An incorrect proof has felled a math paper. There’s not too much to say in a straightforward situation like this one, which we’ve seen before — the result of honest errors, not any malfeasance.
Here’s the abstract for “Spectral mapping theorem for generalized Kato spectrum:”
In this paper, we give an affirmative answer to Mbekhta’s conjecture (Mbekhta, 1990) about the pseudo Fredholm operators in Hilbert space. As a consequence, we characterize pseudo Fredholm operators and we prove that the generalized Kato spectrum satisfies the spectral mapping theorem in the Hilbert spaces setting.
The paper — published in the Journal of Mathematical Analysis and Applications — has been cited twice, according to Thomson Scientific’s Web of Knowledge.
Here’s the retraction note:
This article has been retracted at the request of the Editors-in-Chief because the proof of Theorem 2.2 (and therefore all its implications) is incorrect. It is important to note that this is not considered to be the result of scientific misconduct, but rather honest errors by the authors.
First author Mohamed Amouch, a mathematician at Université Chouaib Doukkali in Morocco, sent us a note from all three authors, which told us they’re still working on the problem:
Unfortunately, the problem is in the proof of theorem 2.2 with the construction of the decomposition of the Hilbert space into two sub-spaces which are invariant under the operator. Hence all its implications are not proved. The Theorem 2.2 remains open and we are still trying to reserve a possible correction. Also, we try to prove spectral mapping theorem for generalized Kato spectrum without using Theorem 2.2.
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This is the best possible reason for a retraction—kudos to the authors! We talked about the issue in journal reading group a few weeks ago. Graduate students everywhere need to be taught about the tradition of paper retraction; whilst fairly common in biomedical sciences, it’s almost unheard of in computer science (maybe related to the lament computer scientists reinvent the wheel every twenty-five years).