Consider the left shift operator $T : \ell^1(\mathbb N) \to \ell^1(\mathbb N) $ by $$T(x_1,x_2….. )=(x_2, x_3 ……..),$$
and also the right shift operator $S : \ell^1(\mathbb N) \to \ell^1(\mathbb N) $ by $$S(x_1,x_2….. )=(0, x_1, x_2 ……..).$$
Can we find an linear operator $R: \ell^1(\mathbb N) \to \ell^1(\mathbb N) $ such that $T=R^2$ or $S=R^2$?
In my opinion, this is intuitively not true because one cannot "shift a vector by half position". But how to prove this? Can anyone help? Thanks!
Best Answer
Suppose that $R^2=T$. Then $\ker R\subseteq\ker T=\Bbb Re_0$, where $e_0=\langle 1,0,0,\ldots\rangle$. Clearly $\ker R$ is non-trivial, so $\ker R=\ker T$. Moreover, $T$ is surjective, so $R$ must also be surjective. In particular, $e_0=Rx$ for some $x\in\ell^1(\Bbb N)\setminus\ker T$, and therefore $R^2x=Re_0=0\ne Tx$.
Now suppose that $R^2=S$, and let $V=\operatorname{ran}S$; clearly $\operatorname{ran}R\supseteq V$. $R$ cannot be surjective, so $\operatorname{ran}R=V$. But then $\operatorname{ran}\left(R\upharpoonright V\right)=\operatorname{ran}S=V$, so for each $x\in\ell^1(\Bbb N)\setminus V$ there must be a $y\in V$ such that $Rx=Ry$. However, $S$ is injective, so $R$ must also be injective.