoint operator is unique

My ideas: is to use fact that set of self-adjoint operators generate vector space.

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  • $\begingroup$ Hint: if $A + B = C + D$, then $A^\dagger + B^\dagger = C^\dagger + D^\dagger $. $\endgroup$ Jun 18, 2022 at 15:59

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If $S_1+A_1=S_2+A_2$, with $S_1,S_2$ selfadjoint and $A_1,A_2$ anti-selfadjoint, then the operator $$ T=S_1-S_2=A_2-A_1 $$ is both selfadjoint and anti-selfadjoint. That is, $T^*=-T^*$, so $T^*=0$.

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