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)

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)