Discussion
For any operator A, i (A+ - A) is :
Answer: A
The operator \(i(A^{+} - A)\) is Hermitian.
Take the Hermitian conjugate, using \((cX)^{+} = c^{*}X^{+}\) and \((A^{+})^{+} = A\):
\[ \begin{aligned} [i(A^{+} - A)]^{+} &= -i(A - A^{+}) \\ &= i(A^{+} - A) \end{aligned} \]
The operator equals its own conjugate, which is the definition of Hermitian. The bracket \(A^{+} - A\) alone is anti-Hermitian, and multiplying an anti-Hermitian operator by i makes it Hermitian. Unitarity (\(U^{+}U = 1\)) and orthogonality are different properties that do not follow for an arbitrary A.
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