Spectrum of an operator $\sigma(L)$

Question

How can I find $\sigma(L)$, $\sigma_{p}(L)$ , $\sigma_{c}(L)$ i , $\sigma_{r}(L)$ for operator $L(x_{1},x_{2},\ldots)=(x_{2},\frac{1}{2}x_{3},\frac{1}{3}x_{4},\ldots)$ ?

Answer 1

First, show that $\|L\|=1$. So $\sigma(L)$ is inside the closed unit disk $\overline{\mathbb D}$.

From what you wrote in your comment, you easily get that if $(L-\lambda I)x=0$, then $$ x_{k+1}=k\lambda^k. $$ The element $(k\lambda^k)_k$ is in $\ell^2(\mathbb N)$ if and only if $|\lambda|<1$. It follows that $$\sigma(L)=\overline{\mathbb D}, \ \ \ \sigma_p(L)=\mathbb D. $$ For any $\lambda\in\mathbb D$, it is easy to check that the range of $L-\lambda I$ contains each element in the canonical basis, so it is dense. Thus $$ \sigma_c(L)=\mathbb T,\ \ \ \ \sigma_r(L)=\varnothing. $$