Let $X$ be a Banach space. We consider the evolution equation:
$$x'(t)=Ax(t), \ \ \ \ \ \ \ t\in \mathbb{R},$$


where $A$ is a bounded operator.


I know that if $X=\mathbb{R^n}$ and $A$ is a matrix, then every bounded nontrivial solution $x(t)$ on $\mathbb{R}$ satisfies $$\inf\_{t\in \mathbb{R}}|x(t)|>0.$$ 
I don't know if this property holds in the case of infinite dimensions where $A$ is a bounded operator, or even for an unbounded operator $A$ generating a $C\_0$-semigroup/group. 



A property of one-parameter groups of operators
In the case of a general $C\_0$-group the assertion is false. Consider the weight $w(s):=\tfrac{1}{1+s^2}, s\in\mathbb{R}$ and the Banach space consisting of all functions $x:\mathbb{R}\to\mathbb{R}$ such that $wx$ is uniformly continuous and the norm $\|x\|:=\sup\_{s\in\mathbb{R}} |w(s)x(s)|$ is finite. Let $T(\cdot)$ be the strongly continous group of left-shifts on this space, i.e. $T(t)x := x(\cdot+t), t\in\mathbb{R}$.


Now if $0\neq x$ is a smooth function with compact support, then $T(\cdot)x$ is a bounded solution of the corresponding evolution equation and it holds
$$ \lim\_{t\to\pm\infty} \|T(t)x\| = 0.$$


