Talk:Invariant subspace problem

Problem vs. conjecture

Why is calling it a conjecture "optimistic"? 86.0.206.49 (talk) 06:28, 21 December 2008 (UTC)

The use of the term "conjecture" usually indicates some confidence that the result is true, despite the lack of a proof. If there is real doubt about the validity, many mathematicians prefer to use the terms "problem" or "question", believing it to be bad form to be in danger of having their "conjecture" proved wrong. (Exceptions abound to this informal rule of usage, however.) -- Spireguy (talk) 03:32, 22 December 2008 (UTC)

Is "complex" Hilbert space necessary?

The article describes the invariant subspace problem as pertaining to complex Hilbert spaces. But isn't the existence of a non-trivial closed invariant subspace equally unknown for a bounded linear operator on a real Hilbert space? (In fact, I wonder if the two problems might be equivalent.)Daqu (talk) 07:18, 20 June 2008 (UTC)

Okay, I see that one would need to require the real dimension to be > 2, since otherwise a rotation in the plane has no non-trivial invariant subspace. Which is like in the complex case, where the complex dimension is required to be > 1. But other than that?Daqu (talk) 18:02, 20 June 2008 (UTC)

. — Preceding unsigned comment added by 213.89.219.79 (talk) 20:35, 29 February 2012 (UTC)

after having seen

Perhaps it could be mentioned that Halmos gave a proof in the same issue of the same journal, after having seen a preprint of Robinson's proof using NSA. This is a well-known fact. It anyone doubts this I could try to look up some references. Katzmik (talk) 11:10, 15 December 2008 (UTC)

Why is this important

Without indulging in OR maybe one could think of something to say about why this problem is of any interest. To a mathematician, the importance of this is almost obvious, or at least is part of the folklore of reductionism, but without specific references to the literature I don't think we are allowed to say much. However, I'd be very surprised if Halmos hasn't written something expository on this.--CSTAR (talk) 19:17, 23 December 2008 (UTC)

Halmos' expository article in Amer Math Monthly was recently deleted from the bibliography here but revived at Criticism of non-standard analysis. Katzmik (talk) 19:20, 23 December 2008 (UTC)

New unsourced material

I removed the following recent addition to the article:

In 2009, Spiros Argyros (National Technical University of Athens) and Richard Haydon (Oxford University) constructed a script-L infinity Banach space on which every operators is a compact perturbation of a multiple of the identity. It follows that every operator on this space has a non-trivial invariant subspace. This is the first known example of a Banach space with this property. Other examples have since been constructed by Spiros Argyros.

This needs a citation to a reliable source. Also, it isn't quite clear: I assume "this property" means "every operator is a compact perturbation of a multiple of the identity", not "every operator on this space has a non-trivial invariant subspace", which is true for e.g. nonseparable Hilbert spaces.

The second property (existence of a non trivial invariant subspace) is consequence of the first, by an old theorem by Aronszajn. I have heard about this result by Argyros and Haydon, but clearly it is not yet published, and maybe one should wait some time before including it. Or one could say that "it was announced by these authors", which is a true verifiable fact. To comment what is said below, this is far more than "interesting", if true. --Bdmy (talk) 23:22, 5 March 2009 (UTC)

The result seems interesting, but as I noted, it needs to be sourced properly; others may have a different opinion as to whether it should be included if it does get a citation. -- Spireguy (talk) 23:07, 5 March 2009 (UTC)

I looked at Tim Gowers' blog for context, and that did make it clear that this is an important result. But I still think it should wait for a published reference. -- Spireguy (talk) 02:20, 6 March 2009 (UTC)

Trojtsky on Read's Operator and Lomonosov's theorem

My spelling is wrong, but can somebody describe the result that of Abramovich's student (whose spelling is difficult for me) on the 4th cycle of Lomonosovia? —Preceding unsigned comment added by Kiefer.Wolfowitz (talk • contribs) 23:18, 27 May 2009 (UTC)

Unclear wording

One paragraph of the article reads:

"To find a "counterexample" to the invariant subspace problem, means to answer affirmatively the following equivalent question: does there exist a bounded linear operator T : H → H such that for every non-zero vector x, the vector space generated by the sequence {Tⁿ(x) : n ≥ 0} is norm dense in H? Such an operator is sometimes called transitive."

Here the phrase "Such an operator" is unclear as to whether it refers to the previous sentence after the word "a", or only after the phrase "such that". It would seem that it must intend the latter. But there is no reason for the wording to be unclear.Daqu (talk) 05:32, 26 September 2010 (UTC)

Origins

It would be very useful for me to find a couple of words on the origins of the problem: when was it formulated and by whom? If it is not known, something like "the problem was already known to X". F121645100408832000 (talk) 10:11, 14 October 2010 (UTC)

Haagerup's contribution

I wonder if it would make sense to add information on the contribution of Haagerup and Schulz on the relative version of the problem in II_1 factors. F121645100408832000 (talk) 10:11, 14 October 2010 (UTC)

Hilbert space case solved?

It is rumoured that Carl Cowen (Purdue) and Eva Gallardo-Gutiérrez (Madrid) have solved the Hilbert space case. The only evidence, so far, is this picture. We'll just have to wait for a preprint. Hanche (talk) 06:53, 28 January 2013 (UTC)

Ah, but there was a gap. Hanche (talk) 14:20, 17 February 2013 (UTC)

Second Bullet of "Known Special Cases" Needs Citation

While the fact is obvious to experts in the field, talk of even any topological vector spaces that are not separable is extremely rare in the literature. I feel like the author had a precise source or straightforward inspired origin for the statement. The source making the remark or the inspiration for it should be cited accordingly. MMmpds (talk) 02:28, 13 April 2019 (UTC)

Solved for Hilbert spaces

Per Enflo just published a proof, should this be added? — SourceIsOpen (talk) 13:16, 28 May 2023 (UTC)