Examples of spectrum of compact operators on the sequence space $l_2$

Question

Suppose $T$ is a compact operator on the sequence space $l_2$, and let $\sigma(T)$ be its spectrum. Is it possible to find a $T \ne 0$ such that $\sigma(T) = \{0\}$?

Also, is it possible to find $T$ such that $\sigma(T) = \{0,1\}$?

Answer 1

In this answer it is shown that spectrum of Volterra operator is $\{0\}$

In this answer it is shown that spectrum of any non trivial projection is $\{0, 1\}$

Answer 2

You don't need $\ell^2$, you can find such operators on $\mathbb{R}^2$. Consider $$\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}.$$

Answer 3

To give a more general answer, given any compact $K\subset\mathbb C$, there exists $T\in B(\ell^2)$ with $\sigma(T)=K$.

Indeed, choose $\{q_j\}$ a countable dense subset of $K$, and define $$ T\{a_n\}=\{q_na_n\}. $$