Timeline for $\|t\| = \sup_{\|z\| \le 1} \|\langle tz,z\rangle\|$ when $t=t^*$
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| S Sep 25, 2021 at 8:06 | history | bounty ended | CommunityBot | ||
| S Sep 25, 2021 at 8:06 | history | notice removed | CommunityBot | ||
| Sep 24, 2021 at 20:26 | vote | accept | Andromeda | ||
| Sep 18, 2021 at 9:15 | answer | added | Andromeda | timeline score: 2 | |
| Sep 17, 2021 at 9:15 | comment | added | Jamie Gabe | Sketch: By functional calculus you write $t=t_+ - t_-$. Suppose $\| t\| = \|t_+\|$ (the case $\| t\| = \|t_-\|$ is similar). Pick $z\in E$ contractive such that $\| t_+^{1/2} z\|$ is close to $\|t_+^{1/2}\|$. Letting $f:[-1,1] \to [0,1]$ be continuous which is 0 on $[-1,0]$ and 1 on $[\epsilon , 1]$, we have $\| \langle t f(t) z, f(t) z\rangle\|$ is close to $\|t\|$. In the standard Hilbert space proof one would take $f = \chi_{[0,1]}$ but a continuous approximation works as well. | |
| S Sep 17, 2021 at 6:18 | history | bounty started | Andromeda | ||
| S Sep 17, 2021 at 6:18 | history | notice added | Andromeda | Draw attention | |
| Sep 16, 2021 at 14:44 | history | edited | Andromeda | CC BY-SA 4.0 |
added 37 characters in body
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| Sep 15, 2021 at 20:24 | comment | added | Andromeda | @NarutakaOZAWA I'm not aware of a proof of this that uses spectral projections. | |
| Sep 15, 2021 at 14:16 | history | edited | Andromeda |
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| Sep 14, 2021 at 23:01 | comment | added | Narutaka OZAWA | Which proof doesn't generalize? Can't you replace a spectral projection with some functional calculus $f(t)$? | |
| Sep 14, 2021 at 20:54 | history | asked | Andromeda | CC BY-SA 4.0 |