Normal Operators
Why
We can abstract the property of self-adjointness.
Definition
An operator $T$ oon an inner product space is
called normal if if
commutes with its adjoints.
In symbols, $T \in \mathcal{L} (V)$ is normal
if
\[
TT^* = T^* T
\]
Every self-adjoint operator is normal.
Suppose $T \in \mathcal{L} (V)$ with $T^* = T$.
Then $TT^* = TT = T^*T$, and so $T$ is
normal.