Suppose that $(T(t))_{t\geq0}$ is a $C_0$semigroup on a Banach space $X$ and assume that there exists $x\in X$ such that $\{T(t)x:\ t\geq0\}$ is dense in $X$. I wonder why the set $\{T(t)x:\ t\geq t_0\}$ is dense for every $t_0>0$. I do not see the reason why this should be true. I tried this with a sequence argument, but it doesn't work. Is there any idea? Maybe by contradiction?
2 Answers
Here is a simple solution which has nothing to do with orbits of semigroups but uses only a simple fact from general topology: If $M$ is dense in a topological space $X$ and $A$ is nowhere dense (that is, $\overline A$ does not have interior points) then $M\setminus A$ remains dense (in fact, if $U$ is any nonempty open set then $U\setminus \overline A$ is open and nonempty since otherwise $U\subseteq \overline A$. Hence $M\cap (U\setminus \overline A)$ is not empty which implies that $U\cap (M\setminus A)$ is not empty.)
To apply this, let $M$ be the dense orbit and $A=\lbrace T(t)x: 0\le t\le t_0\rbrace$ and observe that $A$ is compact (due to the continuity of $t\mapsto T(t)x$) and hence does not have interior points if $X$ is an infinitedimensional Banach space (I guess that you are not interested in finitedimensional spaces because then there aren't any hypercyclic semigroups).

$\begingroup$ Thats a nice option to solve it, but I am looking foward to prove this with sequences? Any suggestion? $\endgroup$ Jan 12, 2017 at 13:14
If $\{T(t)x: t\ge 0\}$ is dense in $X$, then for any $\delta$, there is a $t>t_0$ with $\T(t)xx\<\delta$. Now pick any $y\in X$. There exists $t_1$ with $\T(t_1)xy\<\epsilon/2$. Moreover, there exists $\delta$ such that $\T(t_1)xT(t_1)z\<\epsilon/2$ if $\xz\<\delta$. Now pick $t_2>t_0$ such that $\T(t_2)xx\<\delta$. It follows that $\T(t_1+t_2)xT(t_1)x\<\epsilon/2$, hence $\T(t_1+t_2)xy\<\epsilon$.

$\begingroup$ Thank you very much for this answer. Can this also be wrote down in the language of approximating sequences? $\endgroup$ Jan 12, 2017 at 12:46

$\begingroup$ Why should exists such an $t_2>t_0$? $\endgroup$ Jan 12, 2017 at 12:56

$\begingroup$ Because $\{T(t)x: t\in [0,t_0]\}$ is compact and cannot be dense in a neighborhood of $x$. Therefore the orbit must visit any neighborhood of $x$ at times after $t_0$. $\endgroup$ Jan 13, 2017 at 5:17