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.

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.