Characterization of Normal Operators

Theorem

Let $H$ be a Hilbert space.

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

Then the following are equivalent:

$(1): \qquad AA^* = A^*A$, i.e., $A$ is normal
$(2): \qquad \forall h \in H: \left\Vert{Ah}\right\Vert_H = \left\Vert{A^*h}\right\Vert_H$

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

$(3): \qquad \operatorname{Re} \left({A}\right) \operatorname{Im} \left({A}\right) = \operatorname{Im} \left({A}\right) \operatorname{Re} \left({A}\right)$, i.e., the real and imaginary parts of $A$ commute