Timeline for Equality between operators, on dense subspace, from a quadratic form point of view
Current License: CC BY-SA 4.0
11 events
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| Sep 19 at 8:13 | history | edited | gmvh | CC BY-SA 4.0 |
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| May 30 at 18:15 | comment | added | MathMath | @ChristianRemling Yes, I understand now. Many thanks for pointing that out! | |
| May 30 at 17:27 | comment | added | Christian Remling | Yes, $-\Delta$ on $H^2$ is self-adjoint and we have the functional calculus available to define $f(-\Delta)$, but that's not what you did above, when you give $-\Delta$ the smaller domain $C^{\infty}$. | |
| May 30 at 15:54 | comment | added | MathMath | @ChristianRemling I got confused. The Laplacian is supposed to be self-adjoint, no? Maybe I should be a bit more careful about the dense domain: I was using $C^{\infty}(\Omega)$ not as the domain of $-\Delta$, but I was restricting the attention to the latter to (try to) avoid complications. But, the domain of $-\Delta$ should be the Sobolev space $D(-\Delta) = H^{2}(\Omega)$ for instance. Since $f$ is bounded and continuous, I should be able to define $f(-\Delta)$. Am I doing something wrong? | |
| May 30 at 15:23 | comment | added | Christian Remling | You actually can not use the usual functional calculus to define $f(-\Delta)$ since $-\Delta$ is not self-adjoint of $C^{\infty}$. You could do it by hand, by setting $f(-\Delta)\psi_p = f(\lambda_p)\psi_p$, with $-\Delta\psi_p=\lambda_p\psi_p$. Then, as you noted yourself at the end of your post, (1) with these vectors immediately implies that $A$ is the same operator. | |
| May 30 at 14:06 | comment | added | MathMath | I edited the post! | |
| May 30 at 14:06 | history | edited | MathMath | CC BY-SA 4.0 |
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| May 30 at 14:05 | comment | added | MathMath | @LSpice yes, just realized that. But I can replace it by $C^{\infty}(\Omega)$ instead and the question remains. | |
| May 30 at 14:01 | comment | added | LSpice | What does "rapidly decreasing" mean for a function on a compact set? | |
| May 30 at 12:26 | history | edited | MathMath | CC BY-SA 4.0 |
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| May 30 at 7:26 | history | asked | MathMath | CC BY-SA 4.0 |