# Definition:Real Part (Linear Operator)

## Definition

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:

$\Re A := \dfrac 1 2 \paren {A + A^*}$

## Also denoted as

The real part of $A$ may be denoted by $\map \Re A$, $\map {\mathrm {re} } A$ or $\map {\mathrm {Re} } A$.

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