Spectral/projection-valued measures have very handy applications theoretically, but I got stuck when asked to compute explicitly certain projection-valued measures. Let's focus on the following:
Let $ N: {\mathcal{L}^{2}}([0,1]) \to {\mathcal{L}^{2}}([0,1]) $ be the normal operator defined by $$ \forall f \in {\mathcal{L}^{2}}([0,1]),\forall t \in [0,1]: \quad [N(f)](t) \stackrel{\text{def}}{=} t \cdot f(t). $$ What is the projection-valued measure corresponding to $ N $?
As $ \sigma(N) = [0,1] $, we need a projection-valued measure $ P $ supported on $ [0,1] $. Theoretically, it should be defined just by $$ P(E) = {\chi_{E}}(N), $$ where $ \chi_{E} $ is the characteristic function of $ E \subseteq [0,1] $ and $ {\chi_{E}}(N) $ is obtained via the Borel functional calculus of $ N $.
However, to find $ {\chi_{E}}(N) $, we need to find a sequence of polynomial functions converging to $ \chi_{E} $, which is not an easy job.
I somehow feel that the projection-valued measure is given simply by $ P(E) = {\chi_{E}}(N) $, but I am not sure.
Can someone give a hint on how to compute the spectral/projection-valued measure explicitly?
Thanks!