Got this question for homework, im having troubles to prove one side of it

The question: Prove T is normal if and only if T = T1 + iT2, where T1 and T2 are selfadjoint operators which commute.

$<=$ lets assume we have T1, and T2 such as mentioned, so $\ TT$*$=(T1 + iT2)(T1 + iT2)$* = ... = $(T1+iT2)$*$(T1+iT2)$$=T$*$T$

$=>$ having troubles with that part...

any clue will help thanks!

  • 2
    $\begingroup$ Note that any operator can be written as a sum of self-adjoint and an anti-self-adjoint operator. Normality than easily implies the two commute. $\endgroup$
    – D M
    Apr 20, 2013 at 2:42

1 Answer 1


If $T$ is normal take $ T_1 = (T+T^*)/2 $ and $ T_2 = (T-T^*)/2i $. Clearly you have $ T = T_1+iT_2 $ and $T_1, T_2 $ commute. Now $T_1 $ is clearly self adjoint, $iT_2 $ is skew-adjoint, hence $T_2$ is also self adjoint.


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