# 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 $$fcolon mathbb{C} to mathbb{C}$$?