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.

Answer 2

Only by definitions. Consider the equation $Tx=\lambda x$, i.e. $x_2-\lambda x_1=0$, $x_1-\lambda x_2=0$, $x_4-\lambda x_3=0$, $x_3-\lambda x_4=0$,... We get pairs of equations from which all $x_k$ can be found. For example: $\begin{cases} x_1-\lambda x_2=0,\\x_2-\lambda x_1=0\end{cases}$. This is a homogeneous system of linear equations and it has a nonzero solution if and only if its determinant is zero, i.e. $\begin{vmatrix} 1& -\lambda\\ -\lambda& 1 \end{vmatrix}=0$. From here we have $\lambda=\pm1$ -- this is a point spectrum (you can write out nonzero eigenfunctions corresponding to these numbers).

Further, let $\lambda\ne\pm1$. Consider the operator $T-\lambda I$ and show that this operator has a bounded inverse defined on the whole space. We need to solve equation $Tx-\lambda x=y$, $y\in l_1$. This equation splits into pairs of inhomogeneous systems that have unique solutions and can be found using the Cramer formulas: $$x=\left(\frac{y_2+\lambda y_1}{1-\lambda^2},\frac{y_1+\lambda y_2}{1-\lambda^2},\frac{y_4+\lambda y_3}{1-\lambda^2},\frac{y_3+\lambda y_4}{1-\lambda^2},...\right).$$ Because the $\|x\|\leq\dfrac{2\max\{1,|\lambda|\}}{|1-\lambda^2|}\|y\|$, operator $(T-\lambda I)^{-1}$ is bounded when $\lambda\ne\pm1$, so it's resovent of $T$.