Let be a -algebra with unit .
The general formulation of the result is as follows:
Theorem. Let be a -algebra with unit . Let two normal elements be given and with . Then it follows that .
Lemma. For any we have that is a element of .
Proof. We have for that . And similarly . ∎
With this we can now give a proof the Theorem.
Proof. The condition implies by induction that holds for each . Expanding in power series on both sides yields . This is equivalent to . Set . From the Lemma we obtain that . Since commutes with und with we obtain that
which equals .
Define by . If we substitute in the last estimate we obtain
But is clearly an entire function and therefore Liouville’s theorem implies that for each .
This yields the equality
Comparing the terms of first order for small finishes the proof. ∎
|Date of creation||2013-05-08 21:47:27|
|Last modified on||2013-05-08 21:47:27|
|Last modified by||karstenb (16623)|