Spectrum of an operator on $\ell^1$

Question

I'm supposed to find the spectrum of an operator $T$ on the Banach space $\ell^1$, where $\|x_n\|=\sum_{n=1}^\infty |x_n|$ and $T\{x_n\}=\{x_2,x_1, x_4, x_3, x_6, x_5, x_8,x_7, \ldots \}$.
I have found that the point spectrum of the operator is $\sigma_p=\{1\}$ by analyzing $\{x_2-\lambda x_1, x_1-\lambda x_2, x_4-\lambda x_3, x_3-\lambda x_4, x_6-\lambda x_5,\ldots \} = \{0,0,0,0,0,\ldots \}$ (is this correct?). I've also shown that $\|T\|=1$, so $\sigma _T\subset \{\lambda \in \mathbb{C}, |\lambda |\le 1\}$. How do I proceed to show the whole spectrum?

Answer 1

You could calculate $T^2$ and in the process observe that you haven't got the complete point spectrum. Minimal polynomials also work in infinite dimension.