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Mar 29, 2021 at 17:39 comment added M.G. @Anixx: yup. You should probably ask it in a different question since it's a completely different beast.
Mar 29, 2021 at 17:38 comment added Anixx Okay, so the infinite-dimensional case remains
Mar 29, 2021 at 17:38 vote accept Anixx
Mar 29, 2021 at 17:36 vote accept Anixx
Mar 29, 2021 at 17:37
Mar 29, 2021 at 17:34 comment added M.G. A small roller-coaster for you :-) I missed originally that $W$ is actually a fixed constant matrix, in which case the link I posted previously does complete the finite-dimensional case via a small modification, since $x$ is just a scalar (see the 2nd edit).
Mar 29, 2021 at 17:32 history edited M.G. CC BY-SA 4.0
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Mar 29, 2021 at 17:18 comment added M.G. @Anixx: the first edit was incorrect, so I removed it. The link computes the matrix value at a point only. It is not a priori clear to me how the Jordan Normal Form depends on the matrix coefficients, e.g. continuously, smoothly, holomorphically, via elementary functions etc. in the general finite-dimensional case. But this has probably been already investigated somewhere. Nevertheless the link en.wikipedia.org/wiki/Analytic_function_of_a_matrix contains an explicit formula for the 2x2 case. Apologies about the mishap.
Mar 29, 2021 at 17:16 history edited M.G. CC BY-SA 4.0
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Mar 29, 2021 at 17:12 comment added M.G. @Anixx: no, it only completes my comment about the Jordan Normal Form in the finite-dimensional setting.
Mar 29, 2021 at 17:10 comment added Anixx Does your EDIT1 include the infinite-dimensional case?
Mar 29, 2021 at 17:04 history edited M.G. CC BY-SA 4.0
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Mar 29, 2021 at 16:52 comment added Anixx Just what I observe: there is such element $W$ that $\operatorname{reg}\log (W+x)=\psi(x)$, where $\operatorname{reg}$ is extracting the scalar part.
Mar 29, 2021 at 16:50 comment added M.G. The infinite-dimensional setting is certainly a very interesting case, but there are obviously many issues to be considered. For starters, since you care about the trace, you have to restrict to trace-class operators / nuclear operators. I cannot offer you any insight there at the moment.
Mar 29, 2021 at 16:44 comment added Anixx What about an infinite-dimensional case? I am bearing in mind a commutative algebra where it is not the case (although it is not strictly matrix algebra).
Mar 29, 2021 at 16:40 history answered M.G. CC BY-SA 4.0