Let $S:H\to H$ be a unilateral shift operator. I preferred in Example2.3.2 of Murphy's C*-algebras and operator theory that S has no eigenvalues. While $\{\lambda \in \Bbb C ; |\lambda|<1\} \subset \sigma(S)$. To find spectrum and eigenvalue, we should find a $\lambda$ that satisfies in the equality $S\xi = \lambda\xi$ for some $\xi \in H$. To find spectrum of $S$, the author says
$$S(\alpha_1,\alpha_2,…)=\lambda(\alpha_1,\alpha_2,…) \to $$
$$(\alpha_2,\alpha_3,…)=\lambda(\alpha_1,\alpha_2,…) \to$$
$$\alpha_{n+1} = \lambda\alpha_n~~~~for~every~n \to$$
$$(\alpha_1,\alpha_2,…)=\alpha_1(1,\lambda,\lambda^2,…)$$
So $\lambda \in \sigma(S)$ if $|\lambda|<1$.
I agree with above argument, but I think we can use this argument to show that $S$ has eigenvalue. Where is my mistake? I can not understand the difference between eigenvalue and spectrum. I really confused about it. Please help me. Thanks in advance.
Best Answer
Your definition is not the unilateral shift. If your book gave you $Se_{n}=e_{n+1}$, then that is correct. The coefficient of $e_{1}$ is mapped to become the coefficient of $e_{2}$. So, $$ S(\alpha_{1},\alpha_{2},\alpha_{3},\cdots)=(0,\alpha_1,\alpha_2,\alpha_3,\cdots). $$ The operator $S$ never sends anything to $0$ except $0$ because $\|Sx\|=\|x\|$. The operator you defined in the problem, however, sends $(1,0,0,0,\cdots)$ to $0$. So your interpretation of the operator is not correct, and that's why you're getting a result that contradicts the one you were asked to show.
If the unilateral shift had an eigenvalue $\lambda$, then $\|Sx\|=\|x\|$ would for $|\lambda|=1$. But it has none, which you can show by assuming $$ (0,\alpha_1,\alpha_2,\alpha_3,\cdots)=(\lambda \alpha_1,\lambda \alpha_2,\lambda \alpha_3,\cdots),\\ (0,\overline{\lambda}\alpha_1,\overline{\lambda}\alpha_2,\overline{\lambda}\alpha_3,\cdots) = (\alpha_1,\alpha_2,\alpha_3,\cdots) $$ and concluding that $\alpha_1=0$, and then $\alpha_2=\overline{\lambda}\alpha_1=0$, etc.
You know $\sigma(S)$ is contained in the closed unit disk because $\|S\|=1$. What is interesting is that $\sigma(S)$ is the closed unit disk in $\mathbb{C}$. You have already shown this because your operator is $S^{\star}$, and $\sigma(S)=\sigma(S^{\star})$; you have done this by showing that every $|\lambda| < 1$ is an eigenvalue of $S^{\star}$. It then follows that $\sigma(S)=\sigma(S^{\star})$ is the closed unit disk because the spectrum is closed and contains the open unit disk, and the spectrum must be contained in the closed unit disk.
Finally, to see that your version of the shift operator--which is the backward shift--is the adjoint of the unilateral shift $S$, write $$ \begin{align} (Sx,y) & =(S\sum_{n}(x,e_n)e_n,y) \\ & =(\sum_{n}(x,e_n)e_{n+1},y) \\ & =\sum_{n}(x,e_n)(e_{n+1},y) \\ & = (x,\sum_{n}(y,e_{n+1})e_{n}). \end{align} $$ Hence, $S^{\star}y = \sum_{n}(y,e_{n+1})e_{n}$; this operator maps $e_{1}$ to $0$, $e_{2}$ to $e_{1}$, $e_3$ to $e_2$, etc..