Cfrgtkky

I'm pretty new to this area of study so if there are logical lacune in my proof (I'm sure there are many) please let me know.

This is material I'm self studying. I'm trying to adapt the methods used to show that $ell_{p}$ has hypercyclic operators given here on p.40 Example 2.22.

Let $a>0$ and $|lambda|>1$, on the space $X=L_p(0,infty)$ endowed with the $p$-norm.

The operator

$T: Xto Xquad text{where}quad (Tf)(x)=lambda f(x+a)$

is hypercyclic.

Proof. Let $U$ and $V$ be nonempty open subsets of $L_p(0,infty)$. Since continuous functions with compact support are dense in $L_p(0,infty)$ there exists functions $g(x)in U$ and $N_1in mathbb{N}$ so that $g(x)=0$ if $xnotin [0,N_1]$. Similarly we can find a function $h(x)in V$ where $h(x)=0$ if $xnotin [0,N_2]$. (Now I'm just going to try to find an interger N so that $g(x)$ and $h(x)$ are both zero for $xnotin [0,N]$) Pick $Min mathbb{N}$ such that $acdot Mgeq max{N_1,N_2}$ and let $N$ be the smallest integer such that $Ngeq acdot M$. Then $g(x)$ and $h(x)$ are both zero for $xnotin [0,N]$.

Let $nin mathbb{N}$ be arbitrary such that $acdot n geq N$. Consider the function

$$ z(x) = begin{cases}
g(x) & xin (0,N] \
lambda^{-n}h(x-an) & xin (an,N+an] \
0 & otherwise
end{cases}$$

Note that $(T^nz)(x)=h(x)$ and
$$
|z(x)-g(x)|=lambda^{-n}|h(x-an)|to 0, nto infty
$$
Thus for $n$ sufficently large, $z(x)in U$ and $(T^nz)(x)=h(x)in V$. This shows that $T$ is topologically transitive which implies $T$ hypercyclic since the underlying space is a separable Banach space.