\begin{definition}[Definition:Real Part (Linear Operator)]
Let $\HH$ be a Hilbert space over $\C$.
Let $A \in \map B \HH$ be a bounded linear operator.
Then the '''real part of $A$''' is the Hermitian operator:
:$\Re A := \dfrac 1 2 \paren {A + A^*}$
\end{definition}