Timeline for Is the Invariant Subspace Problem interesting?
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
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| Jan 28, 2013 at 21:38 | comment | added | Delio Mugnolo | @William DeMeo: I'd say this is just some kind of different cultural attitude between functional analysts and number theoreticians. Not all branches of mathematics work in the same way. Graph theoreticians love to propose an insanely high number of conjectures, but I would not say proposing many conjectures is either a necessary or sufficient condition for an interesting branch of maths. (and I do like graph theory). | |
| Jan 21, 2012 at 7:21 | vote | accept | William DeMeo | ||
| Dec 13, 2010 at 0:01 | comment | added | William DeMeo | I'd still like to know of any paper(s) which starts by assuming the (Hilbert space) ISP has a positive/negative answer, and then proves some interesting consequence. | |
| Dec 10, 2010 at 23:46 | vote | accept | William DeMeo | ||
| Jan 21, 2012 at 7:19 | |||||
| Dec 10, 2010 at 23:46 | comment | added | William DeMeo | @Bill: Thank you for your reply. I was familiar with the Enflo and Read work, but not with Argyros-Haydon, so I appreciate learning about that. Your answers 2 & 3 get more to the point though. I was wondering about the "big" problem (linked to in my question): do all bounded linear operator on an infinite-dimensional separable Hilbert space have non-trivial invariant subspaces. If we know that the answer is yes, does it really change things? Given your answer 3, and gowers comment, I think the answer is "yes" and I (gratefully) accept your answer. (Further answers still welcome though.) | |
| Dec 10, 2010 at 20:53 | comment | added | gowers | Related to Bill's answer, perhaps one might say that operators on Hilbert spaces have done more than enough to prove their interest, and the invariant subspace problem shows that we still don't understand them. That's a bit vague, but by "understand" I mean something like, "have a good enough description of a general such operator to answer a question as basic as whether it must have an invariant subspace". I don't think that is quite as circular as it may seem. | |
| Dec 10, 2010 at 20:46 | history | edited | gowers | CC BY-SA 2.5 |
Corrected spelling of Charles Read's surname
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| Dec 10, 2010 at 18:47 | comment | added | Kevin Buzzard | @Gil: $ $ :-) | |
| Dec 10, 2010 at 16:52 | comment | added | Gil Kalai | Kevin, the problem with your comment is that BOTH the first, second and third answers you present can be regarded as good answers to the question "Why is the TPC less interesting than the ISP?" and not only, as you argue, to the question "Why is the TPC more interesting than the ISP?" | |
| Dec 10, 2010 at 16:45 | comment | added | Kevin Buzzard | ...for people working in the general area of distribution of primes. There are plenty of questions in this area---some technical to explain, some far too hard to even approach, some closely related to things like RH. But it's important in any area to have some clear, well-defined, hard, problems, which motivate people to work in the area and which motivate people to formulate other hypotheses which might be easier to work on. | |
| Dec 10, 2010 at 16:43 | comment | added | Kevin Buzzard | I can answer (4). Firstly it is much more elementary than the invariant subspace problem: one can easily explain it to a schoolchild. Secondly it is almost certainly true: indeed, far stronger conjectures have been made about the density of twin primes and computer calculations indicate that these stronger conjectures are probably true. Thirdly a lot of people have worked on it and it's still open. I have no idea who works on ISP but I know for sure that many clever people have thought a lot about twin primes. As a consequence of the last two bullet points, it provides a natural goal.. | |
| Dec 10, 2010 at 16:28 | history | answered | Bill Johnson | CC BY-SA 2.5 |