# Characterization of Normal Operators

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## 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:

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.

## Proof

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $II.2.16$