If $A$ is an operator on $H$. Show that :
(a) $A$ is anti-Hermitian if and only if $iA$ is self-adjoint.
(b) $A-A^{*}$ is anti-Hermitian.
I'm search what means a anti-Hermitian operator and I found that if $S$ is anti-Hermitian for the inner product is: \begin{eqnarray*} \langle Sw,z\rangle = -\langle w,Sz\rangle \end{eqnarray*} For $w,z \in H$.
$A^{*}$ is the adjoint operator of $A$.
So My work for (b) is:
Let $w,s\in H$: \begin{eqnarray*} \langle (A-A^{*})w,z\rangle&=&\langle w,(A-A^{*})^{*}z \rangle \\ &=& \langle w,(A^{*}-A)z\rangle\\ &=&- \langle w,(A-A^{*})z \rangle \end{eqnarray*} So, $A-A^{*}$ is a anti-Hermitian operator. Is this correct?. But I'm confuse with (a) if I take this definition of anti-Hermitian operator.