spectral theory: prove equivalence of "A is essentially maximally accretive" and...

[mona123]

spectral theory: prove equivalence of "A is essentially maximally accretive" and...

hi i want to prove the proposition in page 9 here :http://cermics.enpc.fr/~stoltz/MATHERIALS/nier1.pdf
this is what i tried to do for 2)imply 3)
we proceede by contraposition. let us suppose that for any [FONT=MathJax_Math-italic]a[FONT=MathJax_Main]>[/FONT][FONT=MathJax_Main]0[/FONT][/FONT] we have [FONT=MathJax_Main]([FONT=MathJax_Math-italic]A[/FONT][FONT=MathJax_Main]∗[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Math-italic]a[/FONT][FONT=MathJax_Math-italic]I[/FONT][FONT=MathJax_Math-italic]d[/FONT][FONT=MathJax_Main])[/FONT][/FONT] is not injective hence there exists [FONT=MathJax_Math-italic]u[FONT=MathJax_Main]∈[/FONT][FONT=MathJax_Math-italic]D[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]A[/FONT][FONT=MathJax_Main]∗[/FONT][FONT=MathJax_Main])[/FONT][/FONT] such that [FONT=MathJax_Math-italic]A[FONT=MathJax_Main]∗[/FONT][FONT=MathJax_Math-italic]u[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]a[/FONT][FONT=MathJax_Math-italic]u[/FONT][/FONT] hence [FONT=MathJax_Main]<[FONT=MathJax_Math-italic]A[/FONT][FONT=MathJax_Main]∗[/FONT][FONT=MathJax_Math-italic]u[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]u[/FONT][FONT=MathJax_Main]>=[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]a[/FONT][FONT=MathJax_Main]|[/FONT][FONT=MathJax_Main]|[/FONT][FONT=MathJax_Math-italic]u[/FONT][FONT=MathJax_Main]|[/FONT][FONT=MathJax_Main]|[/FONT][FONT=MathJax_Main]2[/FONT][/FONT] or [FONT=MathJax_Main]<[FONT=MathJax_Math-italic]A[/FONT][FONT=MathJax_Main]∗[/FONT][FONT=MathJax_Math-italic]u[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]u[/FONT][FONT=MathJax_Main]>=<[/FONT][FONT=MathJax_Math-italic]B[/FONT][FONT=MathJax_Main]∗[/FONT][FONT=MathJax_Math-italic]u[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]u[/FONT][FONT=MathJax_Main]>[/FONT][/FONT] but I could not conclude.
can someone please help me .Thanks

 

[stapel]

hi i want to prove the proposition in page 9 here :http://cermics.enpc.fr/~stoltz/MATHERIALS/nier1.pdf

Do you mean the following?

Proposition 3.3.
Let V be a \(\displaystyle \, C^{\infty}\, \) potential on \(\displaystyle \, \mathbb{R}^d,\, \) then the Kramers-Fokker-Planck operator defined on \(\displaystyle \, C_0^{\infty} \left(\mathbb{R}^{2d}\right)\, \) defined by

. . . . .\(\displaystyle K\, :=\, \gamma_0\, \left(-\Delta_v\, +\, \dfrac{1}{4}\, |v|^2\, -\, \dfrac{d}{2}\right)\, +\, X_0,\). . . . .\(\displaystyle (3.1)\)

where

. . . . .\(\displaystyle X_0\, :=\, v\, \cdot\, \delta_x\, -\, \nabla V(x)\, \cdot\, \delta_v\). . . . . . . . . . . . . .\(\displaystyle (3.2)\)

is essentially maximally accretive as soon as \(\displaystyle \, \gamma_0\, >\, 0.\)

If not, which part of the document did you mean? If so, then what do you mean by the rest of your post? For instance:

this is what i tried to do for 2)imply 3)

...what are the "2" and the "3"?

Please be specific. Thank you!

 

[mona123]

Hi
I mean :
The equivalence of the next statements can easily be checked :
1. A is essentially maximally accretive.
2. A is maximally accretive.
3. There exists λ0 > 0 such that A∗ + λ0I is injective.
4. There exists λ1 > 0 such that the range of A + λ1I is dense in H.
If you can help me ,thanks in advance