Motivation of my question: Let $A$ be a bounded selfadjoint operator with spectral measure $E$ and $x$ a vector. Then it is easily seen that the closed linear span of all $A^nx$ ($n\in\mathbb N$) coincides with that of all $E(\Delta)x$, where $\Delta$ runs through all Borel sets. I would like to know if the same holds for normal operators. The inclusion "$\subset$" holds trivially. Just approximate $z^n$ uniformly by simple functions on $\sigma(A)$ (the spectrum of $A$) and use the spectral theorem. The question for the converse inclusion is whether to a disc $D$ one can find a sequence of polyniomials $(p_n)$ which converges at least pointwise to the characteristic function of $D\cap\sigma(A)$ on $\sigma(A)$. If so, $p_n(A)x$ converges to $E(D)x$ and the original claim follows from simple measure theoretic arguments.
Does anyone know an answer to my question?