I'm studying that what set can be the spectrum of operator.
If $\mathcal{H}$ is an infinite dimension Hilbert space and $K$ is a non-empty compact subset of $\mathbb{C}$, show that there is an $A$ in $\mathcal{B}(\mathcal{H})$ such that $\sigma(A) = K$ .
Show that any compact set in $\mathbb{C}$ is the spectrum of an operator. This is related question about this question. But it gave an answer when $\mathcal{H}$ is separable, not the infinite dimension case. But I believe that my question can be reduced to link. How to do that.
In addition, what about this question?
If $K$ is a non-empty compact subset of $\mathbb{C}$, does there exist an operator $A$ in $\mathcal{B}(C[0,1])$ such that $\sigma(A) = K$ .
I saw the result that in Banach space, this property does not hold in general, but I think in this case, we can find this one, but I cannot find this example with above example. can you help me?