Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 10 / 2 / c / Solution

Define the holomorphic function $e$ on the disconnected open set $U$ to be one on $V$ and zero on $W$. Since $e^2=e$, its holomorphic functional calculus value $p={\Theta }_x(e)$ is an idempotent. The spectral mapping theorem and the nonempty spectral intersection with $V$ show $1\in \sigma (p)$; thus $p\ne 0$. These are the Riesz projection values selecting the indicated spectral component.

Part (b) gives $p_n={\Theta }_{x_n}(e)\to p$. If $\sigma (x_n)$ avoided $V$, spectral mapping would give $\sigma (p_n)=\{0\}$. A nonzero idempotent always has $1$ in its spectrum: if $1-p_n$ were invertible, $p_n(1-p_n)=0$ would force $p_n=0$. Thus avoiding $V$ forces $p_n=0$. That is impossible eventually, since $p\ne 0$ and $p_n\to p$. Therefore $\sigma (x_n)\cap V\ne \varnothing$ eventually. This is persistence of a separated spectral component, not a claim that arbitrary individual spectral points vary continuously.