From 84da66243780c4ce87b66772748282a490a3a39b Mon Sep 17 00:00:00 2001 From: Bokuan Li Date: Sun, 11 Oct 2026 14:42:46 -0400 Subject: [PATCH] Thank you Sam. --- src/op/system/os.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/op/system/os.tex b/src/op/system/os.tex index 4150e61..5c55bea 100644 --- a/src/op/system/os.tex +++ b/src/op/system/os.tex @@ -181,7 +181,7 @@ (3): For any $\xi \in H$ with $\norm{\xi}_H = 1$, $\dpn{T\xi, \xi}{H} \le \norm{T}_{B(H)}$, so $w(T) \le \norm{T}_H$. The sharp case is given by $I_H$. - On the other hand, if $w(T) \le 1$, then $2 + T + T^* = 2 + 2\text{Re}(T) \ge 0$ implies that $\norm{\text{Re}(T)}_{B(H)} \le 1$. Similarly, $2 + iT - iT^* = 2 + 2\text{Im}(T) \ge 0$ implies that $\norm{\text{Im}(T)}_{B(H)} \le 1$, so $\norm{T}_{B(H)} \le \norm{\text{Re}(T)}_{B(H)} + \norm{\text{Im}(T)}_{B(H)} \le 2$. Therefore $\norm{T}_{B(H)} \le 2w(T)$. + On the other hand, if $w(T) \le 1$, then $2 + T + T^* = 2 + 2\text{Re}(T) \ge 0$ and $2 - T - T^* = 2 - 2\text{Re}(T) \ge 0$ imply that $\norm{\text{Re}(T)}_{B(H)} \le 1$. Similarly, $\norm{\text{Im}(T)}_{B(H)} \le 1$ as well, so $\norm{T}_{B(H)} \le \norm{\text{Re}(T)}_{B(H)} + \norm{\text{Im}(T)}_{B(H)} \le 2$. Therefore $\norm{T}_{B(H)} \le 2w(T)$. For any $\xi, \eta \in \complex$ with $\norm{(\xi, \eta)}_{\complex^2} = 1$, by \hyperref[Young's inequality]{lemma:young-inequality}, \[