Let $A$ be a Banach Space and $T$ be a bounded operator on $A$. Given that, spectrum of $T$, $\sigma[T]$, is $ F_1 \cup F_2;$ where $ F_1, F_2$ are disjoint closed set in complex plane. Show that there exist topologically complemented subspace $A_1,A_2$ of $A$ such that $A_1,A_2$ are invariant subspace for $T$ and $\sigma(T|A_i)=F_i$ for $i=1,2.$
Until now what I have done is, taking disjoint open set $G_i$ containing $F_i$ and have chosen $f_i= 1_{G_i}-$ the characteristic function on $G_i$, (which are actually analytic on $G_1\cup G_2$). Then I have taken $A_i$ as range of the projection $f_i(T)$, using the functional calculus for $T$.
Using the spectral mapping theorem one can tell that, $\sigma(Tf_i(T))= F_i\cup$ {$0$}. If my guess is correct then I have to show $\sigma(Tf_i(T)|A_i)= F_i$. At this stage I need help.