Definition:Real Part (Linear Operator)

Definition
Let $H$ be a Hilbert space over $\C$.

Let $A \in B \left({H}\right)$ be a bounded linear operator.

Then the real part of $A$ is the self-adjoint operator:


 * $\operatorname{Re} A := \dfrac 1 2 \left({A + A^*}\right)$

The real part of $A$ may be denoted by $\operatorname{Re} \left({A}\right)$, $\operatorname{re} \left({A}\right)$ or $\Re \left({A}\right)$.

This resembles the notation for the real part of a complex number.