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nLab > Latest Changes: Fredholm operator

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 2nd 2013

    I have edited a bit at Fredholm operator. Also started a stubby Fredholm module in the process. But it remains very much unfinished. Have to interrupt now for a bit.

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeApr 2nd 2013
    • (edited Apr 2nd 2013)

    While still short, Fredholm operator is a very good entry!!! (well, I did a part in it, but the improvements are both essential and good and wellcome!)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 11th 2022

    a couple of recent reference items to add to Fredholm operator once the edit functionality is back:

    • Nikolai V. Ivanov, Topological categories related to Fredholm operators: I. Classifying spaces (arXiv:2111.14313)

    • Nikolai V. Ivanov, Topological categories related to Fredholm operators: II. The analytic index (arXiv:2111.15081)

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 24th 2022
    • (edited Apr 4th 2022)

    The entry Fredholm operator is missing any reference for Atkinson’s theorem (i.e. that the definition of Fredholm operators via finite dim ker and coker is equivalent to invertibility up to compact and/or finite-rank operators). This here just to record some.

    The original article is:

    • F. V. Atkinson, The normal solubility of linear equations in normed spaces, Mat. Sb. (N.S.), 28(70):1 (1951), 3–14 (mathnet:msb5589)

    Early streamlined proof is in:

    • Michael Atiyah, Prop. A8 (p. 163) in: K-theory, Harvard Lecture 1964 (notes by D. W. Anderson), Benjamin 1967 (pdf)

    • Gerard J. Murphy, Theorem 2.1 in: Fredholm Index Theory and the Trace, Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences 94A 2 (1994), 161-166 (jstor:20489482 )

    See also

    • A. G. Ramm, A simple proof of the Fredholm alternative and a characterization of the Fredholm operators (arXiv:math/0011133)

    Monographs/texbook accounts include:

    • Bernhelm Booß, p. 35-36 in: Topologie und Analysis, Springer (1977) (doi:10.1007/978-3-642-66752-7)

    • Nigel Higson, John Roe, 2.1.4 in: Analytic K-Homology, Oxford mathematical monographs, Oxford University Press (2000) (ISBN:9780198511762)

    • William Arveson, Theorem 3.3.2 in: A Short Course on Spectral Theory, Graduate Texts in Mathematics, 209, Springer (2002) (doi:10.1007/b97227)

    • CommentRowNumber5.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 25th 2022
    • (edited Mar 25th 2022)

    Also need to add these references:

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 9th 2022
    • (edited May 9th 2022)

    Added pointer to

    and:

    diff, v23, current

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 12th 2023

    Added a reference (would it not be better to split off the material on Fredholm complexes into a separate article?):

    For Fredholm complexes, see

    • Graeme Segal, Fredholm complexes, Quarterly Journal of Mathematics 21:4 (1970), 385–402. doi.

    diff, v28, current

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