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.