Properties of compact operator in $l_2$ commuting with compacts
Can you please explain why from $ ||Ke_i||=||Ke_j||  \forall i,j$
follows that K compact only if K=0?

Let $v$ be any vector in $\ell^2$. 
Then $\langle v,\,Ke_i\rangle=\langle K^*v,\,e_i\rangle \rightarrow 0$ as $i$ goes to infinity, because $K^*v$ is a vector $\ell^2$ as well.
Thus $Ke_i$ converges weakly to zero. Now we know that this sequence is precompact in $\ell^2$ with the strong topology, because $K$ is compact. Let $u$ be any limit point of the sequence: by the above, $u$ is orthogonal to any $v$, thus $u=0$. 
Since $\|Ke_i\|$ is constant, it is equal to $\|u\|=0$. Therefore $Ke_i=0$ for all $i$ thus $K=0$. 
