Timeline for Spectral representation of closed operators with finite spectral bound
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| May 14, 2020 at 19:42 | history | edited | Jacob Lu | CC BY-SA 4.0 |
added 16 characters in body
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| May 13, 2020 at 20:12 | vote | accept | Jacob Lu | ||
| May 13, 2020 at 13:02 | comment | added | Giorgio Metafune | You are right. I just considered norm convergence. | |
| May 13, 2020 at 12:36 | answer | added | Jochen Glueck | timeline score: 4 | |
| May 13, 2020 at 12:10 | comment | added | Jochen Glueck | Alright, I've done the calculation in the one-dimensional case. First note that there's the factor $\frac{1}{2\pi i}$ missing in front of the integral. But this little issue aside, we can use integration by parts and Cauchy's integral formula to see that the formula does indeed hold in the one-dimensional case (and thus also for diagonalizable matrices). I'd suspect that the same works for bounded generators $A$, but I have not checked this in detail. For analytic semigroups, things seem to be more involved since we cannot simply add a large circle on the left of the imaginary axis. | |
| May 13, 2020 at 11:46 | comment | added | Jochen Glueck | @GiorgioMetafune: Well, since the exponential function oscillates along the imaginary axis, there might be a chance for this to converge as an improper integral. I wouldn't be too surprised if it holds, say, in finite dimensions (or even for analytic semigroups). But there are a lot of details to check... | |
| May 13, 2020 at 8:10 | comment | added | Giorgio Metafune | I guess there is no way since the resolvent cannot decay faster than $1/|z|$. The above integral does not converge even for multiplication operators. | |
| May 13, 2020 at 1:07 | comment | added | Jacob Lu | Yes, you are right, the integral cannot be absolutely convergent. My question is whether the integral is well-defined (convergent in the usual functional calculus sense) and equals the semigroup generated by $A$. | |
| May 12, 2020 at 9:02 | comment | added | Jochen Glueck | In which sense would you expect the integral to converge? (Absolute convergence does certainly not hold.) | |
| May 12, 2020 at 4:35 | history | asked | Jacob Lu | CC BY-SA 4.0 |