Timeline for If the cardinality of $B(X)$, the space of operators on $X$, is continuum, must $X$ be separable?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Jun 9, 2018 at 13:24 | vote | accept | ABB | ||
| Jun 9, 2018 at 13:24 | vote | accept | ABB | ||
| Jun 9, 2018 at 13:24 | |||||
| Jun 9, 2018 at 11:48 | comment | added | Taras Banakh | The exists eveb simpler ZFC-example -- the Banach space $C(K)$ over the Alexandroff two-arrows space $K$, see my answer below. | |
| Jun 9, 2018 at 11:15 | answer | added | Taras Banakh | timeline score: 6 | |
| Jun 9, 2018 at 7:59 | comment | added | Tomasz Kania | I think it okay. Then this space will be ZFC counterexample impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/… | |
| Jun 9, 2018 at 6:54 | comment | added | Taras Banakh | @TomekKania The number of weakly compact operators is the number of weakly compact sets times the number of continuous operators mapping the unit ball to a given weakly compact set. The number of weakly compact sets should not exceed $|X^*|^\omega$ and if all weakly compact sets are metrizable (in the weak topology), then the number of continuous operators mapping the unit ball to a given weakly compact set also should not exceed $|X^*|^\omega$. So, for a Banach space $X$ with ``small'' weakly compact sets, the number of weakly compact operators should not exceed $|X^*|^\omega$. Right? | |
| Jun 8, 2018 at 21:47 | comment | added | Tomasz Kania | @TarasBanakh, the problem with spaces $K$ such that every operator on $C(K)$ is of the form $gI+W$ where $g\in C(K)$ and $W$ weakly compact, is that there is no clear way how to count weakly compact operators on them. | |
| Jun 8, 2018 at 4:30 | vote | accept | ABB | ||
| Jun 9, 2018 at 13:24 | |||||
| Jun 7, 2018 at 23:08 | history | edited | Tomasz Kania | CC BY-SA 4.0 |
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| Jun 7, 2018 at 20:37 | history | edited | ABB | CC BY-SA 4.0 |
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| Jun 7, 2018 at 20:07 | answer | added | Tomasz Kania | timeline score: 10 | |
| Jun 7, 2018 at 20:04 | history | edited | ABB | CC BY-SA 4.0 |
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| Jun 7, 2018 at 19:19 | comment | added | Taras Banakh | I hope that a counterexample can be found in papers of Koszmider, for example, in impan.pl/~koszmider/badania/fewsur2.pdf or link.springer.com/article/10.5052%2FRACSAM.2010.19 | |
| Jun 7, 2018 at 19:03 | comment | added | ABB | @TarasBanakh: Edited. | |
| Jun 7, 2018 at 19:02 | history | edited | ABB | CC BY-SA 4.0 |
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| Jun 7, 2018 at 18:51 | comment | added | Taras Banakh | What is $B(X)$? | |
| Jun 7, 2018 at 17:05 | history | asked | ABB | CC BY-SA 4.0 |