TITLE: Operator commutes with spectral projection
QUESTION [1 upvotes]: Let $E$ be the spectral measure to an (unbounded) self-adjoint operator $A$. Is there a sufficient and necessary condition so that for a bounded interval $I$ we have $E_I A= AE_I$?

REPLY [1 votes]: Yes. The necessary and sufficient condition is that $A$ is bounded. 
Indeed, the domain of $AE_I$ is the whole Hilbert space, and the domain of $E_IA$ is the domain $\mathcal D(A)$ of $A$.
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