Timeline for Principal symbol of a non-local operator and Atiyah–Singer index formula
Current License: CC BY-SA 4.0
11 events
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| Oct 10, 2019 at 14:51 | comment | added | Paul Siegel | (And indeed, it doesn't look to me like the Fourier transform of your operator is a polynomial, but nevertheless it looks like you ought to be able to construct a reasonable functional calculus for the operator. I still can't tell if it's Fredholm in the traditional sense, though.) | |
| Oct 10, 2019 at 14:49 | comment | added | Paul Siegel | The approach I'm referring to is explained in detail in chapter 10 of Higson and Roe's Analytic K-homology, culminating in theorem 10.6.5. This shows how to construct a K-homology class from an elliptic operator, and the key point is pseudolocality (for operators constructed via the functional calculus). I don't think this theory applies directly to your operator, but if you can imitate the analysis to produce a class in some sort of K-homology group then you should get an index theorem as a byproduct. | |
| Oct 10, 2019 at 13:47 | comment | added | IgnoranteX | I didn't know about the need to be pseudolocal to work out a classical index theory. Do you have any reference (book&chapter or article) about the topic? And also, would you agree with me that these operators (at least the one in the example) don't have any principal symbol? | |
| Oct 10, 2019 at 12:43 | comment | added | Paul Siegel | If they aren't Fredholm, then one typical way to proceed is to study the algebra generated by the commutators of your preferred class of operators with multiplication operators by continuous functions - your operators will have a Fredholm-like index in the K-theory of this algebra, and often the right way to generalize the Atiyah-Singer index theorem is to calculate this K-theory group. | |
| Oct 10, 2019 at 12:38 | comment | added | Paul Siegel | To make classical index theory work it isn't strictly necessary that the operators are local, just pseudolocal - meaning they should commute with multiplication by a continuous function modulo compact operators. It doesn't look like this quite holds for the operators that you introduced, which makes me wonder if these operator are even Fredholm - do you know one way or the other? | |
| Oct 10, 2019 at 12:09 | history | edited | IgnoranteX | CC BY-SA 4.0 |
added 105 characters in body
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| Oct 10, 2019 at 12:04 | comment | added | IgnoranteX | You are right. By "Hörmander symbol" I mean that the symbol is "in the class $S_{1,0}^m$ of Hörmander", using the wording of Wikipedia. I hope there is no big ambiguity in this definition. | |
| Oct 9, 2019 at 21:22 | comment | added | LSpice | The Wikipedia page refers to Hörmander, and uses the word 'symbol' in the discussion of linear differential operators with constant coefficients, but seems never explicitly to mention "Hörmander symbol". | |
| Oct 9, 2019 at 21:21 | history | edited | LSpice | CC BY-SA 4.0 |
Minor proofreading
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| Oct 9, 2019 at 21:10 | review | First posts | |||
| Oct 9, 2019 at 22:17 | |||||
| Oct 9, 2019 at 21:05 | history | asked | IgnoranteX | CC BY-SA 4.0 |