commuting with compacts

Question

Let $S$ be a shift on $\ell_2$. Is there any non-zero compact operator $K$, commuting with $S$? If yes, how to find a bounded operator $T\in\mathcal{B}(\ell_2)$ which does not commute with any non-zero compact $K\in\mathcal{B}(\ell_2)$?

Answer 1

If you mean with $l_2$ the hilbert space $l_2(\mathbb{Z})$, and the shift defined by $Se_i = e_{i+1}$. Then if we assume it is true, we get

$$ \| Ke_{i} \| = \| SKe_{i} \| = \|KSe_{i}\| = \| Ke_{i+1} \| $$

By induction we then have $\|Ke_{i}\| = \|Ke_{j}\|$ for all $i,j \in \mathbb{Z}$. Therefore $K$ can only be compact if $K = 0$.

So the only compact operator commuting with $S$ is the zero operator.