Timeline for Spectral Radius and Spectral Norm for Markov Operators
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Mar 17, 2023 at 17:29 | vote | accept | Sam OT | ||
| Mar 16, 2023 at 10:47 | history | edited | Sam OT | CC BY-SA 4.0 |
Added/clarified details
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| Mar 16, 2023 at 10:31 | comment | added | Sam OT | Good to know that $\rho(P) = \rho(P^\star)$, though. I knew that $\| P \| = \| P^\star \|$, but I actually thought that it wasn't true for $\rho$. Thanks! | |
| Mar 16, 2023 at 10:30 | comment | added | Sam OT | To add to (iv), I thought one typically looked at $P$ on $L^2$ restricted to non-constant functions when looking at spectral gaps. Indeed, $P$ always has constant functions as eigenfunctions with eigenvalue $1$, so $\rho(P) = 1$ if you allow constant functions. || I'll add the above to the body of the question, as well as make one other correction: $\Omega$ should be $\{ x \in \mathbb R^n \mid \sum_i x_i = 0 \}$. But, I guess that doesn't really change anything. | |
| Mar 16, 2023 at 10:28 | comment | added | Sam OT | @JochenGlueck Certainly! (i) $P$ is the transition matrix of a Markov chain on $\Omega = \mathbb R^n$, so $P(x,y)$ is probability of being at $y \in \mathbb R^n$ after one step started at $x$; I thought people call these "operators" when $\Omega$ is continuous, not discrete, but maybe I was mistaken... (ii) Irreducibility here means that one can hit any positive-volume set. (iii) That is given near the end of the question; see the three bullets. (iv) I do not know what that means, sorry... 😬 (v) This is a fairly standard abuse, in Markov chains at least: $(Pf)(x) := \int_\Omega f(y) P(x,dy)$. | |
| Mar 15, 2023 at 10:19 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Mar 15, 2023 at 9:29 | answer | added | DRJ | timeline score: 2 | |
| Mar 14, 2023 at 19:00 | comment | added | Jochen Glueck | By the way, one actually has $\rho(P) = \rho(P^*)$ for every bounded linear operator $P$ on a Hilbert space. | |
| Mar 14, 2023 at 18:59 | comment | added | Jochen Glueck | Could you please clarify a few points in your post? (i) What do you mean by a Markov operator on the state space $\Omega = \mathbb{R}^n$? In particular, on which space does the operator act a priori? (ii) Which definition of irreducibility do you use? (iii) What do you mean by 'discretely supported'? (iv) Why is $B$ invariant under the adjoint $P^*$? (v) You seem to refer to three different objects by $P$: the Markov operator with which you start; its restriction to $L^2(\Omega,\pi)$; and, in the three bullet points, to somekind of integral kernel of $P$. Could you clarify the notation? | |
| Mar 14, 2023 at 15:17 | history | asked | Sam OT | CC BY-SA 4.0 |