Anti-Hermitian Operator

An operator is anti-hermitian if and only if

B^+=−B^\begin{gather*} \hat{B}^+=-\hat{B} \end{gather*}

note that if you do

A^=iB^\begin{gather*} \hat{A}=i\hat{B} \end{gather*}

then A^\hat{A} is Hermitian

Any operator T^\hat{T} can be written as linear combination of Hermitian and anti-Hermitian operator.

T^=A^+iB^A^=T^+T^+2B^=T^−T^+2i\begin{gather*} \hat{T}=\hat{A}+i\hat{B}\\ \hat{A}=\frac{\hat{T}+\hat{T}^+}{2}\\ \hat{B}=\frac{\hat{T}-\hat{T}^+}{2i} \end{gather*}

Note that this is true for regular complex numbers