This theorem is true or false in infinite dimensions?

Consider a vector space $E$ with finite dimension and linear map $A: E \rightarrow E$. The following statements are equivalent:

• $x'(t)=A \circ x(t)$ defines an attractor.

• All eigenvalues of $A$ have negative real part.

• There are $\alpha, \beta, c, C>0$ and $k\geq0$ such that$$c|t^k|e^{-\alpha t}\parallel v\parallel \leq \parallel e^{tA}(v)\parallel\leq Ce^{-\beta t}\parallel v \parallel, \ \ \ v\in E$$

And, if $x'(t)=A\circ x(t)$ defines an attractor, then $$\displaystyle \lim_{t\rightarrow - \infty}\parallel e^{tA}(v) \parallel=+\infty , \ \ \ \ \ \forall v\in E$$

I have the proof for finite dimension, but I use the Jordan canonical form theorem. Clearly, I can't use this argument in infinite dimension.

Consider a linear map $A$ and $\phi :\mathbb{R}\times E \rightarrow E$ the main flow of differential equation $x'(t)=A\circ x(t)$. This equation defines an attractor when, $$\displaystyle\lim_{t\rightarrow\infty}\phi(t,v)=0$$ $\forall v \in E,$ fixed.

• I think by "application" you mean "linear application", that is (in English) "linear map". Also, could you define "attractor"? – Lior Silberman Apr 21 '14 at 0:43
• Now, I define an attractor. Is it clear? @LiorSilberman – Henfe Apr 21 '14 at 1:13
• My question is in disagreement with something? – Henfe Apr 21 '14 at 2:18

As stated the Theorem is false, even for bounded self-adjoint operators on Hilbert space. Take an operator with eigenvalue $-1/k$ on the $k$th basis vector. All the spectrum has negative real parts, wich does imply $\lim_{t\to-\infty}\Vert e^{At} v \Vert = \infty$ for all (non-zero) vectors $v$, but the more precise bound above that doesn't hold -- there is no uniform exponential rate of decay.