Timeline for On the limit set of eigenvalues of banded Toeplitz Hessenberg matrices
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| Aug 27, 2016 at 20:32 | comment | added | Twi | 2Per Alexandersson: Thanks for the interesting references. I read them. I tried to find a differential operator $L$ of fixed order and polynomial coefficients such that the characteristic polynomial $p_{n}$ of a $n\times n$ banded Toeplitz matrix satisfies the eigenvalue equation $Lp_{n}=\lambda_{n}p_{n}$ for all $n$ sufficiently large. However, except the tridiagonal matrix, where the polynomials $p_{n}$ are related to Chebyshev polynomials of the 2nd kind, I found nothing. Now, I doubt that such operator $L$ exists in general (but I have no proof for this claim). | |
| Jun 19, 2016 at 23:34 | comment | added | Per Alexandersson | Also, I suspect that this is connected to root measures obtained from linear differential operators, see e.g. diva-portal.org/smash/get/diva2:190169/FULLTEXT01.pdf (in particular, page 80 states these are trees), perhaps it is possible to show that one can construct a differential operator with same limit measure as the Toeplitz case? One reason to study Toeplitz matrices is in fact to better understand such differential operators, so it would not be surprising if there is a stronger connection... | |
| Jun 19, 2016 at 23:03 | comment | added | Per Alexandersson | I searched a bit, and I think you might get some more intuition from this article, using pseudo-spectra: people.maths.ox.ac.uk/trefethen/publication/PDF/1992_53.pdf | |
| Jun 19, 2016 at 21:54 | comment | added | Twi | I realize that I need not to mention the relation with Toeplitz matrices. The reason why I did that is: 1. I think it might be useful for a possible proof of the statement. 2. I made the pictures by plotting the eigenvalues of $T_{n}(b)$, with $n=300$. Maybe I could use another title. Of course, one can think about all the generalizations as you mentioned. Nevertheless, I would like to stay at this "simple case of $b$" (or your formulation, $P(z)=\lambda z$) for some time since I have no idea how to solve even this problem. | |
| Jun 19, 2016 at 16:10 | history | answered | Per Alexandersson | CC BY-SA 3.0 |