Timeline for Perron-Frobenius and Markov chains on countable state space
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 5, 2021 at 19:46 | vote | accept | Landauer | ||
| Apr 5, 2021 at 19:21 | comment | added | Jochen Glueck | @Martinique: That's a very good question. I've been wondering about this for several years... :-) (Which doesn't necessarily mean that it's difficult, though; maybe I'm just missing something more or less obvious.) On a more applied note, though, one might remark that such a situation will be only rarely encountered in practice: every general theorem I am aware of which shows that certain spectral values are isolated, in fact also implies that these spectral values are poles. | |
| Apr 5, 2021 at 19:15 | comment | added | Landauer | Thanks, I will have a closer look at the appendix. Regarding my first comment: not assuming point spectrum or a pole of the resolvent, can an operator as described in the question have isolated spectrum on the unit circle that is no point spectrum? | |
| Apr 5, 2021 at 19:10 | comment | added | Jochen Glueck | @Martinique: Now my answer to your first comment: Poles of the resolvent are always eigenvalues (see for instance Proposition A.3.2(a)). (Or did I misunderstand your first comment, and you wanted to know something else?) | |
| Apr 5, 2021 at 19:06 | comment | added | Jochen Glueck | @Martinique: Let my first answer your second comment: The results in Appendix A of the thesis are all formulated for operators on infinite-dimensional spaces. Matrices do not occur in Appendix A (except for Remark A.2.2). The existence of the rank-$2$ generalized eigenvector is the content of Proposition A.2.3(b). The argument from the post above is then given in Proposition A.2.4(a). The relation to poles of the resolvent and to the associated spectral projection is given in Proposition A.3.2(d). | |
| Apr 5, 2021 at 18:20 | comment | added | Landauer | I also wonder why I should be able to find this rank $2$ generalized eigenvector actually. Is there a theorem that ensure this?-I am asking because in your thesis you state a lot of things for matrices which have a nice finite Jordan decomposition, but here with the operator? | |
| Apr 5, 2021 at 17:47 | vote | accept | Landauer | ||
| Apr 5, 2021 at 18:15 | |||||
| Apr 5, 2021 at 17:28 | comment | added | Landauer | thanks a lot. Btw. do you know if an isolated point in the spectrum on the boundary has to be point spectrum? | |
| Apr 5, 2021 at 17:23 | history | answered | Jochen Glueck | CC BY-SA 4.0 |