Let $X$ be a Banach space. $A\in B(X)$ is a bounded operator. we can define $e^{tA}$ by

$$e^{tA}=\sum_{k=0}^{+\infty}\frac{t^kA^k}{k!}$$ I am interested in this property:

If $x\in X$, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$, then we have necessarily $$\inf_{t\in \mathbb{R}}|e^{tA}x|>0 \ \ \ \ \ or \ \ \ \ e^{tA}x=0 \ \ (i.e. \ \ x=0).$$

This property is clear in the scalar case $A=a\in \mathbb{C}$. Because $t\mapsto e^{ta}x$ is bounded on $\mathbb{R}$ if and only if $Re(a)=0$, and then $\inf_{t\in \mathbb{R}}|e^{ta}x|=|x|$.

This property is also true if $X$ is finite dimensional i.e if $A$ is a matrix, and was answered here in math stack exchange.

So my question is: "**Does this property hold if $X$ is infinite dimensional ?** "