Operator has no eigenvalues

Hello everyone

Show: The operator T has no eigenvalues

$\displaystyle T:l^2(\mathbb{C}) \to l^2(\mathbb{C})$

$\displaystyle Tx:=\sum^{\infty}_{k=1} \frac{x_k}{k+1}e_{k+1}$ , where $\displaystyle x = (x_k)_{k \in \mathbb{N}}$

Proof:

Iff $\displaystyle \lambda \not= 0 $ it follows that $\displaystyle x_1 = 0 $ and $\displaystyle x_{k+1} = \frac{x_k}{\lambda(k+1)} $ if $\displaystyle k \ge 1 $

$\displaystyle \Rightarrow x = 0 $

I don't get it: how do we know that $\displaystyle x_1 = 0 $ ?

Help would be much appreciated

Thanks

Rapha