Characterization of Normal Operators
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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
Proof
Sources
- 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next) $II.2.16$