Timeline for The Invariant Subspace Problem: examples
Current License: CC BY-SA 2.5
12 events
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| Nov 16, 2023 at 12:48 | comment | added | The Amplitwist |
The link to maths.leeds.ac.uk in a comment above seems to be broken, but a copy of the PDF is saved at the Wayback Machine. The full citation of the article is as follows: Partington, Jonathan R.; Pozzi, Elodie, Universal shifts and composition operators, Oper. Matrices 5, No. 3, 455-467 (2011). Zbl 1244.47007.
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| Jul 4, 2020 at 7:27 | comment | added | William DeMeo | +1 good question! See also, my question | |
| May 8, 2011 at 12:16 | vote | accept | Andrey Rekalo | ||
| May 4, 2011 at 22:56 | answer | added | Manfred Sauter | timeline score: 5 | |
| Jan 24, 2011 at 12:54 | history | edited | Andrey Rekalo | CC BY-SA 2.5 |
Specified the question
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| Jan 6, 2011 at 20:10 | comment | added | Andrey Rekalo | @Zen Harper: Thanks for the comment! | |
| Jan 6, 2011 at 1:22 | comment | added | Zen Harper | Sorry I can't find the reference - maybe an expert can supply it? Since about 1990(?), the general invariant subspace problem is known to be equivalent to a special case. Let $L^2_a(D)$ be the Bergman space of analytic functions on the unit disc $D = \{ |z|<1 \}$, with squared norm the area integral $\| f \|^2 = \frac{1}{\pi}\int\int_D |f(z)|^2 dA(z)$, and the linear operator $M$ is just $(Mf)(z) = z f(z)$. Similarly to David Feldman's comment, it is not $M$ itself, but the restriction of $M$ to subspaces, which is important; but the subspaces themselves have no simple description. | |
| Dec 20, 2010 at 10:25 | comment | added | Andrey Rekalo | @ David: This is interesting, thanks. | |
| Dec 20, 2010 at 8:31 | comment | added | David Feldman | This paper maths.leeds.ac.uk/~pmt6jrp/op_de_composition_rev.pdf gives examples of concrete operators which "all their invariant subspaces have themselves have non-trivial invariant subspaces" implies that every bounded operator on Hilbert space has an invariant subspace. Of course you might complain that the operator is concrete, but not its invariant subspaces. | |
| Dec 19, 2010 at 18:28 | comment | added | Andrey Rekalo | @ Andres: Thank you for the reference. | |
| Dec 19, 2010 at 18:08 | comment | added | Andrés E. Caicedo | Andrey, there are very general positive results, so I do not think a "concrete" candidate is known. There is a nice recent paper with good references to the state of the art on the problem: B. S. Yadav, "The Present State and Heritages of the Invariant Subspace Problem", Milan j. math. 73 (2005), 289–316. | |
| Dec 19, 2010 at 17:48 | history | asked | Andrey Rekalo | CC BY-SA 2.5 |