Projection Valued Measure Question - Spectral Theorem

Question

My question is related to the spectral theorem. Suppose you have a projection valued (spectral) measure, $E$, from the complex numbers, $\mathbb{C}$, to the projections on some Hilbert space $\mathcal{H}$, such that the support of $E$ is a compact subset $K \subseteq \mathbb{C}$. If one consider $\int_K z dE(z) := N$, then this is a normal operator in $\mathcal{B}(\mathcal{H})$. By the spectral theorem, there exists a projection valued measure, say $F$, such that $N = \int_{\sigma(N)} z dF(z)$, where $\sigma(N)$ denotes the spectrum of $N$. My question is does $F = E$? I think an analogous question is, does $\sigma(N) = K$? I think I have shown that $\sigma(N) \subseteq K$, but don't know how to show equality, or even if it is equal. Thanks for your help!

Answer 1

Yes on both counts -- this is the uniqueness part of the spectral theorem. See e.g. pp. 423-424 of Fell-Doran for a careful statement and proof.