Our professor put this one in our exam, taking it out along the way though because it seemed too tricky. Still we wasted nearly an hour on it and can't stop thinking about a solution.
What we have: The left shift $L : \ell^p \to \ell^p$ $$L(x_1,x_2,x_3,\ldots) = (x_2,x_3,\ldots)$$
and another operator $T$. We should prove that if $TL=LT$, then $T$ is continuous.
We had defined subspaces $$ X_k = \{ (x_i) : x_i = 0 \text{ for } i>k \} $$ and seen that these are $T$-invariant and the restrictions $T : X_k \to X_k$ continuous (obvious). The hint was to use closed-graph-theorem to show that $T$ is continuous. Of course we can truncate any sequence to then lie in $X_k$, however I do not see how convergence of the truncated sequences relates to convergence of the images under $T$.
Any help please?