Suppose for any postive operator $T \in B(H) $,we have a real bounded Borel function $f_{T}$ on $\sigma(T)$.
If we have a non-trivial projection $P$on a Hilbert space $H$ and $k>0$.Let $P^{\perp}=1-P$ .$f$ is a real bounded borel functional on $\sigma(kP+P^{\perp})$,how to show that there exist two functions $g_P,h_P:(0,\infty)\to (0,\infty)$ such that $f(kP+P^{\perp})=g_P(k)P+h_P(k)P^{\perp}$?
Best Answer
The spectrum of $kP+P^\perp$ is $\{1,k\}$, so the spectral measure is given by $E(\{k\})=P$, $E(\{1\})=P^\perp$. So, for any Borel function $f$, $$ f(kP+P^\perp)=\int_{\{1,k\}} f(\lambda)\,dE(\lambda)=f(1)P^\perp+f(k)P. $$