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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.

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    $\begingroup$ The question is not really meaningful for a semigroup when you can only consider $t\ge 0$, and of course $\inf_{t\ge 0} \|x(t)\|=0$ wouldn't be unexpected. $\endgroup$
    – Christian Remling
    Commented May 7, 2014 at 15:46
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    $\begingroup$ Have you tried to reproduce the proof for finite dimesional spaces in the case of a general Banach space ? The arguments may differ but you should either be able to adapt it or to pinpoint crucial steps that may help you find a counter-example. $\endgroup$
    – Loïc Teyssier
    Commented May 7, 2014 at 18:34
  • $\begingroup$ The two answers to this question: mathoverflow.net/q/164182/12898 will give you full and detailed answer. $\endgroup$
    – András Bátkai
    Commented May 7, 2014 at 19:07
  • $\begingroup$ @Andr My problem is I want a $C_0$ group $T(t)$ or a function $e^{tA}$ and a vector $x \in X$ such that $T(t)x$ is bounded on $\mathbb{R}$, nontrivial and $\inf_{t \in \mathbb{R} }|T(t)x|=0$. It seems like I cannot modify the nilpotent translation semigroup you gave to make it a group. I am interest in the solutions on the whole real line, since $\inf_{t \geq 0 }|T(t)x|=0$ is of course simply verified by exponentially stable orbits. Thanks ! $\endgroup$
    – user144542
    Commented May 8, 2014 at 0:26
  • $\begingroup$ My first comment was on the semigroup case, where you can have many weird things happening. For bounded $A$ you can modify the finite dimensional case. I will return to your question of the $C_0$-group case soon. $\endgroup$
    – András Bátkai
    Commented May 8, 2014 at 7:22

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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 $$ \lim_{t\to\pm\infty} \|T(t)x\| = 0.$$

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