Timeline for Sum of two unbounded self-adjoint operators
Current License: CC BY-SA 3.0
9 events
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| Aug 22, 2022 at 17:25 | comment | added | Christian Remling | @SMS: No, for an analogous criterion for $x\notin\sigma (A+B)$, you would need $E$ to be zero on a neighborhood of $\{ (s,t): s+t=x\}$. | |
| Aug 22, 2022 at 17:13 | comment | added | SMS | So then, we should also have $x + y \notin \sigma(A + B)$ if and only if $E(M) = 0$ for some set of the form $M = (x - \delta_1, x + \delta_1) \times (y - \delta_2, y + \delta_2)$. Do I understand this correctly? Thanks! | |
| Aug 22, 2022 at 13:50 | comment | added | Christian Remling | @SMS: I think there's a similar description here: $x\notin\sigma (A)$ if and only if $E(M)=0$ for some set of the form $M=(x-\delta,x+\delta)\times\mathbb R$. | |
| Aug 21, 2022 at 20:32 | comment | added | SMS | @ChristianRemling Is there any way to read the spectra of $A$ and $B$ from these two variable projection valued measures? In the single variable case, we know that Spec$(A) = $ supp$ E_\lambda$, where $A = \int \lambda dE_\lambda$. | |
| Apr 16, 2015 at 6:48 | comment | added | ifw | @amano $ikX$ is relatively $\Delta$ compact, in particular, it is relatively $\Delta$ bounded with relative bound $0$. Then see my answer below. | |
| Apr 16, 2015 at 0:09 | vote | accept | amano | ||
| Apr 15, 2015 at 19:03 | comment | added | Christian Remling | @amano: This definitely works because you can use the Fourier transform to (simultaneously) make multiplication operators out of both of them. (So you in fact don't need any general theory to make $A+B$ self-adjoint on a suitable domain.) | |
| Apr 15, 2015 at 18:56 | comment | added | amano | Very nice answer! Suppose we take the following example: $A = \Delta$, the Laplacian on the torus, and $B = ik\frac{\partial}{\partial x_1}$, where $k \in \mathbb{R}$. Clearly they commute. But I am not sure whether they strongly commute in the sense you describe. Could you please shed a little light on this point? Thanks! | |
| Apr 15, 2015 at 18:39 | history | answered | Christian Remling | CC BY-SA 3.0 |