Does a projection valued measure (PVM) induce a PVM on a generic subspace of the Hilbert space?

Question

Let $E:{\cal B}(X) \to Pr({\cal H})$ be a projection valued measure (PVM), where ${\cal B}(X)$ is the Borel $\sigma$-algebra of a suitable topological space $X$ and $Pr({\cal H})$ is the set of orthogonal projections of a Hilbert space $\cal H$. Let $\cal H'$ be a generic subspace of $\cal H$. Does $E$ induce in a natural way a PVM $E':{\cal B}(X) \to Pr({\cal H'})$?