Characterization of Normal Operators

Theorem
Let $\HH$ be a Hilbert space.

Let $A$ be a bounded linear operator on $\HH$.

Then the following are equivalent:


 * $(1): \quad A A^* = A^* A$, that is, $A$ is normal
 * $(2): \quad \forall h \in H: \norm {A h}_\HH = \norm {A^*h}_\HH$

where:
 * $A^*$ denotes the adjoint of $A$
 * $\norm {\, \cdot\,}$ denotes the norm

If $\HH$ is a Hilbert space over $\C$, these are also equivalent to:


 * $(3): \quad \map \Re A \map \Im A = \map \Im A \map \Re A$, that is, the real and imaginary parts of $A$ commute.