Given an infinite dimensional Hilbert space $H$. Let $A\subseteq \mathbb{C}$ be closed and bounded. Construct a bounded linear operator $S$ on $H$ such that $\sigma(S)=A$, where $\sigma(S)$ is the spectrum of $S$.
I can not find how to approach this question. What are compacts set in $\mathbb{C}$? and which operator can be good choice?
On
First suppose that $H$ is separable. Fix an orthronormal basis $\{e_n\}$ of $H$. Fix a countable set $B$ of $A$ and enumerate the elements of $B$ by $\lambda_n$. Consider the operator $M\colon H\to H$ defined by $Me_n = \lambda_n e_n$. Show that the spectrum of $M$ is in fact $A$. Then modify this to work for any infinite-dimensional Hilbert space.
If $H$ is separable, then $H$ is isomorphic to $L^2(A,\mu)$, where $A$ is the given compact set, and $\mu$ a regular probability measure. So we will construct an example here.
Let $f$ be a function which is $0$ everywhere outside $A$, and is non zero inside $A$. Then $M_f$ has spectrum equal to $A$.