Let $L\colon \ell^p \rightarrow \ell^p$ such that $L((a_n)_n)= \left(\frac{1}{n}*a_n\right)_n$.
1) Determine $L'$(adjoint operator), $\ker(L)$, $\ker(L')$, $\operatorname{rg}(L)$, $\operatorname{rg}(L')$ as well as $\operatorname{cl}(\operatorname{rg}(L))$ and $\operatorname{cl}(\operatorname{rg}(L'))$.
2) Let $K$ be a bounded linear operator. Using $L$, show that $\operatorname{cl}(\operatorname{rg}(K')) \subseteq \ker(K)^\perp$ but equality does not hold in general.
3) $K$ surjective $\rightarrow$ $K'$ injective, but $\leftarrow$ does not hold i.g.
4) $K'$ surjective $\rightarrow$ $K$ injective, but $\leftarrow$ does not hold i.g.
So far, I know: $\langle Lx,y \rangle = \langle x,L'y\rangle$ and so it follows $L=L'$. The kernel of $L$ is $(0)$. Then, from the lecture, I get $\operatorname{cl}(\operatorname{rg}(L))= \ker(L')_\perp = (here) \ker(L)_\perp = (0)_\perp = \ell^p$. How can I find $\operatorname{rg}(L)$? I need it to solve $2)$, $3)$ and $4)$.
I appreciate any help.