Find a bounded linear operator whose spectrum is equal to a nonempty compact subset.

Question

Given a nonempty compact subset $K$ of $\mathbb{C}$, I would like to find a bounded linear operator $T: \ell^2(\mathbb{N}) \rightarrow \ell^2(\mathbb{N})$ whose spectrum is equal to $K$, i.e. $\sigma_{L(\ell^2(\mathbb{N}))}(T)=K$. Any ideas would be appreciated.

Answer 1

Hint: Let $(c_n)$ be a countable dense set of $K$ and define $T(e_n)= c_ne_n$. There has an obvious extension of this to a bounded operator. Show that each $c_n$ is an eigen value and that $T-\lambda I$ is invertible if $\lambda \notin K$.