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

This needs considerable tedious hard slog to complete it. In particular: Which specific norm do we link to here? To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $II.2.16$