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Fixed sign error.
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Bokuan Li
2026-10-18 19:57:47 -04:00
parent d09736a42e
commit 6dbdd2396d
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src/op/system/os.tex
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@@ -33,7 +33,7 @@
Assume without loss of generality that $\seq{x_n} \subset S_{sa}$, so that $\sigma_A(x_n) \subset \real$ for all $n \in \natp$. By \hyperref[upper semicontinuity of the spectrum]{proposition:spectrum-continuous}, $\lim_{n \to \infty}\inf(\sigma_A(x_n)) \ge 0$. Therefore Assume without loss of generality that $\seq{x_n} \subset S_{sa}$, so that $\sigma_A(x_n) \subset \real$ for all $n \in \natp$. By \hyperref[upper semicontinuity of the spectrum]{proposition:spectrum-continuous}, $\lim_{n \to \infty}\inf(\sigma_A(x_n)) \ge 0$. Therefore
\[ \[
x = \limv{n}x_n = \limv{n}[x_n + \inf(\sigma_A(x_n)) \wedge 0] \in \ol{S_+} x = \limv{n}x_n = \limv{n}[x_n - \inf(\sigma_A(x_n)) \wedge 0] \in \ol{S_+}
\] \]
by \autoref{corollary:spectrum-characterisation-iff}. by \autoref{corollary:spectrum-characterisation-iff}.