Spectral decomposition of unbounded normal operator and commutativity


Let $ H $ be a Hilbert space, $ N $ be an unbounded normal operator on $ H $, and $ E $ be the spectral measure of $ N $. If a bounded linear operator $ S $ on $ H $ satisfies that $ SN subseteq NS $, then $ E(omega)S = SE(omega) $ for all Borel subset $ omega subseteq mathbb{C} $. My questions are:

  1. Under the same assumptions, can we say that $ Sf(N) subseteq f(N)S $ for all Borel functions $ fcolon mathbb{C} to mathbb{C} $? Here $ f(N) $ denotes a Borel functional calculus of $ N $; i.e., $ f(N) = int f ,dE $.
  2. How about unbounded operators? For example, if $ S $ is a densely defined closed operator on $ H $ satisfying $ SN subseteq NS $, can we say that $ E(omega)S = SE(omega) $ for all Borel subset $ omega subseteq mathbb{C} $? If so, can we say that $ Sf(N) subseteq f(N)S $ for all Borel functions $ fcolon mathbb{C} to mathbb{C} $?