Added a handful of examples.
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Bokuan Li
2026-06-02 21:39:59 -04:00
parent 42433c40ca
commit 2c1169e55a
8 changed files with 133 additions and 3 deletions

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@@ -26,10 +26,15 @@
\sigma_A(x) \subset \complex \setminus e^{i\theta}[0, \infty)
\]
then $x \in G_0(A)$.
\begin{enumerate}
\item There exists $y \in A$ such that $x = \exp(y)$.
\item $x \in G_0(A)$.
\end{enumerate}
\end{proposition}
\begin{proof}
Since there exists a branch of the logarithm $\ell: \complex \setminus e^{i\theta}[0, \infty) \to \complex$, there exists $y \in A$ such that $x = \exp(y)$. In which case, $x \in G_0(A)$ by \autoref{proposition:functional-calculus-exp}.
(1): Since there exists a branch of the logarithm $\ell: \complex \setminus e^{i\theta}[0, \infty) \to \complex$, there exists $y \in A$ such that $x = \exp(y)$.
(2): \autoref{proposition:functional-calculus-exp}.
\end{proof}