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Let $T$ be an arbitrary operator on a finite dimensional inner product space $(V,\langle\,,\,\rangle)$. Set $R=(1/2)(T^*+T)$, $S=(1/2)i(-T+T^*)$. Prove that if $T=R_1+iS_1$, where $R_1$, $S_1$ are self-adjoint, then $R_1=R$, $S_1=S$

($T^*$ is the adjoint of $T$)

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    $T^*=R_1^*-iS_1^*=R_1-iS_1$. Now solve for $R_1$ and $S_1$.2012-11-30

2 Answers 2