Q: Existence of a linear operator with a given spectrum I know that a linear operator defined as $T: l^2 \rightarrow l^2$ as $Tx = y$
where $x = (a_j), y = (b_j)$ and $b_j = c_ja_j$ where $(c_j)$ are dense in $[0,1]$.
Then $\sigma(T) =[0,1]$ and $\sigma_p$$(T)=\{c_i:i \in \Bbb{N}\}$.
Using this, how can I do the following:
Show the existence of a Linear operator on $l^2$ whose eigen values are dense in a given compact set $K$ $\subset$ $\Bbb{C}$ and $\sigma(T) = K$.
Here $\sigma(T)$ denotes the spectrum of $T$ and $\sigma_p(T)$ denotes the point spectrum of $T$
