Consider the following operator on $\ell\_{\mathbb{N}^\*}^2(\mathbb{C})$ defined as:

$$ T(x\_1,x\_2,\cdots)=(0,x\_2,x\_3,\cdots).$$

> 
> I want prove that 
>  $$\sigma(T)=\{0,1\}.$$
> 
> 
> 

I have the following idea: let us consider the right shift
$$S\_r(x\_1,x\_2,\cdots)=(0,x\_1,x\_2,x\_3,\cdots).$$
If I show that $T$ is the orthonal projection on the range of $S\_r$, then clearly
$$\sigma(T)=\{0,1\}.$$

Because, it is well known that the spectrum of an Orthogonal Projection Operator is equal to $\{0,1\}$.

Note that 
$$T=S\_rT\_\ell,$$
where
$$S\_\ell(x\_1,x\_2,\cdots)=(x\_2,x\_3,\cdots).$$