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Add additionial proof needed for proseminar

This commit is contained in:
Anton Mosich 2022-06-17 11:37:48 +02:00
parent 6fb48c603f
commit 51c35107d5
Signed by: Flugschwein
GPG Key ID: 9303E1C32E3A14A0
1 changed files with 6 additions and 3 deletions
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LinAlg2.tex
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@ -3967,9 +3967,12 @@ Wir haben eine echte Verallgemeinerung.
\im(\alpha^\dagger) = \im(\alpha^\dagger \circ \alpha)
\end{aligned}
\end{equation}
\tl UE\br\,:
$\ker(\alpha) \subseteq \ker(\alpha^\dagger \circ \alpha) \subseteq \ker(\alpha \circ \alpha^\dagger
\circ \alpha) = \ker(\alpha) \implies \ker(\alpha) = \ker(\alpha \circ \alpha^\dagger \circ \alpha)$
$\ker(\alpha) \subseteq \ker(\alpha^\dagger \circ \alpha) \subseteq
\ker(\alpha \circ \alpha^\dagger \circ \alpha)
= \ker(\alpha) \implies \ker(\alpha) = \ker(\alpha^\dagger \circ \alpha)$
$\im(\alpha) \subseteq \im(\alpha \circ \alpha^\dagger) \subseteq
\im(\alpha \circ \alpha^\dagger \circ \alpha) = \im(\alpha) \implies
\im(\alpha) = \im(\alpha \circ \alpha^\dagger)$
\item $\nu := \alpha^\dagger \circ \alpha$ erfüllt $\nu \circ \nu$ und ist selbstadjungiert
? für $\nu' := \alpha \circ \alpha^\dagger$ \\
$\implies \underbrace{\ker(\nu)}_{=\ker(\alpha)} \bot \im(\nu)$