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:

$A^*$ 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.


Proof


Sources