Let $T$ be a bounded operator and $A=\sigma(T)$ be its spectrum. Let $A^n \subset A$ be sequence of subsets s.t $A^n \rightarrow A$ (in compact open topology so $x\in A$ belongs to all but finetly many of $A^n$). Let $\Pi^n$ be the characteristic function of $A^n$.
I want to understand few things.
1- First of all does $\Pi^n(T)$ make sense for non normal operators? Borel functional calculus as far as I know only works for normal operators but since characteristic function is more simpler I was wondering if one can define this. And if it does, does the functional calculus hold true that is do we have $\sigma(\Pi^nT) = \Pi^n\sigma(T)$
2- If it does,does $\Pi^n(T) \rightarrow T$
If for instance we were dealing with continuous functional calculus the mapping $f \rightarrow f(T)$ is continuous therefore if $f^n\rightarrow f$ one would have $f^n(T)\rightarrow f(T)$
My knowledge of functional calculus is basic so any references which may lead to the answer is also very welcome.