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The invariant subspace problem is still a problem

Recently we reported that Eva Gallarda and Carl Cowen had announced they had a proof of the invariant subspace conjecture for Hilbert spaces.

Well, yesterday they announced at the blog Café Matemático that there was a problem with their proof:

On 10 December 2012, we submitted a paper “Rota’s Universal Operators and Invariant Subspaces in Hilbert Spaces” for publication, and we spoke about it several times before the more formal announcement at the RSME meeting in Santiago de Compostela on 25 January 2013. By that time, the paper had been read and no problems found by several other mathematicians. We have heard nothing so far from the journal to which it was submitted.
We regret to inform you, however, that a gap in our proof was discovered after the announcement at Santiago. After working for the past few days to bridge the gap, so far unsuccessfully, we are today formally withdrawing our submission to the journal.

We will, of course, continue to work to bridge the gap. At this point, Carl plans to contribute a talk to the Southeast Analysis Meeting (SEAM) to be held at Blacksburg, Virginia on 15, 16 March 2013 with the title as above.

So far at least, there have been no errors found in the paper besides the erroneous assertion that the work included in the paper proved the Invariant Subspace Theorem, while in fact it did not. For this reason, we plan to submit a paper by mid-March, either a paper that claims to prove the Invariant Subspace Theorem if we can bridge the gap or a paper substantially the same as the paper submitted earlier, but without claims beyond what we have actually proved correct. In the latter case, the manuscript will be made available to those interested about that time. If we believe we have proved the result, no submission, no announcement, and no manuscript will be made available until after the new manuscript has been reviewed by several mathematicians.

So that’s a bit of a bummer, but it doesn’t sound like they’re left with nothing.

We’ll keep you updated if any more info appears.

via Julia Collins (aka @haggismaths) on Twitter

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