Timeline for Self-adjoint operator with pure point spectrum
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 21, 2023 at 19:10 | comment | added | Christian Remling | @M.González: In that case $\sigma_{ac}=\sigma_{cont}=\sigma_{ess}=\sigma=[-1,1]$, $\sigma_{pp}=\{ 0\}$. | |
| Dec 21, 2023 at 16:29 | comment | added | M.González | @Christian Remling: If $H$ is the orthogonal sum $L_2(-1,1)\oplus \mathcal{C}$ and $A(f(t),z)= (tf(t),0)$, which is the continuous spectrum of $A$? | |
| Dec 21, 2023 at 15:41 | comment | added | Christian Remling | @M.Gonzalez: Since $R(A-t)^{\perp}=N(A-t)$ for $t\in\mathbb R$, this last condition repeats the first one, so this set is $\sigma(A)\setminus\sigma_p(A)$ (with $\sigma_p$ denoting the eigenvalues), and it's not even a spectrum (closed) in general. | |
| Dec 21, 2023 at 10:28 | comment | added | M.González | @Christian Remling: For me, $z$ in the continuous spectrum of $A$ means $A-zI$ injective with proper dense range. | |
| Dec 20, 2023 at 20:40 | review | Close votes | |||
| Jan 4 at 3:12 | |||||
| Dec 20, 2023 at 20:17 | comment | added | Christian Remling | @M.González: I don't think this is what the OP means by "continuous spectrum" (a vector is in the subspace corresponding to continuous spectrum if its spectral measure is a continuous measure on $\mathbb R$ (= has no atoms)). What you call continuous spectrum is more commonly referred to as "essential spectrum". | |
| Dec 20, 2023 at 20:15 | comment | added | Christian Remling | Yes, this is true. I think your question would have been better suited for math.stackexchange.com , though. | |
| Dec 20, 2023 at 19:26 | comment | added | M.González | The operator $A:\ell_2\to\ell_2$ defined by $A(x_n)=(x_n/n)$ is compact, self-adjoint, has an orthonormal basis of eigenvectors (the unit vector basis $(e_i)$), but $0$ is in the continuous spectrum of $A$. | |
| Dec 20, 2023 at 11:41 | history | edited | gmvh |
Added top-level tags
|
|
| S Dec 20, 2023 at 10:28 | review | First questions | |||
| Dec 20, 2023 at 11:42 | |||||
| S Dec 20, 2023 at 10:28 | history | asked | user3476591 | CC BY-SA 4.0 |