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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]