Spectral Decomposition of Function-of-Normal-Operator In Arveson's book A Short Course on Spectral Theory, on page 64 (section on spectral measures) the author mentions the usual spectral decomposition of a normal operator $N$ as $$N=\sum_{\lambda \in \sigma (N)}\lambda \cdot E_{\lambda}$$ where $E_\lambda$ is the projection into the eigenspace of $\lambda$. He then goes to say

Functions of $N$ can also be expressed in a similar way: $$f(N)=\sum_{\lambda \in \sigma (N)}f(\lambda) \cdot E_{\lambda}$$

Why is this true? Can you please give context and explain the significance of this representation?
Thanks in advance!
 A: The discussion at that point in the book was focused on a normal complex matrix $A$ with spectrum (eigenvalues in this case) $\sigma(A)=\{\lambda_{1},\cdots,\lambda_{k}\}$. Normal matrices are unitarily equivalently to diagonal matrices. Equivalently, if $\{ \lambda_{j}\}_{j=1}^{k}$ are the distinct eigenvalues of $A$, then there is an orthonormal basis of eigenvectors
$$
                 \bigcup_{j=1}^{k}\{ e_{j,1},e_{j,2},\cdots,e_{j,k_{j}}\}
$$
where $\{ e_{j,1},\cdots,e_{j,k_{j}}\}$ are eigenvectors of $A$ with a common eigenvector $\lambda_{j}$. In this context, the spectral projection $E_{\lambda_{j}}$ is the orthogonal projection onto the eigenspace of $A$ with eigenvalue $\lambda_{j}$; more precisely, in terms of the inner-product $(\cdot,\cdot)$ on $\mathbb{C}^{N}$:
$$
        E_{\lambda_{j}}x = (x,e_{j,1})e_{j,1}+\cdots+(x,e_{j,k_{j}})e_{j,k_{j}}.
$$
So $AE_{\lambda_{j}}=\lambda_{j}E_{\lambda_{j}}$ for all $j$, and
$$
     A = \sum_{j} \lambda_{j}E_{\lambda_{j}} = \sum_{\lambda\in\sigma(A)}\lambda E_{\lambda}.
$$
These projections are selfadjoint, mutually orthogonal, and a partition of the identity $I$:
$$
                       E_{\lambda}^{2}=E_{\lambda}^{\star}=E_{\lambda},\\
                       E_{\lambda}E_{\lambda'}=0,\;\; \lambda\ne \lambda',\\
                    \sum_{\lambda\in\sigma} E_{\lambda}= I,\\
                       A = \sum_{\lambda\in\sigma}\lambda E_{\lambda}.
$$
The decomposition satisfying these last four properties is the Spectral Theorem for a normal matrix $A$. This spectral decomposition is unique, and can be stated in terms of operators only, without mention of a basis. Because of these properties,
$$
                   A^{n} = \sum_{\lambda\in\sigma}\lambda^{n}E_{\lambda}
$$
which easily extends to any complex function $F$ on $\sigma(A)$:
$$
                             F(A) = \sum_{\lambda}F(\lambda)E_{\lambda}.
$$
It's easy to check that $F(A)G(A)=(FG)(A)$ using properties from the Spectral Theorem (or spectral decomposition.) And, $1(A)=I$. The mapping from such functions $F$ to matrices is an algebraic homomorphism from the algebra of complex functions on $\sigma$ to the algebra of matrices generated by $A$.
